Chapter 15
Numerical Methods for Fractional Differential Equations
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This chapter is the fractional-calculus core of the book. After completing it, the reader should be able to:
recognize the principal fractional integrals and derivatives used in modern analysis and applications;
distinguish Riemann–Liouville, Caputo, Grunwald–Letnikov, Weyl, Marchaud, Riesz, Hadamard, Hilfer, Katugampola, and Caputo–Katugampola operators;
understand why the Caputo derivative is preferred for classical initial conditions;
derive the Caputo and Caputo–Katugampola derivatives of power functions;
transform Caputo and Caputo–Katugampola differential equations into Volterra integral equations;
implement L1, L1-2, Alikhanov, L2, L3, Grunwald–Letnikov, Lubich convolution quadrature, and PECE-type methods;
derive cubic and quintic Chebyshev–Lobatto panel formulas for singular memory integrals;
evaluate Caputo and Caputo–Katugampola kernels by exact incomplete-beta panel weights;
compare accuracy, stability, smoothness requirements, and computational costs of classical and high-order methods;
build residual-minimization objectives for fractional ODEs and PDEs using accurate quadrature;
understand fractional clocks and stretched-time prototype solutions.
Why Fractional Derivatives?
Fractional derivatives model memory, hereditary effects, viscoelasticity, anomalous diffusion, long-range temporal dependence, and nonlocal transport. The simplest Caputo fractional initial-value problem is {}^CD_{0+}^{\alpha}u(t)=f(t,u(t)), \qquad u(0)=u_0, \qquad 0<\alpha<1. Unlike an integer-order derivative, the value of {}^CD_{0+}^{\alpha}u(t) depends on the full history 0<s<t. This is the numerical difficulty and the modeling strength.
For N time steps, direct history summation typically costs O(N^2). High-order fractional solvers must therefore balance accuracy, memory, quadrature, stability, and long-time efficiency.
Fractional Integrals
For \alpha>0, the left Riemann–Liouville fractional integral is (I_{a+}^{\alpha}f)(t) = \frac{1}{\Gamma(\alpha)} \int_a^t (t-s)^{\alpha-1}f(s)\,\dd s.
The right fractional integral is (I_{b-}^{\alpha}f)(t) = \frac{1}{\Gamma(\alpha)} \int_t^b (s-t)^{\alpha-1}f(s)\,\dd s.
For sufficiently integrable f and \alpha,\beta>0, I_{a+}^{\alpha}I_{a+}^{\beta}f = I_{a+}^{\alpha+\beta}f.
Using the definitions, I_{a+}^{\alpha}I_{a+}^{\beta}f(t) = \frac{1}{\Gamma(\alpha)\Gamma(\beta)} \int_a^t (t-s)^{\alpha-1} \int_a^s (s-r)^{\beta-1}f(r)\,\dd r\,\dd s. Interchange the order of integration: = \frac{1}{\Gamma(\alpha)\Gamma(\beta)} \int_a^t f(r) \int_r^t (t-s)^{\alpha-1}(s-r)^{\beta-1} \dd s\,\dd r. With s=r+(t-r)\xi, the inner integral becomes (t-r)^{\alpha+\beta-1} B(\alpha,\beta) = (t-r)^{\alpha+\beta-1} \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}. Thus I_{a+}^{\alpha}I_{a+}^{\beta}f(t) = \frac{1}{\Gamma(\alpha+\beta)} \int_a^t (t-r)^{\alpha+\beta-1}f(r)\,\dd r.
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Main Fractional Derivatives
Let n-1<\alpha<n, n\in\mathbb N. The left Riemann–Liouville derivative is (D_{a+}^{\alpha}f)(t) = \frac{\dd^n}{\dd t^n} I_{a+}^{n-\alpha}f(t) = \frac{1}{\Gamma(n-\alpha)} \frac{\dd^n}{\dd t^n} \int_a^t (t-s)^{n-\alpha-1}f(s)\,\dd s.
Let n-1<\alpha<n. The Caputo derivative is ({}^CD_{a+}^{\alpha}f)(t) = I_{a+}^{n-\alpha}f^{(n)}(t) = \frac{1}{\Gamma(n-\alpha)} \int_a^t (t-s)^{n-\alpha-1}f^{(n)}(s)\,\dd s. For 0<\alpha<1, ({}^CD_{a+}^{\alpha}f)(t) = \frac{1}{\Gamma(1-\alpha)} \int_a^t (t-s)^{-\alpha}f'(s)\,\dd s.
The Grunwald–Letnikov derivative is formally ({}^{GL}D_{a+}^{\alpha}f)(t) = \lim_{h\to0^+} h^{-\alpha} \sum_{k=0}^{\lfloor(t-a)/h\rfloor} (-1)^k{\alpha\choose k}f(t-kh), where {\alpha\choose k} = \frac{\alpha(\alpha-1)\cdots(\alpha-k+1)}{k!}.
For functions on the whole line with suitable decay, the Weyl fractional derivative is a translation-invariant fractional derivative. In Fourier variables it is characterized by \widehat{D_W^{\alpha}f}(\xi)=(i\xi)^\alpha \widehat f(\xi), with the branch chosen according to the left or right orientation.
For 0<\alpha<1, the Marchaud derivative on the real line is (D_M^\alpha f)(t) = \frac{\alpha}{\Gamma(1-\alpha)} \int_0^\infty \frac{f(t)-f(t-s)}{s^{1+\alpha}}\,\dd s, whenever the integral is meaningful.
The Riesz fractional derivative is the symmetric fractional derivative whose Fourier symbol is \widehat{(-\Delta)^{\alpha/2}f}(\xi)=|\xi|^\alpha\widehat f(\xi). In one dimension it is often written as a symmetric combination of left and right fractional derivatives.
The Hadamard fractional integral is (I_{a+}^{\alpha,H}f)(t) = \frac{1}{\Gamma(\alpha)} \int_a^t \left(\log\frac{t}{s}\right)^{\alpha-1} f(s)\frac{\dd s}{s}. The Hadamard derivative is obtained by applying powers of t\frac{\dd}{\dd t} to a Hadamard fractional integral.
For n-1<\alpha<n and 0\le\beta\le1, the Hilfer derivative is (D_{a+}^{\alpha,\beta}f)(t) = I_{a+}^{\beta(n-\alpha)} \frac{\dd^n}{\dd t^n} I_{a+}^{(1-\beta)(n-\alpha)}f(t). It interpolates between Riemann–Liouville and Caputo-type behavior.
For 0<\alpha<1, one common Caputo–Fabrizio derivative uses a nonsingular exponential kernel: ({}^{CF}D_{a+}^{\alpha}f)(t) = \frac{M(\alpha)}{1-\alpha} \int_a^t \exp\!\left[-\frac{\alpha}{1-\alpha}(t-s)\right] f'(s)\,\dd s. Here M(\alpha) is a normalization.
One Atangana–Baleanu Caputo-type derivative uses a Mittag–Leffler kernel: ({}^{ABC}D_{a+}^{\alpha}f)(t) = \frac{B(\alpha)}{1-\alpha} \int_a^t E_\alpha\!\left( -\frac{\alpha}{1-\alpha}(t-s)^\alpha \right)f'(s)\,\dd s.
For \alpha>0 and \rho>0, the Katugampola fractional integral is (I_{a+}^{\alpha,\rho}f)(t) = \frac{\rho^{1-\alpha}}{\Gamma(\alpha)} \int_a^t s^{\rho-1} (t^\rho-s^\rho)^{\alpha-1} f(s)\,\dd s.
Let n-1<\alpha<n, \rho>0, and define \mathcal D_\rho=t^{1-\rho}\frac{\dd}{\dd t}. The Caputo–Katugampola derivative is ({}^CD_{a+}^{\alpha,\rho}f)(t) = I_{a+}^{n-\alpha,\rho} \left(\mathcal D_\rho^n f\right)(t). For 0<\alpha<1, ({}^CD_{a+}^{\alpha,\rho}f)(t) = \frac{\rho^\alpha}{\Gamma(1-\alpha)} \int_a^t (t^\rho-s^\rho)^{-\alpha}f'(s)\,\dd s.
For 0<\alpha<1, the factor \rho^\alpha is essential. It makes the operator consistent with the stretched variable y=t^\rho. In that variable, Caputo–Katugampola differentiation becomes a scaled Caputo derivative.
Caputo versus Riemann–Liouville
For 0<\alpha<1, D_{a+}^{\alpha}f(t) = \frac{1}{\Gamma(1-\alpha)} \frac{\dd}{\dd t} \int_a^t (t-s)^{-\alpha}f(s)\,\dd s, whereas {}^CD_{a+}^{\alpha}f(t) = \frac{1}{\Gamma(1-\alpha)} \int_a^t (t-s)^{-\alpha}f'(s)\,\dd s. They are related by {}^CD_{a+}^{\alpha}f(t) = D_{a+}^{\alpha}\bigl[f(t)-f(a)\bigr]. Thus {}^CD_{a+}^{\alpha}1=0, while the Riemann–Liouville derivative of a constant is generally singular: D_{a+}^{\alpha}1 = \frac{(t-a)^{-\alpha}}{\Gamma(1-\alpha)}.
The Caputo derivative allows classical initial conditions: u(0)=u_0, \qquad u'(0)=u_1, \qquad \ldots This is why Caputo models are dominant in many physical fractional differential equations. Riemann–Liouville derivatives are powerful analytically, but their initial conditions are fractional and less directly measurable.
Power Functions and Fractional Clocks
Let 0<\alpha<1 and \beta>-1. If \beta=0, then {}^CD_{0+}^{\alpha}1=0. If \beta>0, then {}^CD_{0+}^{\alpha}t^\beta = \frac{\Gamma(\beta+1)}{\Gamma(\beta+1-\alpha)} t^{\beta-\alpha}.
For \beta>0, \frac{\dd}{\dd s}s^\beta=\beta s^{\beta-1}. Therefore {}^CD_{0+}^{\alpha}t^\beta = \frac{\beta}{\Gamma(1-\alpha)} \int_0^t (t-s)^{-\alpha}s^{\beta-1}\,\dd s. Set s=t\xi. Then {}^CD_{0+}^{\alpha}t^\beta = \frac{\beta t^{\beta-\alpha}}{\Gamma(1-\alpha)} \int_0^1 (1-\xi)^{-\alpha}\xi^{\beta-1}\,\dd \xi. The integral is B(\beta,1-\alpha) = \frac{\Gamma(\beta)\Gamma(1-\alpha)} {\Gamma(\beta+1-\alpha)}. Since \beta\Gamma(\beta)=\Gamma(\beta+1), the formula follows.
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Let 0<\alpha<1, \rho>0, and \beta>0. Then {}^CD_{0+}^{\alpha,\rho}t^\beta = \rho^\alpha \frac{\Gamma(\beta/\rho+1)} {\Gamma(\beta/\rho+1-\alpha)} t^{\beta-\alpha\rho}. Also, {}^CD_{0+}^{\alpha,\rho}1=0.
Set y=t^\rho, \qquad v(y)=f(y^{1/\rho}). For 0<\alpha<1, {}^CD_{0+}^{\alpha,\rho}f(t) = \rho^\alpha\,{}^CD_{0+}^{\alpha}v(y). If f(t)=t^\beta, then v(y)=y^{\beta/\rho}. By the Caputo power formula, {}^CD_y^\alpha y^{\beta/\rho} = \frac{\Gamma(\beta/\rho+1)} {\Gamma(\beta/\rho+1-\alpha)} y^{\beta/\rho-\alpha}. Returning to t, y^{\beta/\rho-\alpha} = t^{\beta-\alpha\rho}. Multiplication by \rho^\alpha gives the result.
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A fractional clock is a time transformation adapted to fractional memory. For the Caputo derivative, \tau_\alpha(t)=\frac{t^\alpha}{\Gamma(1+\alpha)} satisfies {}^CD_{0+}^{\alpha}\tau_\alpha(t)=1. For the Caputo–Katugampola derivative, \tau_{\alpha,\rho}(t) = \frac{t^{\alpha\rho}} {\rho^\alpha\Gamma(1+\alpha)} satisfies {}^CD_{0+}^{\alpha,\rho}\tau_{\alpha,\rho}(t)=1.
Volterra Integral Form
The Caputo IVP {}^CD_{0+}^{\alpha}u(t)=f(t,u(t)), \qquad u(0)=u_0, \qquad 0<\alpha<1, is equivalent to the Volterra integral equation u(t) = u_0+ \frac{1}{\Gamma(\alpha)} \int_0^t (t-s)^{\alpha-1}f(s,u(s))\,\dd s.
If u is sufficiently smooth, then {}^CD_{0+}^{\alpha}u=f(t,u), \qquad u(0)=u_0, is equivalent to u(t) = u_0+ I_{0+}^{\alpha}f(\cdot,u(\cdot))(t).
For Caputo derivatives, I_{0+}^{\alpha}{}^CD_{0+}^{\alpha}u(t)=u(t)-u(0). Applying I_{0+}^{\alpha} to the differential equation gives u(t)-u_0 = I_{0+}^{\alpha}f(\cdot,u(\cdot))(t). Conversely, applying {}^CD_{0+}^{\alpha} to the integral equation recovers the differential equation under the usual regularity assumptions.
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For the Caputo–Katugampola equation {}^CD_{0+}^{\alpha,\rho}u(t)=f(t,u(t)), \qquad u(0)=u_0, the Volterra form is u(t) = u_0+ \frac{\rho^{1-\alpha}}{\Gamma(\alpha)} \int_0^t s^{\rho-1} (t^\rho-s^\rho)^{\alpha-1} f(s,u(s))\,\dd s.
The L1 Scheme
Let t_n=nh, \qquad u_n\approx u(t_n). For 0<\alpha<1, the L1 approximation is {}^CD_{0+}^{\alpha}u(t_n) \approx \frac{1}{h^\alpha\Gamma(2-\alpha)} \sum_{k=0}^{n-1} a_k \left( u_{n-k}-u_{n-k-1} \right), where a_k=(k+1)^{1-\alpha}-k^{1-\alpha}. Equivalently, {}^CD_{0+}^{\alpha}u(t_n) \approx \frac{1}{\Gamma(1-\alpha)} \sum_{j=0}^{n-1} \frac{u_{j+1}-u_j}{h} \int_{t_j}^{t_{j+1}}(t_n-s)^{-\alpha}\,\dd s. The integral is exact: \int_{t_j}^{t_{j+1}}(t_n-s)^{-\alpha}\,\dd s = \frac{(t_n-t_j)^{1-\alpha}-(t_n-t_{j+1})^{1-\alpha}} {1-\alpha}.
If u is sufficiently smooth away from t=0, the L1 approximation has classical local accuracy of order O(h^{2-\alpha}) for smooth data. Reduced order may occur when the solution has an initial weak singularity.
On each interval [t_j,t_{j+1}], L1 replaces u'(s) by the derivative of the linear interpolant. The interpolation error for u' is O(h) locally, and integration against the weakly singular kernel produces an accumulated O(h^{2-\alpha}) estimate under sufficient regularity. Initial singularities weaken the smoothness needed for this estimate, explaining the order reduction.
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Caputo–Katugampola L1 Scheme
For Caputo–Katugampola derivatives, the natural time variable is y=t^\rho. Let y_n=nh_y, \qquad t_n=y_n^{1/\rho}, \qquad v(y)=u(y^{1/\rho}). Since {}^CD_{0+}^{\alpha,\rho}u(t) = \rho^\alpha\,{}^CD_{0+}^{\alpha}v(y), the L1 scheme in the stretched variable becomes {}^CD_{0+}^{\alpha,\rho}u(t_n) \approx \frac{\rho^\alpha}{h_y^\alpha\Gamma(2-\alpha)} \sum_{k=0}^{n-1} a_k \left( u(t_{n-k})-u(t_{n-k-1}) \right), with a_k=(k+1)^{1-\alpha}-k^{1-\alpha}. This is a clean way to build \rho-adapted memory grids.
Grunwald–Letnikov and Shifted GL Methods
The basic Grunwald weights are g_k^{(\alpha)}=(-1)^k{\alpha\choose k}, \qquad g_0^{(\alpha)}=1, with recurrence g_k^{(\alpha)} = \left(1-\frac{\alpha+1}{k}\right)g_{k-1}^{(\alpha)}. A left GL approximation is D_{0+}^{\alpha}u(t_n) \approx h^{-\alpha} \sum_{k=0}^{n} g_k^{(\alpha)}u_{n-k}. Shifted GL formulas improve stability for certain fractional diffusion equations: D_{0+}^{\alpha}u(t_n) \approx h^{-\alpha} \sum_{k=0}^{n+p} g_k^{(\alpha)}u_{n-k+p}, where p is a shift.
L1-2, Alikhanov, L2, and L3 Ideas
Higher-order formulas approximate u'(s) or u(s) by higher-degree interpolants on each time panel and integrate the weakly singular kernel exactly or accurately.
- L1.
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Piecewise linear interpolation of u. Equivalent to piecewise constant approximation of u'. Simple and robust.
- L1-2.
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A correction of L1 using higher-order information, often based on quadratic interpolation on early panels and improved interpolation of u'. It increases the order for smooth solutions.
- Alikhanov / L2-1\sigma.
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A widely used second-order-type method evaluating the Caputo derivative at an off-step point t_{n+\sigma}, commonly with \sigma=1-\frac{\alpha}{2}. It uses quadratic interpolation and achieves high accuracy for sufficiently smooth solutions.
- L2.
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A quadratic interpolation family. It approximates the history integrand by degree two polynomials and integrates the fractional kernel.
- L3.
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A cubic interpolation family. It uses degree three information and requires more smoothness, but can be significantly more accurate on smooth solutions.
A unifying construction is: {}^CD_{0+}^{\alpha}u(t_n) = \frac{1}{\Gamma(1-\alpha)} \sum_{j=0}^{n-1} \int_{t_j}^{t_{j+1}} (t_n-s)^{-\alpha}u'(s)\,\dd s. Approximate u on each panel by a polynomial p_j: u(s)\approx p_j(s), \qquad u'(s)\approx p_j'(s). Then {}^CD_{0+}^{\alpha}u(t_n) \approx \frac{1}{\Gamma(1-\alpha)} \sum_{j=0}^{n-1} \int_{t_j}^{t_{j+1}} (t_n-s)^{-\alpha}p_j'(s)\,\dd s. This single formula contains L1, L2, L3, and Chebyshev–Beta panel rules.
Lubich Convolution Quadrature
Lubich convolution quadrature starts from a linear multistep method with generating polynomial \delta(\zeta). The fractional weights \omega_n are defined by \left(\frac{\delta(\zeta)}{h}\right)^\alpha = \sum_{n=0}^{\infty}\omega_n\zeta^n. For a Riemann–Liouville-type convolution operator, one writes D_h^\alpha u_n = \sum_{j=0}^{n}\omega_j u_{n-j}. For backward Euler, \delta(\zeta)=1-\zeta. For BDF2, \delta(\zeta)=\frac32-2\zeta+\frac12\zeta^2. Higher-order BDF-based convolution quadratures are systematic but require careful stability and starting corrections.
Caption.
Lubich convolution quadrature weights
multistep generating polynomial \delta(\zeta), step size h, order \alpha, number of steps N
Expand
\item \left(\delta(\zeta)/h\right)^\alpha \item = \item \sum_{n=0}^{N}\omega_n\zeta^n+O(\zeta^{N+1}) \item
Use \omega_n in the convolution
\item D_h^\alpha u_n=\sum_{j=0}^{n}\omega_j u_{n-j} \item
Return \omega_0,\ldots,\omega_N
PECE Predictor–Corrector Methods
The Caputo IVP is equivalent to u(t) = u_0+ \frac{1}{\Gamma(\alpha)} \int_0^t (t-s)^{\alpha-1}f(s,u(s))\,\dd s. A PECE method uses a predictor, evaluates f, and then corrects using a fractional Adams–Moulton-type rule.
A typical predictor has the form u_{n+1}^{P} = u_0+ \frac{1}{\Gamma(\alpha)} \sum_{j=0}^{n} b_{j,n+1} f(t_j,u_j), where b_{j,n+1} = \int_{t_j}^{t_{j+1}} (t_{n+1}-s)^{\alpha-1}\,\dd s. The corrector has the form u_{n+1} = u_0+ \frac{1}{\Gamma(\alpha)} \left[ a_{n+1,n+1}f(t_{n+1},u_{n+1}^{P}) + \sum_{j=0}^{n} a_{j,n+1}f(t_j,u_j) \right].
Fast Convolution and Short-Memory Principle
Direct convolution has cost O(N^2). Fast methods approximate the kernel by exponentials: (t-s)^{-\alpha} \approx \sum_{\ell=1}^{M}w_\ell e^{-\eta_\ell(t-s)}. Then history variables satisfy recurrences: H_\ell^{n+1} = e^{-\eta_\ell h}H_\ell^n+ \int_{t_n}^{t_{n+1}} e^{-\eta_\ell(t_{n+1}-s)}u'(s)\,\dd s. This reduces the cost to approximately O(MN) with memory O(M).
The short-memory principle truncates old history: \int_0^t (t-s)^{-\alpha}u'(s)\,\dd s \approx \int_{t-L}^{t} (t-s)^{-\alpha}u'(s)\,\dd s. This is efficient but must be used with caution because power-law memory decays slowly.
Chebyshev–Lobatto Panel Quadrature
We now derive high-accuracy panel formulas for Caputo and Caputo–Katugampola memory integrals. Let [a_j,b_j] be a time panel with h_j=b_j-a_j, \qquad s=a_j+h_j\theta, \qquad 0\le\theta\le1. Choose Chebyshev–Lobatto nodes on the reference interval: \theta_m^{(q)} = \frac{1-\cos(m\pi/q)}{2}, \qquad m=0,\ldots,q. The physical nodes are s_{j,m}=a_j+h_j\theta_m^{(q)}. Let \ell_m^{(q)}(\theta) be the Lagrange cardinal polynomial at the reference nodes. Approximate u(a_j+h_j\theta) \approx p_j(\theta)= \sum_{m=0}^{q} u(s_{j,m})\ell_m^{(q)}(\theta). Then \frac{\dd u}{\dd s}(a_j+h_j\theta) \approx \frac1{h_j} \sum_{m=0}^{q} u(s_{j,m}) \frac{\dd \ell_m^{(q)}}{\dd\theta}(\theta).
For the Caputo derivative at t_n, C_{j}^{(q)}(t_n) = \int_{a_j}^{b_j} (t_n-s)^{-\alpha}u'(s)\,\dd s is approximated by C_{j}^{(q)}(t_n) \approx \sum_{m=0}^{q} u(s_{j,m}) W_{j,m}^{C,q}(t_n), where W_{j,m}^{C,q}(t_n) = \int_0^1 (t_n-a_j-h_j\theta)^{-\alpha} \frac{\dd \ell_m^{(q)}}{\dd\theta}(\theta)\,\dd\theta. Thus {}^CD_{0+}^{\alpha}u(t_n) \approx \frac{1}{\Gamma(1-\alpha)} \sum_{j=0}^{n-1} \sum_{m=0}^{q} u(s_{j,m}) W_{j,m}^{C,q}(t_n).
For the Caputo–Katugampola derivative, W_{j,m}^{CK,q}(t_n) = \int_0^1 \left(t_n^\rho-(a_j+h_j\theta)^\rho\right)^{-\alpha} \frac{\dd \ell_m^{(q)}}{\dd\theta}(\theta)\,\dd\theta, and {}^CD_{0+}^{\alpha,\rho}u(t_n) \approx \frac{\rho^\alpha}{\Gamma(1-\alpha)} \sum_{j=0}^{n-1} \sum_{m=0}^{q} u(s_{j,m}) W_{j,m}^{CK,q}(t_n).
Incomplete-Beta Weights for the Caputo Kernel
If \frac{\dd \ell_m^{(q)}}{\dd\theta}(\theta) = \sum_{r=0}^{q-1} d_{m,r}^{(q)}\theta^r, then W_{j,m}^{C,q}(t) = \sum_{r=0}^{q-1} d_{m,r}^{(q)} \int_0^1 (t-a_j-h_j\theta)^{-\alpha}\theta^r\,\dd\theta. Equivalently, in physical variables, W_{j,m}^{C,q}(t) = \sum_{r=0}^{q-1} \frac{d_{m,r}^{(q)}}{h_j^{r+1}} I_{r}^{C}(a_j,b_j;t), where I_r^C(a,b;t) = \int_a^b (t-s)^{-\alpha}(s-a)^r\,\dd s. Expanding (s-a)^r=\sum_{\ell=0}^{r}{r\choose \ell}(-a)^{r-\ell}s^\ell, we get I_r^C(a,b;t) = \sum_{\ell=0}^{r} {r\choose \ell}(-a)^{r-\ell} t^{\ell+1-\alpha} \left[ B_{b/t}(\ell+1,1-\alpha) - B_{a/t}(\ell+1,1-\alpha) \right], where B_z(p,q) is the incomplete beta function: B_z(p,q)=\int_0^z \xi^{p-1}(1-\xi)^{q-1}\,\dd \xi.
Incomplete-Beta Weights for the Caputo–Katugampola Kernel
For Caputo–Katugampola, I_r^{CK}(a,b;t) = \int_a^b (t^\rho-s^\rho)^{-\alpha}(s-a)^r\,\dd s. Using the same expansion of (s-a)^r, it remains to evaluate \int_a^b (t^\rho-s^\rho)^{-\alpha}s^\ell\,\dd s. With z=\left(\frac{s}{t}\right)^\rho, one obtains \int_a^b (t^\rho-s^\rho)^{-\alpha}s^\ell\,\dd s = \frac{1}{\rho} t^{\ell+1-\alpha\rho} \left[ B_{(b/t)^\rho} \left(\frac{\ell+1}{\rho},1-\alpha\right) - B_{(a/t)^\rho} \left(\frac{\ell+1}{\rho},1-\alpha\right) \right]. Therefore I_r^{CK}(a,b;t) = \sum_{\ell=0}^{r} {r\choose \ell}(-a)^{r-\ell} \frac{t^{\ell+1-\alpha\rho}}{\rho} \left[ B_{(b/t)^\rho} \left(\frac{\ell+1}{\rho},1-\alpha\right) - B_{(a/t)^\rho} \left(\frac{\ell+1}{\rho},1-\alpha\right) \right]. The panel weights are W_{j,m}^{CK,q}(t) = \sum_{r=0}^{q-1} \frac{d_{m,r}^{(q)}}{h_j^{r+1}} I_r^{CK}(a_j,b_j;t).
Cubic Chebyshev–Lobatto Formula
For q=3, the reference nodes are \theta_0=0, \qquad \theta_1=\frac14, \qquad \theta_2=\frac34, \qquad \theta_3=1. The derivatives of the Lagrange basis functions are: \ell_0'(\theta) = -16\theta^2+\frac{64}{3}\theta-\frac{19}{3}, \ell_1'(\theta) = 32\theta^2-\frac{112}{3}\theta+8, \ell_2'(\theta) = -32\theta^2+\frac{80}{3}\theta-\frac{8}{3}, \ell_3'(\theta) = 16\theta^2-\frac{32}{3}\theta+1. Hence d^{(3)} = \begin{pmatrix} -19/3&64/3&-16\\ 8&-112/3&32\\ -8/3&80/3&-32\\ 1&-32/3&16 \end{pmatrix}, where row m gives \ell_m'(\theta)=d_{m,0}+d_{m,1}\theta+d_{m,2}\theta^2. The cubic Caputo panel contribution is C_j^{(3)}(t_n) = \sum_{m=0}^{3}u(s_{j,m}) \sum_{r=0}^{2} \frac{d_{m,r}^{(3)}}{h_j^{r+1}} I_r^C(a_j,b_j;t_n). The cubic Caputo–Katugampola panel contribution is CK_j^{(3)}(t_n) = \sum_{m=0}^{3}u(s_{j,m}) \sum_{r=0}^{2} \frac{d_{m,r}^{(3)}}{h_j^{r+1}} I_r^{CK}(a_j,b_j;t_n).
Quintic Chebyshev–Lobatto Formula
For q=5, the nodes are \theta_0=0, \qquad \theta_1=\frac{3-\sqrt5}{8}, \qquad \theta_2=\frac{5-\sqrt5}{8}, \theta_3=\frac{5+\sqrt5}{8}, \qquad \theta_4=\frac{3+\sqrt5}{8}, \qquad \theta_5=1. The basis derivatives have the form \ell_m'(\theta) = d_{m,0}^{(5)} +d_{m,1}^{(5)}\theta +d_{m,2}^{(5)}\theta^2 +d_{m,3}^{(5)}\theta^3 +d_{m,4}^{(5)}\theta^4. They are explicitly: \begin{aligned} \ell_0'&=-17+\frac{832}{5}\theta-\frac{2544}{5}\theta^2+\frac{3072}{5}\theta^3-256\theta^4,\\ \ell_1'&=12+4\sqrt5-(184+40\sqrt5)\theta +\left(\frac{3696}{5}+\frac{432\sqrt5}{5}\right)\theta^2\\ &\quad+\left(-\frac{5376}{5}-\frac{256\sqrt5}{5}\right)\theta^3 +512\theta^4,\\ \ell_2'&=-4-\frac{4\sqrt5}{5} +\left(\frac{584}{5}+\frac{104\sqrt5}{5}\right)\theta +\left(-\frac{2928}{5}-\frac{336\sqrt5}{5}\right)\theta^2\\ &\quad+\left(\frac{4864}{5}+\frac{256\sqrt5}{5}\right)\theta^3 -512\theta^4,\\ \ell_3'&=-4+\frac{4\sqrt5}{5} +\left(\frac{584}{5}-\frac{104\sqrt5}{5}\right)\theta +\left(-\frac{2928}{5}+\frac{336\sqrt5}{5}\right)\theta^2\\ &\quad+\left(\frac{4864}{5}-\frac{256\sqrt5}{5}\right)\theta^3 -512\theta^4,\\ \ell_4'&=12-4\sqrt5+(-184+40\sqrt5)\theta +\left(\frac{3696}{5}-\frac{432\sqrt5}{5}\right)\theta^2\\ &\quad+\left(-\frac{5376}{5}+\frac{256\sqrt5}{5}\right)\theta^3 +512\theta^4,\\ \ell_5'&=1-32\theta+\frac{1008}{5}\theta^2-\frac{2048}{5}\theta^3+256\theta^4. \end{aligned} The quintic Caputo panel contribution is C_j^{(5)}(t_n) = \sum_{m=0}^{5}u(s_{j,m}) \sum_{r=0}^{4} \frac{d_{m,r}^{(5)}}{h_j^{r+1}} I_r^C(a_j,b_j;t_n). The quintic Caputo–Katugampola panel contribution is CK_j^{(5)}(t_n) = \sum_{m=0}^{5}u(s_{j,m}) \sum_{r=0}^{4} \frac{d_{m,r}^{(5)}}{h_j^{r+1}} I_r^{CK}(a_j,b_j;t_n).
For a fixed panel [a_j,b_j], the q-degree Chebyshev–Lobatto panel formula integrates exactly the Caputo or Caputo–Katugampola memory contribution whenever u is a polynomial of degree at most q on that panel.
If u is a polynomial of degree at most q, then its Chebyshev–Lobatto interpolant on the panel is exactly u. Therefore the derivative p_j' equals u' exactly on the panel. Since the singular kernel integrals defining the weights are evaluated exactly through incomplete-beta expressions, the panel contribution is exact.
□
Comparison of Fractional Differentiation Methods
| Method | Main idea | Strength | Limitation |
|---|---|---|---|
| L1 | piecewise linear history | simple, robust baseline | low order for smooth data |
| L1-2 | corrected L1 / quadratic information | higher accuracy | start-up details matter |
| Alikhanov / L2-1\sigma | quadratic interpolation at shifted time | high accuracy for smooth solutions | smoothness and starting corrections |
| L2 | quadratic panel interpolation | systematic construction | more weights |
| L3 | cubic panel interpolation | high order on smooth data | sensitive near singular starts |
| GL | fractional binomial convolution | simple for uniform grids | stability and boundary issues |
| Lubich CQ | multistep generating functions | systematic high order | convolution weights and start-up corrections |
| PECE | Volterra predictor–corrector | natural for nonlinear IVPs | direct history cost |
| Fast convolution | kernel compression | long-time efficiency | kernel approximation error |
| Chebyshev–Beta cubic | cubic Lobatto interpolation plus exact beta weights | accurate residual evaluation | more complex weights |
| Chebyshev–Beta quintic | quintic Lobatto interpolation plus exact beta weights | very accurate for smooth panels | needs smoothness and careful implementation |
Examples
Example 1: Caputo derivative of a constant
Since \frac{\dd}{\dd s}1=0, we have {}^CD_{0+}^{\alpha}1=0. For Caputo–Katugampola, {}^CD_{0+}^{\alpha,\rho}1=0.
Example 2: Fractional relaxation
Consider {}^CD_{0+}^{\alpha}u(t)=-\lambda u(t), \qquad u(0)=u_0. The solution is u(t)=u_0E_\alpha(-\lambda t^\alpha), where E_\alpha(z)=\sum_{k=0}^{\infty}\frac{z^k}{\Gamma(\alpha k+1)} is the Mittag–Leffler function.
For Caputo–Katugampola relaxation, {}^CD_{0+}^{\alpha,\rho}u(t)=-\lambda u(t), \qquad u(0)=u_0, the stretched variable gives u(t) = u_0E_\alpha \left( -\frac{\lambda}{\rho^\alpha}t^{\alpha\rho} \right).
Example 3: Fractional diffusion
A time-fractional diffusion model is {}^CD_{0+}^{\alpha}u(x,t)=\kappa u_{xx}(x,t). A standard method combines L1 in time with centered finite differences in space: \frac{1}{h^\alpha\Gamma(2-\alpha)} \sum_{k=0}^{n-1}a_k \left( U_i^{n-k}-U_i^{n-k-1} \right) = \kappa \frac{U_{i-1}^n-2U_i^n+U_{i+1}^n}{\Delta x^2}. This is implicit in U^n.
Example 4: Residual minimization
Let u(x,t;\theta) be a parameterized approximation for a fractional PDE: \mathcal R(x,t;\theta) = {}^CD_{0+}^{\alpha}u(x,t;\theta) - \mathcal N_x[u](x,t;\theta). A residual objective is J(\theta) = \int_0^T\int_\Omega |\mathcal R(x,t;\theta)|^2\,\dd x\,\dd t. The time-fractional derivative inside \mathcal R should be evaluated with a high-accuracy history rule, for example cubic or quintic Chebyshev–Beta weights. Then the outer objective should also be evaluated by high-order quadrature.
Fractional Residual Minimization and Fractional Clocks
A fractional-clock ansatz has the form u(x,t)=U(x,\tau(t)), where, for Caputo models, \tau(t)=c_1t^\alpha+c_2t^{2\alpha}+c_3t^{3\alpha}. For Caputo–Katugampola models, a natural clock is \tau(t)=c_1t^{\alpha\rho}+c_2t^{2\alpha\rho}+c_3t^{3\alpha\rho}. A residual-minimization objective is J(\theta) = \int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t + \lambda P(\theta). For numerical stability, one may add anti-degeneracy penalties, for example P(\theta)= \frac{1}{1+|c_1|^2} + \frac{1}{1+|\text{wave speed}|^2}, depending on the model.
Practical Guidelines
- Use L1.
-
When a reliable baseline is needed and first-level accuracy is acceptable.
- Use Alikhanov or L2-1\sigma.
-
When the solution is smooth enough and second-order accuracy is needed.
- Use Lubich convolution quadrature.
-
When systematic high-order convolution weights are desired, especially for linear or mildly nonlinear problems.
- Use PECE.
-
For nonlinear fractional IVPs when the Volterra form is natural and moderate problem sizes are acceptable.
- Use fast convolution.
-
For long-time simulations where direct history summation is too expensive.
- Use Chebyshev–Beta cubic/quintic.
-
For high-accuracy residual evaluation, parameter optimization, and fractional PDE diagnostics where exact singular-kernel panel integration is valuable.
- Use Caputo–Katugampola.
-
When the memory geometry is better described by t^\rho-s^\rho, or when a stretched fractional clock is part of the model.
Exercises
The following exercise bank is intentionally large. Basic problems test definitions and computations. Starred exercises require derivations, numerical analysis, or implementation design. Problems marked \star, \star\star, and \star\star\star are progressively harder.
Basic problems
Compute {}^CD_{0+}^{\alpha}1.
Compute D_{0+}^{\alpha}1 for 0<\alpha<1.
Derive {}^CD_{0+}^{\alpha}t^\beta = \frac{\Gamma(\beta+1)}{\Gamma(\beta+1-\alpha)}t^{\beta-\alpha} for \beta>0.
Derive the Caputo–Katugampola derivative of t^\beta.
Show that {}^CD_{0+}^{\alpha}\frac{t^\alpha}{\Gamma(1+\alpha)}=1.
Show that {}^CD_{0+}^{\alpha,\rho} \frac{t^{\alpha\rho}}{\rho^\alpha\Gamma(1+\alpha)} = 1.
Write the Volterra integral form of {}^CD_{0+}^{\alpha}u=f(t,u), \qquad u(0)=u_0.
Write the Volterra integral form of {}^CD_{0+}^{\alpha,\rho}u=f(t,u), \qquad u(0)=u_0.
Derive the L1 weights a_k=(k+1)^{1-\alpha}-k^{1-\alpha}.
Write pseudocode for evaluating the L1 approximation at t_n.
Derive the recurrence for Grunwald weights g_k^{(\alpha)}= \left(1-\frac{\alpha+1}{k}\right)g_{k-1}^{(\alpha)}.
For backward Euler, write the Lubich generating function for fractional weights.
Write \delta(\zeta) for BDF2 convolution quadrature.
Explain why PECE methods are based on the Volterra integral form.
State the short-memory principle and its main risk.
Explain how a sum-of-exponentials approximation speeds up history evaluation.
Write the cubic Chebyshev–Lobatto nodes on [0,1].
Write the quintic Chebyshev–Lobatto nodes on [0,1].
Verify the four cubic derivative basis polynomials \ell_m'(\theta).
Define the incomplete beta function B_z(p,q).
Derive the beta expression for \int_a^b(t-s)^{-\alpha}s^\ell\,\dd s.
Derive the beta expression for \int_a^b(t^\rho-s^\rho)^{-\alpha}s^\ell\,\dd s.
Write a residual-minimization objective for a fractional differential equation.
Write the solution of {}^CD_{0+}^{\alpha}u=-\lambda u, \qquad u(0)=u_0.
Compare L1, Lubich, PECE, and Chebyshev–Beta evaluation in one paragraph.
Intermediate problems \star
Prove the referenced result.
Prove {}^CD_{a+}^{\alpha}f = D_{a+}^{\alpha}[f-f(a)] for 0<\alpha<1.
Prove the referenced result.
Prove the referenced result.
Prove the referenced result.
Show that the Caputo–Katugampola derivative becomes a scaled Caputo derivative in the variable y=t^\rho.
Prove the referenced result under smoothness assumptions.
Explain why solutions of fractional IVPs often have weak singularities near t=0.
Explain the role of \sigma=1-\alpha/2 in Alikhanov-type schemes.
Derive a quadratic panel formula for the Caputo derivative using three interpolation nodes.
Prove that the cubic Chebyshev–Beta panel formula is exact for cubic polynomials on each panel.
Prove that the quintic Chebyshev–Beta panel formula is exact for quintic polynomials on each panel.
Derive W_{j,m}^{C,q}(t) = \sum_{r=0}^{q-1} \frac{d_{m,r}^{(q)}}{h_j^{r+1}} I_r^C(a_j,b_j;t).
Derive the corresponding Caputo–Katugampola beta weight.
Compute the first four Lubich weights for backward Euler convolution quadrature.
Derive the first three BDF2 convolution quadrature weights for a fractional order \alpha.
Derive the PECE predictor weights from the Volterra integral equation.
Derive a corrector formula using piecewise linear interpolation of f(t,u(t)).
Derive the recurrence for one exponential history variable.
Estimate the truncation error caused by discarding history older than L.
Derive the L1 finite-difference scheme for a time-fractional diffusion equation.
Explain why implicit spatial discretization is common in fractional diffusion.
Derive the gradient of a quadrature-based residual objective with respect to parameters.
For u(x,t)=U(x,t^\alpha/\Gamma(1+\alpha)), compute the Caputo derivative formally for simple U.
Compare L1, Alikhanov, Lubich, PECE, fast convolution, and Chebyshev–Beta methods in terms of accuracy and cost.
Advanced problems \star\star
Study starting corrections for high-order Lubich convolution quadrature when the solution has initial singularities.
Derive how a graded mesh t_n=T(n/N)^r can recover accuracy for weakly singular fractional solutions.
Prove an error estimate for the Alikhanov L2-1\sigma formula under smoothness assumptions.
Construct a cubic L3-type formula using polynomial interpolation on uniform panels.
Verify all six quintic Chebyshev–Lobatto derivative polynomials.
Design an implementation of the quintic Caputo–Katugampola beta-weight formula.
Analyze numerical cancellation in incomplete-beta weight differences when a panel is far from t_n.
Analyze the behavior of the last-panel weights as b_j\to t_n.
Design a sum-of-exponentials approximation for the kernel t^{-\alpha} on [\Delta t,T].
Compare the computational complexity of direct history summation and fast convolution.
Derive fractional Adams–Bashforth and Adams–Moulton weights from the Volterra form.
Study convergence of PECE methods for nonlinear fractional IVPs.
Derive Newton’s method for an implicit fractional time-stepping system.
Construct a residual-minimization objective for a time-fractional nonlinear PDE.
Construct a residual-minimization objective involving a Caputo–Katugampola derivative.
Derive a numerical method for {}^CD_{0+}^{\alpha}u+\omega^2u=0.
Design a method for a_1{}^CD^\alpha u+a_2{}^CD^\beta u=f(t,u).
Formulate a quadrature method for a distributed-order derivative \int_0^1 \mu(\alpha){}^CD^\alpha u\,\dd\alpha.
Discuss numerical challenges for variable-order Caputo derivatives {}^CD^{\alpha(t)}u(t).
Compare Riesz, spectral fractional Laplacian, and integral fractional Laplacian discretizations.
Discuss boundary conditions for space-fractional diffusion problems.
Derive how kernel approximation error propagates into the fractional derivative.
Design an adaptive time-step strategy for a Caputo fractional IVP.
Explain when a small fractional residual implies a small solution error.
Design a benchmark comparing L1, Alikhanov, Lubich, PECE, and Chebyshev–Beta methods on a known exact solution.
Design a benchmark for Caputo–Katugampola methods using an exact power-function solution.
Research-level problems \star\star\star
Develop an error theory for the quintic Chebyshev–Beta Caputo–Katugampola formula.
Design start-up corrections for Chebyshev–Beta panel formulas when u(t)\sim t^\gamma near zero.
Develop a posteriori estimates connecting fractional residual norms to solution errors.
Combine high-order Chebyshev–Beta panel formulas with fast convolution.
Design an adaptive panel strategy that refines where the memory integrand is least smooth.
Formulate residual minimization for a time-fractional KdV-type equation using Chebyshev–Beta Caputo evaluation.
Formulate residual diagnostics for a time-fractional nonlinear Schrodinger equation.
Develop a high-order numerical method for a fractional Duffing oscillator and compare with PECE.
Design a Caputo or Caputo–Katugampola numerical residual method for a fractional Binet-type orbital model.
Derive first-order optimality conditions for a Caputo fractional optimal control problem.
Develop adjoint gradients for parameter estimation in fractional differential equations.
Compare fractional physics-informed neural networks with Chebyshev–Beta residual minimization.
Design a method for a PDE with both Caputo time derivative and fractional Laplacian.
Develop a solver for distributed-order diffusion and analyze the order quadrature.
Propose a definition and numerical method for a variable-order Caputo–Katugampola derivative.
Design preconditioners for implicit fractional time-stepping systems.
Develop a parallel-in-time history algorithm for fractional differential equations.
Design a GPU implementation of fast fractional convolution.
Create certified benchmark problems with exact Caputo and Caputo–Katugampola solutions.
Design reproducible experiments comparing all methods in this chapter, including accuracy, runtime, memory, and residual norms.
Exercise solutions
The solutions below are converted from the uploaded LaTeX solution file. Open an accordion to view the full solution.
This chapter gives detailed worked solutions for the exercises in Chapter 15. Each solution includes the problem formulation, the method, the mathematical derivation, the conclusion, and a diagnostic comment.
Exercise 15.1
Problem formulation.
Compute {}^CD_{0+}^{\alpha}1.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
For 0<\alpha<1, the Caputo derivative is {}^CD_{0+}^{\alpha}f(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}f'(s)\,\dd s. If f(t)=t^\beta, \beta>0, then f'(s)=\beta s^{\beta-1}. Thus {}^CD_{0+}^{\alpha}t^\beta = \frac{\beta}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}s^{\beta-1}\,\dd s. Set s=t\xi. Then {}^CD_{0+}^{\alpha}t^\beta = \frac{\beta t^{\beta-\alpha}}{\Gamma(1-\alpha)} B(\beta,1-\alpha) = \frac{\Gamma(\beta+1)}{\Gamma(\beta+1-\alpha)}t^{\beta-\alpha}. For f(t)=1, f'(t)=0, so {}^CD^\alpha1=0.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.2
Problem formulation.
Compute D_{0+}^{\alpha}1 for 0<\alpha<1.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.3
Problem formulation.
Derive {}^CD_{0+}^{\alpha}t^\beta = \frac{\Gamma(\beta+1)}{\Gamma(\beta+1-\alpha)}t^{\beta-\alpha} for \beta>0.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
For 0<\alpha<1, the Caputo derivative is {}^CD_{0+}^{\alpha}f(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}f'(s)\,\dd s. If f(t)=t^\beta, \beta>0, then f'(s)=\beta s^{\beta-1}. Thus {}^CD_{0+}^{\alpha}t^\beta = \frac{\beta}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}s^{\beta-1}\,\dd s. Set s=t\xi. Then {}^CD_{0+}^{\alpha}t^\beta = \frac{\beta t^{\beta-\alpha}}{\Gamma(1-\alpha)} B(\beta,1-\alpha) = \frac{\Gamma(\beta+1)}{\Gamma(\beta+1-\alpha)}t^{\beta-\alpha}. For f(t)=1, f'(t)=0, so {}^CD^\alpha1=0.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.4
Problem formulation.
Derive the Caputo–Katugampola derivative of t^\beta.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
For 0<\alpha<1, the Caputo–Katugampola derivative is {}^CD_{0+}^{\alpha,\rho}f(t) = \frac{\rho^\alpha}{\Gamma(1-\alpha)} \int_0^t(t^\rho-s^\rho)^{-\alpha}f'(s)\,\dd s. Let y=t^\rho and v(y)=f(y^{1/\rho}). Then {}^CD_{0+}^{\alpha,\rho}f(t) = \rho^\alpha\,{}^CD_{0+}^{\alpha}v(y). For f(t)=t^\beta, v(y)=y^{\beta/\rho}, so {}^CD_{0+}^{\alpha,\rho}t^\beta = \rho^\alpha \frac{\Gamma(\beta/\rho+1)} {\Gamma(\beta/\rho+1-\alpha)} t^{\beta-\alpha\rho}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.5
Problem formulation.
Show that {}^CD_{0+}^{\alpha}\frac{t^\alpha}{\Gamma(1+\alpha)}=1.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
For 0<\alpha<1, the Caputo–Katugampola derivative is {}^CD_{0+}^{\alpha,\rho}f(t) = \frac{\rho^\alpha}{\Gamma(1-\alpha)} \int_0^t(t^\rho-s^\rho)^{-\alpha}f'(s)\,\dd s. Let y=t^\rho and v(y)=f(y^{1/\rho}). Then {}^CD_{0+}^{\alpha,\rho}f(t) = \rho^\alpha\,{}^CD_{0+}^{\alpha}v(y). For f(t)=t^\beta, v(y)=y^{\beta/\rho}, so {}^CD_{0+}^{\alpha,\rho}t^\beta = \rho^\alpha \frac{\Gamma(\beta/\rho+1)} {\Gamma(\beta/\rho+1-\alpha)} t^{\beta-\alpha\rho}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.6
Problem formulation.
Show that {}^CD_{0+}^{\alpha,\rho} \frac{t^{\alpha\rho}}{\rho^\alpha\Gamma(1+\alpha)} = 1.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
For 0<\alpha<1, the Caputo–Katugampola derivative is {}^CD_{0+}^{\alpha,\rho}f(t) = \frac{\rho^\alpha}{\Gamma(1-\alpha)} \int_0^t(t^\rho-s^\rho)^{-\alpha}f'(s)\,\dd s. Let y=t^\rho and v(y)=f(y^{1/\rho}). Then {}^CD_{0+}^{\alpha,\rho}f(t) = \rho^\alpha\,{}^CD_{0+}^{\alpha}v(y). For f(t)=t^\beta, v(y)=y^{\beta/\rho}, so {}^CD_{0+}^{\alpha,\rho}t^\beta = \rho^\alpha \frac{\Gamma(\beta/\rho+1)} {\Gamma(\beta/\rho+1-\alpha)} t^{\beta-\alpha\rho}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.7
Problem formulation.
Write the Volterra integral form of {}^CD_{0+}^{\alpha}u=f(t,u), \qquad u(0)=u_0.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
For the Caputo IVP {}^CD_{0+}^{\alpha}u(t)=f(t,u(t)), \qquad u(0)=u_0, apply the fractional integral I^\alpha. Since I^\alpha{}^CD^\alpha u(t)=u(t)-u(0), we obtain u(t)=u_0+\frac{1}{\Gamma(\alpha)} \int_0^t(t-s)^{\alpha-1}f(s,u(s))\,\dd s. This is the Volterra integral formulation.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.8
Problem formulation.
Write the Volterra integral form of {}^CD_{0+}^{\alpha,\rho}u=f(t,u), \qquad u(0)=u_0.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
For 0<\alpha<1, the Caputo–Katugampola derivative is {}^CD_{0+}^{\alpha,\rho}f(t) = \frac{\rho^\alpha}{\Gamma(1-\alpha)} \int_0^t(t^\rho-s^\rho)^{-\alpha}f'(s)\,\dd s. Let y=t^\rho and v(y)=f(y^{1/\rho}). Then {}^CD_{0+}^{\alpha,\rho}f(t) = \rho^\alpha\,{}^CD_{0+}^{\alpha}v(y). For f(t)=t^\beta, v(y)=y^{\beta/\rho}, so {}^CD_{0+}^{\alpha,\rho}t^\beta = \rho^\alpha \frac{\Gamma(\beta/\rho+1)} {\Gamma(\beta/\rho+1-\alpha)} t^{\beta-\alpha\rho}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.9
Problem formulation.
Derive the L1 weights a_k=(k+1)^{1-\alpha}-k^{1-\alpha}.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Start from {}^CD_{0+}^{\alpha}u(t_n) = \frac{1}{\Gamma(1-\alpha)} \sum_{j=0}^{n-1} \int_{t_j}^{t_{j+1}}(t_n-s)^{-\alpha}u'(s)\,\dd s. Approximate u'(s)\approx\frac{u_{j+1}-u_j}{h}. The panel integral is \int_{t_j}^{t_{j+1}}(t_n-s)^{-\alpha}\,\dd s = \frac{(t_n-t_j)^{1-\alpha}-(t_n-t_{j+1})^{1-\alpha}}{1-\alpha}. Using \Gamma(2-\alpha)=(1-\alpha)\Gamma(1-\alpha), one obtains {}^CD_{0+}^{\alpha}u(t_n) \approx \frac{1}{h^\alpha\Gamma(2-\alpha)} \sum_{k=0}^{n-1}a_k(u_{n-k}-u_{n-k-1}), where a_k=(k+1)^{1-\alpha}-k^{1-\alpha}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.10
Problem formulation.
Write pseudocode for evaluating the L1 approximation at t_n.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Start from {}^CD_{0+}^{\alpha}u(t_n) = \frac{1}{\Gamma(1-\alpha)} \sum_{j=0}^{n-1} \int_{t_j}^{t_{j+1}}(t_n-s)^{-\alpha}u'(s)\,\dd s. Approximate u'(s)\approx\frac{u_{j+1}-u_j}{h}. The panel integral is \int_{t_j}^{t_{j+1}}(t_n-s)^{-\alpha}\,\dd s = \frac{(t_n-t_j)^{1-\alpha}-(t_n-t_{j+1})^{1-\alpha}}{1-\alpha}. Using \Gamma(2-\alpha)=(1-\alpha)\Gamma(1-\alpha), one obtains {}^CD_{0+}^{\alpha}u(t_n) \approx \frac{1}{h^\alpha\Gamma(2-\alpha)} \sum_{k=0}^{n-1}a_k(u_{n-k}-u_{n-k-1}), where a_k=(k+1)^{1-\alpha}-k^{1-\alpha}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.11
Problem formulation.
Derive the recurrence for Grunwald weights g_k^{(\alpha)}= \left(1-\frac{\alpha+1}{k}\right)g_{k-1}^{(\alpha)}.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.12
Problem formulation.
For backward Euler, write the Lubich generating function for fractional weights.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
For 0<\alpha<1, the Caputo–Katugampola derivative is {}^CD_{0+}^{\alpha,\rho}f(t) = \frac{\rho^\alpha}{\Gamma(1-\alpha)} \int_0^t(t^\rho-s^\rho)^{-\alpha}f'(s)\,\dd s. Let y=t^\rho and v(y)=f(y^{1/\rho}). Then {}^CD_{0+}^{\alpha,\rho}f(t) = \rho^\alpha\,{}^CD_{0+}^{\alpha}v(y). For f(t)=t^\beta, v(y)=y^{\beta/\rho}, so {}^CD_{0+}^{\alpha,\rho}t^\beta = \rho^\alpha \frac{\Gamma(\beta/\rho+1)} {\Gamma(\beta/\rho+1-\alpha)} t^{\beta-\alpha\rho}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.13
Problem formulation.
Write \delta(\zeta) for BDF2 convolution quadrature.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.14
Problem formulation.
Explain why PECE methods are based on the Volterra integral form.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
For the Caputo IVP {}^CD_{0+}^{\alpha}u(t)=f(t,u(t)), \qquad u(0)=u_0, apply the fractional integral I^\alpha. Since I^\alpha{}^CD^\alpha u(t)=u(t)-u(0), we obtain u(t)=u_0+\frac{1}{\Gamma(\alpha)} \int_0^t(t-s)^{\alpha-1}f(s,u(s))\,\dd s. This is the Volterra integral formulation.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.15
Problem formulation.
State the short-memory principle and its main risk.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.16
Problem formulation.
Explain how a sum-of-exponentials approximation speeds up history evaluation.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.17
Problem formulation.
Write the cubic Chebyshev–Lobatto nodes on [0,1].
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
On a panel [a,b], approximate u by a Chebyshev–Lobatto interpolant p(\theta)=\sum_{m=0}^{q}u(s_m)\ell_m^{(q)}(\theta), \qquad s=a+(b-a)\theta. Then u'(s)\approx \frac1{b-a}\sum_{m=0}^{q}u(s_m)(\ell_m^{(q)})'(\theta). If (\ell_m^{(q)})'(\theta)=\sum_{r=0}^{q-1}d_{m,r}^{(q)}\theta^r, the Caputo panel weight is W_{m}^{C,q}(t) = \sum_{r=0}^{q-1}\frac{d_{m,r}^{(q)}}{(b-a)^{r+1}} I_r^C(a,b;t), where I_r^C(a,b;t)=\int_a^b(t-s)^{-\alpha}(s-a)^r\,\dd s. The corresponding moments are evaluated exactly by incomplete beta functions.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.18
Problem formulation.
Write the quintic Chebyshev–Lobatto nodes on [0,1].
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
On a panel [a,b], approximate u by a Chebyshev–Lobatto interpolant p(\theta)=\sum_{m=0}^{q}u(s_m)\ell_m^{(q)}(\theta), \qquad s=a+(b-a)\theta. Then u'(s)\approx \frac1{b-a}\sum_{m=0}^{q}u(s_m)(\ell_m^{(q)})'(\theta). If (\ell_m^{(q)})'(\theta)=\sum_{r=0}^{q-1}d_{m,r}^{(q)}\theta^r, the Caputo panel weight is W_{m}^{C,q}(t) = \sum_{r=0}^{q-1}\frac{d_{m,r}^{(q)}}{(b-a)^{r+1}} I_r^C(a,b;t), where I_r^C(a,b;t)=\int_a^b(t-s)^{-\alpha}(s-a)^r\,\dd s. The corresponding moments are evaluated exactly by incomplete beta functions.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.19
Problem formulation.
Verify the four cubic derivative basis polynomials \ell_m'(\theta).
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.20
Problem formulation.
Define the incomplete beta function B_z(p,q).
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
On a panel [a,b], approximate u by a Chebyshev–Lobatto interpolant p(\theta)=\sum_{m=0}^{q}u(s_m)\ell_m^{(q)}(\theta), \qquad s=a+(b-a)\theta. Then u'(s)\approx \frac1{b-a}\sum_{m=0}^{q}u(s_m)(\ell_m^{(q)})'(\theta). If (\ell_m^{(q)})'(\theta)=\sum_{r=0}^{q-1}d_{m,r}^{(q)}\theta^r, the Caputo panel weight is W_{m}^{C,q}(t) = \sum_{r=0}^{q-1}\frac{d_{m,r}^{(q)}}{(b-a)^{r+1}} I_r^C(a,b;t), where I_r^C(a,b;t)=\int_a^b(t-s)^{-\alpha}(s-a)^r\,\dd s. The corresponding moments are evaluated exactly by incomplete beta functions.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.21
Problem formulation.
Derive the beta expression for \int_a^b(t-s)^{-\alpha}s^\ell\,\dd s.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
For 0<\alpha<1, the Caputo derivative is {}^CD_{0+}^{\alpha}f(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}f'(s)\,\dd s. If f(t)=t^\beta, \beta>0, then f'(s)=\beta s^{\beta-1}. Thus {}^CD_{0+}^{\alpha}t^\beta = \frac{\beta}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}s^{\beta-1}\,\dd s. Set s=t\xi. Then {}^CD_{0+}^{\alpha}t^\beta = \frac{\beta t^{\beta-\alpha}}{\Gamma(1-\alpha)} B(\beta,1-\alpha) = \frac{\Gamma(\beta+1)}{\Gamma(\beta+1-\alpha)}t^{\beta-\alpha}. For f(t)=1, f'(t)=0, so {}^CD^\alpha1=0.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.22
Problem formulation.
Derive the beta expression for \int_a^b(t^\rho-s^\rho)^{-\alpha}s^\ell\,\dd s.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
For 0<\alpha<1, the Caputo–Katugampola derivative is {}^CD_{0+}^{\alpha,\rho}f(t) = \frac{\rho^\alpha}{\Gamma(1-\alpha)} \int_0^t(t^\rho-s^\rho)^{-\alpha}f'(s)\,\dd s. Let y=t^\rho and v(y)=f(y^{1/\rho}). Then {}^CD_{0+}^{\alpha,\rho}f(t) = \rho^\alpha\,{}^CD_{0+}^{\alpha}v(y). For f(t)=t^\beta, v(y)=y^{\beta/\rho}, so {}^CD_{0+}^{\alpha,\rho}t^\beta = \rho^\alpha \frac{\Gamma(\beta/\rho+1)} {\Gamma(\beta/\rho+1-\alpha)} t^{\beta-\alpha\rho}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.23
Problem formulation.
Write a residual-minimization objective for a fractional differential equation.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.24
Problem formulation.
Write the solution of {}^CD_{0+}^{\alpha}u=-\lambda u, \qquad u(0)=u_0.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.25
Problem formulation.
Compare L1, Lubich, PECE, and Chebyshev–Beta evaluation in one paragraph.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Start from {}^CD_{0+}^{\alpha}u(t_n) = \frac{1}{\Gamma(1-\alpha)} \sum_{j=0}^{n-1} \int_{t_j}^{t_{j+1}}(t_n-s)^{-\alpha}u'(s)\,\dd s. Approximate u'(s)\approx\frac{u_{j+1}-u_j}{h}. The panel integral is \int_{t_j}^{t_{j+1}}(t_n-s)^{-\alpha}\,\dd s = \frac{(t_n-t_j)^{1-\alpha}-(t_n-t_{j+1})^{1-\alpha}}{1-\alpha}. Using \Gamma(2-\alpha)=(1-\alpha)\Gamma(1-\alpha), one obtains {}^CD_{0+}^{\alpha}u(t_n) \approx \frac{1}{h^\alpha\Gamma(2-\alpha)} \sum_{k=0}^{n-1}a_k(u_{n-k}-u_{n-k-1}), where a_k=(k+1)^{1-\alpha}-k^{1-\alpha}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.26
Problem formulation.
Prove the referenced result.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.27
Problem formulation.
Prove {}^CD_{a+}^{\alpha}f = D_{a+}^{\alpha}[f-f(a)] for 0<\alpha<1.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
For 0<\alpha<1, the Caputo derivative is {}^CD_{0+}^{\alpha}f(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}f'(s)\,\dd s. If f(t)=t^\beta, \beta>0, then f'(s)=\beta s^{\beta-1}. Thus {}^CD_{0+}^{\alpha}t^\beta = \frac{\beta}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}s^{\beta-1}\,\dd s. Set s=t\xi. Then {}^CD_{0+}^{\alpha}t^\beta = \frac{\beta t^{\beta-\alpha}}{\Gamma(1-\alpha)} B(\beta,1-\alpha) = \frac{\Gamma(\beta+1)}{\Gamma(\beta+1-\alpha)}t^{\beta-\alpha}. For f(t)=1, f'(t)=0, so {}^CD^\alpha1=0.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.28
Problem formulation.
Prove the referenced result.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
For 0<\alpha<1, the Caputo derivative is {}^CD_{0+}^{\alpha}f(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}f'(s)\,\dd s. If f(t)=t^\beta, \beta>0, then f'(s)=\beta s^{\beta-1}. Thus {}^CD_{0+}^{\alpha}t^\beta = \frac{\beta}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}s^{\beta-1}\,\dd s. Set s=t\xi. Then {}^CD_{0+}^{\alpha}t^\beta = \frac{\beta t^{\beta-\alpha}}{\Gamma(1-\alpha)} B(\beta,1-\alpha) = \frac{\Gamma(\beta+1)}{\Gamma(\beta+1-\alpha)}t^{\beta-\alpha}. For f(t)=1, f'(t)=0, so {}^CD^\alpha1=0.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.29
Problem formulation.
Prove the referenced result.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
For 0<\alpha<1, the Caputo–Katugampola derivative is {}^CD_{0+}^{\alpha,\rho}f(t) = \frac{\rho^\alpha}{\Gamma(1-\alpha)} \int_0^t(t^\rho-s^\rho)^{-\alpha}f'(s)\,\dd s. Let y=t^\rho and v(y)=f(y^{1/\rho}). Then {}^CD_{0+}^{\alpha,\rho}f(t) = \rho^\alpha\,{}^CD_{0+}^{\alpha}v(y). For f(t)=t^\beta, v(y)=y^{\beta/\rho}, so {}^CD_{0+}^{\alpha,\rho}t^\beta = \rho^\alpha \frac{\Gamma(\beta/\rho+1)} {\Gamma(\beta/\rho+1-\alpha)} t^{\beta-\alpha\rho}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.30
Problem formulation.
Prove the referenced result.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
For 0<\alpha<1, the Caputo derivative is {}^CD_{0+}^{\alpha}f(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}f'(s)\,\dd s. If f(t)=t^\beta, \beta>0, then f'(s)=\beta s^{\beta-1}. Thus {}^CD_{0+}^{\alpha}t^\beta = \frac{\beta}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}s^{\beta-1}\,\dd s. Set s=t\xi. Then {}^CD_{0+}^{\alpha}t^\beta = \frac{\beta t^{\beta-\alpha}}{\Gamma(1-\alpha)} B(\beta,1-\alpha) = \frac{\Gamma(\beta+1)}{\Gamma(\beta+1-\alpha)}t^{\beta-\alpha}. For f(t)=1, f'(t)=0, so {}^CD^\alpha1=0.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.31
Problem formulation.
Show that the Caputo–Katugampola derivative becomes a scaled Caputo derivative in the variable y=t^\rho.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
For 0<\alpha<1, the Caputo–Katugampola derivative is {}^CD_{0+}^{\alpha,\rho}f(t) = \frac{\rho^\alpha}{\Gamma(1-\alpha)} \int_0^t(t^\rho-s^\rho)^{-\alpha}f'(s)\,\dd s. Let y=t^\rho and v(y)=f(y^{1/\rho}). Then {}^CD_{0+}^{\alpha,\rho}f(t) = \rho^\alpha\,{}^CD_{0+}^{\alpha}v(y). For f(t)=t^\beta, v(y)=y^{\beta/\rho}, so {}^CD_{0+}^{\alpha,\rho}t^\beta = \rho^\alpha \frac{\Gamma(\beta/\rho+1)} {\Gamma(\beta/\rho+1-\alpha)} t^{\beta-\alpha\rho}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.32
Problem formulation.
Prove the referenced result under smoothness assumptions.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Start from {}^CD_{0+}^{\alpha}u(t_n) = \frac{1}{\Gamma(1-\alpha)} \sum_{j=0}^{n-1} \int_{t_j}^{t_{j+1}}(t_n-s)^{-\alpha}u'(s)\,\dd s. Approximate u'(s)\approx\frac{u_{j+1}-u_j}{h}. The panel integral is \int_{t_j}^{t_{j+1}}(t_n-s)^{-\alpha}\,\dd s = \frac{(t_n-t_j)^{1-\alpha}-(t_n-t_{j+1})^{1-\alpha}}{1-\alpha}. Using \Gamma(2-\alpha)=(1-\alpha)\Gamma(1-\alpha), one obtains {}^CD_{0+}^{\alpha}u(t_n) \approx \frac{1}{h^\alpha\Gamma(2-\alpha)} \sum_{k=0}^{n-1}a_k(u_{n-k}-u_{n-k-1}), where a_k=(k+1)^{1-\alpha}-k^{1-\alpha}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.33
Problem formulation.
Explain why solutions of fractional IVPs often have weak singularities near t=0.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.34
Problem formulation.
Explain the role of \sigma=1-\alpha/2 in Alikhanov-type schemes.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.35
Problem formulation.
Derive a quadratic panel formula for the Caputo derivative using three interpolation nodes.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
For 0<\alpha<1, the Caputo derivative is {}^CD_{0+}^{\alpha}f(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}f'(s)\,\dd s. If f(t)=t^\beta, \beta>0, then f'(s)=\beta s^{\beta-1}. Thus {}^CD_{0+}^{\alpha}t^\beta = \frac{\beta}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}s^{\beta-1}\,\dd s. Set s=t\xi. Then {}^CD_{0+}^{\alpha}t^\beta = \frac{\beta t^{\beta-\alpha}}{\Gamma(1-\alpha)} B(\beta,1-\alpha) = \frac{\Gamma(\beta+1)}{\Gamma(\beta+1-\alpha)}t^{\beta-\alpha}. For f(t)=1, f'(t)=0, so {}^CD^\alpha1=0.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.36
Problem formulation.
Prove that the cubic Chebyshev–Beta panel formula is exact for cubic polynomials on each panel.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
On a panel [a,b], approximate u by a Chebyshev–Lobatto interpolant p(\theta)=\sum_{m=0}^{q}u(s_m)\ell_m^{(q)}(\theta), \qquad s=a+(b-a)\theta. Then u'(s)\approx \frac1{b-a}\sum_{m=0}^{q}u(s_m)(\ell_m^{(q)})'(\theta). If (\ell_m^{(q)})'(\theta)=\sum_{r=0}^{q-1}d_{m,r}^{(q)}\theta^r, the Caputo panel weight is W_{m}^{C,q}(t) = \sum_{r=0}^{q-1}\frac{d_{m,r}^{(q)}}{(b-a)^{r+1}} I_r^C(a,b;t), where I_r^C(a,b;t)=\int_a^b(t-s)^{-\alpha}(s-a)^r\,\dd s. The corresponding moments are evaluated exactly by incomplete beta functions.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.37
Problem formulation.
Prove that the quintic Chebyshev–Beta panel formula is exact for quintic polynomials on each panel.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
On a panel [a,b], approximate u by a Chebyshev–Lobatto interpolant p(\theta)=\sum_{m=0}^{q}u(s_m)\ell_m^{(q)}(\theta), \qquad s=a+(b-a)\theta. Then u'(s)\approx \frac1{b-a}\sum_{m=0}^{q}u(s_m)(\ell_m^{(q)})'(\theta). If (\ell_m^{(q)})'(\theta)=\sum_{r=0}^{q-1}d_{m,r}^{(q)}\theta^r, the Caputo panel weight is W_{m}^{C,q}(t) = \sum_{r=0}^{q-1}\frac{d_{m,r}^{(q)}}{(b-a)^{r+1}} I_r^C(a,b;t), where I_r^C(a,b;t)=\int_a^b(t-s)^{-\alpha}(s-a)^r\,\dd s. The corresponding moments are evaluated exactly by incomplete beta functions.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.38
Problem formulation.
Derive W_{j,m}^{C,q}(t) = \sum_{r=0}^{q-1} \frac{d_{m,r}^{(q)}}{h_j^{r+1}} I_r^C(a_j,b_j;t).
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
On a panel [a,b], approximate u by a Chebyshev–Lobatto interpolant p(\theta)=\sum_{m=0}^{q}u(s_m)\ell_m^{(q)}(\theta), \qquad s=a+(b-a)\theta. Then u'(s)\approx \frac1{b-a}\sum_{m=0}^{q}u(s_m)(\ell_m^{(q)})'(\theta). If (\ell_m^{(q)})'(\theta)=\sum_{r=0}^{q-1}d_{m,r}^{(q)}\theta^r, the Caputo panel weight is W_{m}^{C,q}(t) = \sum_{r=0}^{q-1}\frac{d_{m,r}^{(q)}}{(b-a)^{r+1}} I_r^C(a,b;t), where I_r^C(a,b;t)=\int_a^b(t-s)^{-\alpha}(s-a)^r\,\dd s. The corresponding moments are evaluated exactly by incomplete beta functions.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.39
Problem formulation.
Derive the corresponding Caputo–Katugampola beta weight.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
For 0<\alpha<1, the Caputo–Katugampola derivative is {}^CD_{0+}^{\alpha,\rho}f(t) = \frac{\rho^\alpha}{\Gamma(1-\alpha)} \int_0^t(t^\rho-s^\rho)^{-\alpha}f'(s)\,\dd s. Let y=t^\rho and v(y)=f(y^{1/\rho}). Then {}^CD_{0+}^{\alpha,\rho}f(t) = \rho^\alpha\,{}^CD_{0+}^{\alpha}v(y). For f(t)=t^\beta, v(y)=y^{\beta/\rho}, so {}^CD_{0+}^{\alpha,\rho}t^\beta = \rho^\alpha \frac{\Gamma(\beta/\rho+1)} {\Gamma(\beta/\rho+1-\alpha)} t^{\beta-\alpha\rho}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.40
Problem formulation.
Compute the first four Lubich weights for backward Euler convolution quadrature.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
For 0<\alpha<1, the Caputo–Katugampola derivative is {}^CD_{0+}^{\alpha,\rho}f(t) = \frac{\rho^\alpha}{\Gamma(1-\alpha)} \int_0^t(t^\rho-s^\rho)^{-\alpha}f'(s)\,\dd s. Let y=t^\rho and v(y)=f(y^{1/\rho}). Then {}^CD_{0+}^{\alpha,\rho}f(t) = \rho^\alpha\,{}^CD_{0+}^{\alpha}v(y). For f(t)=t^\beta, v(y)=y^{\beta/\rho}, so {}^CD_{0+}^{\alpha,\rho}t^\beta = \rho^\alpha \frac{\Gamma(\beta/\rho+1)} {\Gamma(\beta/\rho+1-\alpha)} t^{\beta-\alpha\rho}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.41
Problem formulation.
Derive the first three BDF2 convolution quadrature weights for a fractional order \alpha.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 15.42
Problem formulation.
Derive the PECE predictor weights from the Volterra integral equation.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
For the Caputo IVP {}^CD_{0+}^{\alpha}u(t)=f(t,u(t)), \qquad u(0)=u_0, apply the fractional integral I^\alpha. Since I^\alpha{}^CD^\alpha u(t)=u(t)-u(0), we obtain u(t)=u_0+\frac{1}{\Gamma(\alpha)} \int_0^t(t-s)^{\alpha-1}f(s,u(s))\,\dd s. This is the Volterra integral formulation.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.43
Problem formulation.
Derive a corrector formula using piecewise linear interpolation of f(t,u(t)).
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.44
Problem formulation.
Derive the recurrence for one exponential history variable.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.45
Problem formulation.
Estimate the truncation error caused by discarding history older than L.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 15.46
Problem formulation.
Derive the L1 finite-difference scheme for a time-fractional diffusion equation.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Start from {}^CD_{0+}^{\alpha}u(t_n) = \frac{1}{\Gamma(1-\alpha)} \sum_{j=0}^{n-1} \int_{t_j}^{t_{j+1}}(t_n-s)^{-\alpha}u'(s)\,\dd s. Approximate u'(s)\approx\frac{u_{j+1}-u_j}{h}. The panel integral is \int_{t_j}^{t_{j+1}}(t_n-s)^{-\alpha}\,\dd s = \frac{(t_n-t_j)^{1-\alpha}-(t_n-t_{j+1})^{1-\alpha}}{1-\alpha}. Using \Gamma(2-\alpha)=(1-\alpha)\Gamma(1-\alpha), one obtains {}^CD_{0+}^{\alpha}u(t_n) \approx \frac{1}{h^\alpha\Gamma(2-\alpha)} \sum_{k=0}^{n-1}a_k(u_{n-k}-u_{n-k-1}), where a_k=(k+1)^{1-\alpha}-k^{1-\alpha}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.47
Problem formulation.
Explain why implicit spatial discretization is common in fractional diffusion.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Always test the result on the scalar model equation or Fourier mode and verify the amplification factor satisfies the stated stability bound.
Exercise 15.48
Problem formulation.
Derive the gradient of a quadrature-based residual objective with respect to parameters.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.49
Problem formulation.
For u(x,t)=U(x,t^\alpha/\Gamma(1+\alpha)), compute the Caputo derivative formally for simple U.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
For 0<\alpha<1, the Caputo derivative is {}^CD_{0+}^{\alpha}f(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}f'(s)\,\dd s. If f(t)=t^\beta, \beta>0, then f'(s)=\beta s^{\beta-1}. Thus {}^CD_{0+}^{\alpha}t^\beta = \frac{\beta}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}s^{\beta-1}\,\dd s. Set s=t\xi. Then {}^CD_{0+}^{\alpha}t^\beta = \frac{\beta t^{\beta-\alpha}}{\Gamma(1-\alpha)} B(\beta,1-\alpha) = \frac{\Gamma(\beta+1)}{\Gamma(\beta+1-\alpha)}t^{\beta-\alpha}. For f(t)=1, f'(t)=0, so {}^CD^\alpha1=0.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.50
Problem formulation.
Compare L1, Alikhanov, Lubich, PECE, fast convolution, and Chebyshev–Beta methods in terms of accuracy and cost.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Start from {}^CD_{0+}^{\alpha}u(t_n) = \frac{1}{\Gamma(1-\alpha)} \sum_{j=0}^{n-1} \int_{t_j}^{t_{j+1}}(t_n-s)^{-\alpha}u'(s)\,\dd s. Approximate u'(s)\approx\frac{u_{j+1}-u_j}{h}. The panel integral is \int_{t_j}^{t_{j+1}}(t_n-s)^{-\alpha}\,\dd s = \frac{(t_n-t_j)^{1-\alpha}-(t_n-t_{j+1})^{1-\alpha}}{1-\alpha}. Using \Gamma(2-\alpha)=(1-\alpha)\Gamma(1-\alpha), one obtains {}^CD_{0+}^{\alpha}u(t_n) \approx \frac{1}{h^\alpha\Gamma(2-\alpha)} \sum_{k=0}^{n-1}a_k(u_{n-k}-u_{n-k-1}), where a_k=(k+1)^{1-\alpha}-k^{1-\alpha}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.51
Problem formulation.
Study starting corrections for high-order Lubich convolution quadrature when the solution has initial singularities.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 15.52
Problem formulation.
Derive how a graded mesh t_n=T(n/N)^r can recover accuracy for weakly singular fractional solutions.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.53
Problem formulation.
Prove an error estimate for the Alikhanov L2-1\sigma formula under smoothness assumptions.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 15.54
Problem formulation.
Construct a cubic L3-type formula using polynomial interpolation on uniform panels.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.55
Problem formulation.
Verify all six quintic Chebyshev–Lobatto derivative polynomials.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
On a panel [a,b], approximate u by a Chebyshev–Lobatto interpolant p(\theta)=\sum_{m=0}^{q}u(s_m)\ell_m^{(q)}(\theta), \qquad s=a+(b-a)\theta. Then u'(s)\approx \frac1{b-a}\sum_{m=0}^{q}u(s_m)(\ell_m^{(q)})'(\theta). If (\ell_m^{(q)})'(\theta)=\sum_{r=0}^{q-1}d_{m,r}^{(q)}\theta^r, the Caputo panel weight is W_{m}^{C,q}(t) = \sum_{r=0}^{q-1}\frac{d_{m,r}^{(q)}}{(b-a)^{r+1}} I_r^C(a,b;t), where I_r^C(a,b;t)=\int_a^b(t-s)^{-\alpha}(s-a)^r\,\dd s. The corresponding moments are evaluated exactly by incomplete beta functions.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.56
Problem formulation.
Design an implementation of the quintic Caputo–Katugampola beta-weight formula.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
For 0<\alpha<1, the Caputo–Katugampola derivative is {}^CD_{0+}^{\alpha,\rho}f(t) = \frac{\rho^\alpha}{\Gamma(1-\alpha)} \int_0^t(t^\rho-s^\rho)^{-\alpha}f'(s)\,\dd s. Let y=t^\rho and v(y)=f(y^{1/\rho}). Then {}^CD_{0+}^{\alpha,\rho}f(t) = \rho^\alpha\,{}^CD_{0+}^{\alpha}v(y). For f(t)=t^\beta, v(y)=y^{\beta/\rho}, so {}^CD_{0+}^{\alpha,\rho}t^\beta = \rho^\alpha \frac{\Gamma(\beta/\rho+1)} {\Gamma(\beta/\rho+1-\alpha)} t^{\beta-\alpha\rho}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.57
Problem formulation.
Analyze numerical cancellation in incomplete-beta weight differences when a panel is far from t_n.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
On a panel [a,b], approximate u by a Chebyshev–Lobatto interpolant p(\theta)=\sum_{m=0}^{q}u(s_m)\ell_m^{(q)}(\theta), \qquad s=a+(b-a)\theta. Then u'(s)\approx \frac1{b-a}\sum_{m=0}^{q}u(s_m)(\ell_m^{(q)})'(\theta). If (\ell_m^{(q)})'(\theta)=\sum_{r=0}^{q-1}d_{m,r}^{(q)}\theta^r, the Caputo panel weight is W_{m}^{C,q}(t) = \sum_{r=0}^{q-1}\frac{d_{m,r}^{(q)}}{(b-a)^{r+1}} I_r^C(a,b;t), where I_r^C(a,b;t)=\int_a^b(t-s)^{-\alpha}(s-a)^r\,\dd s. The corresponding moments are evaluated exactly by incomplete beta functions.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Always test the result on the scalar model equation or Fourier mode and verify the amplification factor satisfies the stated stability bound.
Exercise 15.58
Problem formulation.
Analyze the behavior of the last-panel weights as b_j\to t_n.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.59
Problem formulation.
Design a sum-of-exponentials approximation for the kernel t^{-\alpha} on [\Delta t,T].
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.60
Problem formulation.
Compare the computational complexity of direct history summation and fast convolution.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.61
Problem formulation.
Derive fractional Adams–Bashforth and Adams–Moulton weights from the Volterra form.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
For the Caputo IVP {}^CD_{0+}^{\alpha}u(t)=f(t,u(t)), \qquad u(0)=u_0, apply the fractional integral I^\alpha. Since I^\alpha{}^CD^\alpha u(t)=u(t)-u(0), we obtain u(t)=u_0+\frac{1}{\Gamma(\alpha)} \int_0^t(t-s)^{\alpha-1}f(s,u(s))\,\dd s. This is the Volterra integral formulation.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.62
Problem formulation.
Study convergence of PECE methods for nonlinear fractional IVPs.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 15.63
Problem formulation.
Derive Newton’s method for an implicit fractional time-stepping system.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.64
Problem formulation.
Construct a residual-minimization objective for a time-fractional nonlinear PDE.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.65
Problem formulation.
Construct a residual-minimization objective involving a Caputo–Katugampola derivative.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
For 0<\alpha<1, the Caputo–Katugampola derivative is {}^CD_{0+}^{\alpha,\rho}f(t) = \frac{\rho^\alpha}{\Gamma(1-\alpha)} \int_0^t(t^\rho-s^\rho)^{-\alpha}f'(s)\,\dd s. Let y=t^\rho and v(y)=f(y^{1/\rho}). Then {}^CD_{0+}^{\alpha,\rho}f(t) = \rho^\alpha\,{}^CD_{0+}^{\alpha}v(y). For f(t)=t^\beta, v(y)=y^{\beta/\rho}, so {}^CD_{0+}^{\alpha,\rho}t^\beta = \rho^\alpha \frac{\Gamma(\beta/\rho+1)} {\Gamma(\beta/\rho+1-\alpha)} t^{\beta-\alpha\rho}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.66
Problem formulation.
Derive a numerical method for {}^CD_{0+}^{\alpha}u+\omega^2u=0.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.67
Problem formulation.
Design a method for a_1{}^CD^\alpha u+a_2{}^CD^\beta u=f(t,u).
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
On a panel [a,b], approximate u by a Chebyshev–Lobatto interpolant p(\theta)=\sum_{m=0}^{q}u(s_m)\ell_m^{(q)}(\theta), \qquad s=a+(b-a)\theta. Then u'(s)\approx \frac1{b-a}\sum_{m=0}^{q}u(s_m)(\ell_m^{(q)})'(\theta). If (\ell_m^{(q)})'(\theta)=\sum_{r=0}^{q-1}d_{m,r}^{(q)}\theta^r, the Caputo panel weight is W_{m}^{C,q}(t) = \sum_{r=0}^{q-1}\frac{d_{m,r}^{(q)}}{(b-a)^{r+1}} I_r^C(a,b;t), where I_r^C(a,b;t)=\int_a^b(t-s)^{-\alpha}(s-a)^r\,\dd s. The corresponding moments are evaluated exactly by incomplete beta functions.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.68
Problem formulation.
Formulate a quadrature method for a distributed-order derivative \int_0^1 \mu(\alpha){}^CD^\alpha u\,\dd\alpha.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 15.69
Problem formulation.
Discuss numerical challenges for variable-order Caputo derivatives {}^CD^{\alpha(t)}u(t).
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
For 0<\alpha<1, the Caputo derivative is {}^CD_{0+}^{\alpha}f(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}f'(s)\,\dd s. If f(t)=t^\beta, \beta>0, then f'(s)=\beta s^{\beta-1}. Thus {}^CD_{0+}^{\alpha}t^\beta = \frac{\beta}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}s^{\beta-1}\,\dd s. Set s=t\xi. Then {}^CD_{0+}^{\alpha}t^\beta = \frac{\beta t^{\beta-\alpha}}{\Gamma(1-\alpha)} B(\beta,1-\alpha) = \frac{\Gamma(\beta+1)}{\Gamma(\beta+1-\alpha)}t^{\beta-\alpha}. For f(t)=1, f'(t)=0, so {}^CD^\alpha1=0.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 15.70
Problem formulation.
Compare Riesz, spectral fractional Laplacian, and integral fractional Laplacian discretizations.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.71
Problem formulation.
Discuss boundary conditions for space-fractional diffusion problems.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Distinguish forward error from backward error; a small residual does not imply a small forward error when the problem is ill-conditioned.
Exercise 15.72
Problem formulation.
Derive how kernel approximation error propagates into the fractional derivative.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 15.73
Problem formulation.
Design an adaptive time-step strategy for a Caputo fractional IVP.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
For 0<\alpha<1, the Caputo derivative is {}^CD_{0+}^{\alpha}f(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}f'(s)\,\dd s. If f(t)=t^\beta, \beta>0, then f'(s)=\beta s^{\beta-1}. Thus {}^CD_{0+}^{\alpha}t^\beta = \frac{\beta}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}s^{\beta-1}\,\dd s. Set s=t\xi. Then {}^CD_{0+}^{\alpha}t^\beta = \frac{\beta t^{\beta-\alpha}}{\Gamma(1-\alpha)} B(\beta,1-\alpha) = \frac{\Gamma(\beta+1)}{\Gamma(\beta+1-\alpha)}t^{\beta-\alpha}. For f(t)=1, f'(t)=0, so {}^CD^\alpha1=0.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.74
Problem formulation.
Explain when a small fractional residual implies a small solution error.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 15.75
Problem formulation.
Design a benchmark comparing L1, Alikhanov, Lubich, PECE, and Chebyshev–Beta methods on a known exact solution.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Start from {}^CD_{0+}^{\alpha}u(t_n) = \frac{1}{\Gamma(1-\alpha)} \sum_{j=0}^{n-1} \int_{t_j}^{t_{j+1}}(t_n-s)^{-\alpha}u'(s)\,\dd s. Approximate u'(s)\approx\frac{u_{j+1}-u_j}{h}. The panel integral is \int_{t_j}^{t_{j+1}}(t_n-s)^{-\alpha}\,\dd s = \frac{(t_n-t_j)^{1-\alpha}-(t_n-t_{j+1})^{1-\alpha}}{1-\alpha}. Using \Gamma(2-\alpha)=(1-\alpha)\Gamma(1-\alpha), one obtains {}^CD_{0+}^{\alpha}u(t_n) \approx \frac{1}{h^\alpha\Gamma(2-\alpha)} \sum_{k=0}^{n-1}a_k(u_{n-k}-u_{n-k-1}), where a_k=(k+1)^{1-\alpha}-k^{1-\alpha}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.76
Problem formulation.
Design a benchmark for Caputo–Katugampola methods using an exact power-function solution.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
For 0<\alpha<1, the Caputo–Katugampola derivative is {}^CD_{0+}^{\alpha,\rho}f(t) = \frac{\rho^\alpha}{\Gamma(1-\alpha)} \int_0^t(t^\rho-s^\rho)^{-\alpha}f'(s)\,\dd s. Let y=t^\rho and v(y)=f(y^{1/\rho}). Then {}^CD_{0+}^{\alpha,\rho}f(t) = \rho^\alpha\,{}^CD_{0+}^{\alpha}v(y). For f(t)=t^\beta, v(y)=y^{\beta/\rho}, so {}^CD_{0+}^{\alpha,\rho}t^\beta = \rho^\alpha \frac{\Gamma(\beta/\rho+1)} {\Gamma(\beta/\rho+1-\alpha)} t^{\beta-\alpha\rho}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.77
Problem formulation.
Develop an error theory for the quintic Chebyshev–Beta Caputo–Katugampola formula.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
For 0<\alpha<1, the Caputo–Katugampola derivative is {}^CD_{0+}^{\alpha,\rho}f(t) = \frac{\rho^\alpha}{\Gamma(1-\alpha)} \int_0^t(t^\rho-s^\rho)^{-\alpha}f'(s)\,\dd s. Let y=t^\rho and v(y)=f(y^{1/\rho}). Then {}^CD_{0+}^{\alpha,\rho}f(t) = \rho^\alpha\,{}^CD_{0+}^{\alpha}v(y). For f(t)=t^\beta, v(y)=y^{\beta/\rho}, so {}^CD_{0+}^{\alpha,\rho}t^\beta = \rho^\alpha \frac{\Gamma(\beta/\rho+1)} {\Gamma(\beta/\rho+1-\alpha)} t^{\beta-\alpha\rho}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 15.78
Problem formulation.
Design start-up corrections for Chebyshev–Beta panel formulas when u(t)\sim t^\gamma near zero.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
On a panel [a,b], approximate u by a Chebyshev–Lobatto interpolant p(\theta)=\sum_{m=0}^{q}u(s_m)\ell_m^{(q)}(\theta), \qquad s=a+(b-a)\theta. Then u'(s)\approx \frac1{b-a}\sum_{m=0}^{q}u(s_m)(\ell_m^{(q)})'(\theta). If (\ell_m^{(q)})'(\theta)=\sum_{r=0}^{q-1}d_{m,r}^{(q)}\theta^r, the Caputo panel weight is W_{m}^{C,q}(t) = \sum_{r=0}^{q-1}\frac{d_{m,r}^{(q)}}{(b-a)^{r+1}} I_r^C(a,b;t), where I_r^C(a,b;t)=\int_a^b(t-s)^{-\alpha}(s-a)^r\,\dd s. The corresponding moments are evaluated exactly by incomplete beta functions.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.79
Problem formulation.
Develop a posteriori estimates connecting fractional residual norms to solution errors.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 15.80
Problem formulation.
Combine high-order Chebyshev–Beta panel formulas with fast convolution.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
On a panel [a,b], approximate u by a Chebyshev–Lobatto interpolant p(\theta)=\sum_{m=0}^{q}u(s_m)\ell_m^{(q)}(\theta), \qquad s=a+(b-a)\theta. Then u'(s)\approx \frac1{b-a}\sum_{m=0}^{q}u(s_m)(\ell_m^{(q)})'(\theta). If (\ell_m^{(q)})'(\theta)=\sum_{r=0}^{q-1}d_{m,r}^{(q)}\theta^r, the Caputo panel weight is W_{m}^{C,q}(t) = \sum_{r=0}^{q-1}\frac{d_{m,r}^{(q)}}{(b-a)^{r+1}} I_r^C(a,b;t), where I_r^C(a,b;t)=\int_a^b(t-s)^{-\alpha}(s-a)^r\,\dd s. The corresponding moments are evaluated exactly by incomplete beta functions.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 15.81
Problem formulation.
Design an adaptive panel strategy that refines where the memory integrand is least smooth.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
On a panel [a,b], approximate u by a Chebyshev–Lobatto interpolant p(\theta)=\sum_{m=0}^{q}u(s_m)\ell_m^{(q)}(\theta), \qquad s=a+(b-a)\theta. Then u'(s)\approx \frac1{b-a}\sum_{m=0}^{q}u(s_m)(\ell_m^{(q)})'(\theta). If (\ell_m^{(q)})'(\theta)=\sum_{r=0}^{q-1}d_{m,r}^{(q)}\theta^r, the Caputo panel weight is W_{m}^{C,q}(t) = \sum_{r=0}^{q-1}\frac{d_{m,r}^{(q)}}{(b-a)^{r+1}} I_r^C(a,b;t), where I_r^C(a,b;t)=\int_a^b(t-s)^{-\alpha}(s-a)^r\,\dd s. The corresponding moments are evaluated exactly by incomplete beta functions.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.82
Problem formulation.
Formulate residual minimization for a time-fractional KdV-type equation using Chebyshev–Beta Caputo evaluation.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
For 0<\alpha<1, the Caputo derivative is {}^CD_{0+}^{\alpha}f(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}f'(s)\,\dd s. If f(t)=t^\beta, \beta>0, then f'(s)=\beta s^{\beta-1}. Thus {}^CD_{0+}^{\alpha}t^\beta = \frac{\beta}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}s^{\beta-1}\,\dd s. Set s=t\xi. Then {}^CD_{0+}^{\alpha}t^\beta = \frac{\beta t^{\beta-\alpha}}{\Gamma(1-\alpha)} B(\beta,1-\alpha) = \frac{\Gamma(\beta+1)}{\Gamma(\beta+1-\alpha)}t^{\beta-\alpha}. For f(t)=1, f'(t)=0, so {}^CD^\alpha1=0.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.83
Problem formulation.
Formulate residual diagnostics for a time-fractional nonlinear Schrodinger equation.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.84
Problem formulation.
Develop a high-order numerical method for a fractional Duffing oscillator and compare with PECE.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 15.85
Problem formulation.
Design a Caputo or Caputo–Katugampola numerical residual method for a fractional Binet-type orbital model.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
For 0<\alpha<1, the Caputo–Katugampola derivative is {}^CD_{0+}^{\alpha,\rho}f(t) = \frac{\rho^\alpha}{\Gamma(1-\alpha)} \int_0^t(t^\rho-s^\rho)^{-\alpha}f'(s)\,\dd s. Let y=t^\rho and v(y)=f(y^{1/\rho}). Then {}^CD_{0+}^{\alpha,\rho}f(t) = \rho^\alpha\,{}^CD_{0+}^{\alpha}v(y). For f(t)=t^\beta, v(y)=y^{\beta/\rho}, so {}^CD_{0+}^{\alpha,\rho}t^\beta = \rho^\alpha \frac{\Gamma(\beta/\rho+1)} {\Gamma(\beta/\rho+1-\alpha)} t^{\beta-\alpha\rho}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.86
Problem formulation.
Derive first-order optimality conditions for a Caputo fractional optimal control problem.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
For 0<\alpha<1, the Caputo derivative is {}^CD_{0+}^{\alpha}f(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}f'(s)\,\dd s. If f(t)=t^\beta, \beta>0, then f'(s)=\beta s^{\beta-1}. Thus {}^CD_{0+}^{\alpha}t^\beta = \frac{\beta}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}s^{\beta-1}\,\dd s. Set s=t\xi. Then {}^CD_{0+}^{\alpha}t^\beta = \frac{\beta t^{\beta-\alpha}}{\Gamma(1-\alpha)} B(\beta,1-\alpha) = \frac{\Gamma(\beta+1)}{\Gamma(\beta+1-\alpha)}t^{\beta-\alpha}. For f(t)=1, f'(t)=0, so {}^CD^\alpha1=0.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 15.87
Problem formulation.
Develop adjoint gradients for parameter estimation in fractional differential equations.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.88
Problem formulation.
Compare fractional physics-informed neural networks with Chebyshev–Beta residual minimization.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
On a panel [a,b], approximate u by a Chebyshev–Lobatto interpolant p(\theta)=\sum_{m=0}^{q}u(s_m)\ell_m^{(q)}(\theta), \qquad s=a+(b-a)\theta. Then u'(s)\approx \frac1{b-a}\sum_{m=0}^{q}u(s_m)(\ell_m^{(q)})'(\theta). If (\ell_m^{(q)})'(\theta)=\sum_{r=0}^{q-1}d_{m,r}^{(q)}\theta^r, the Caputo panel weight is W_{m}^{C,q}(t) = \sum_{r=0}^{q-1}\frac{d_{m,r}^{(q)}}{(b-a)^{r+1}} I_r^C(a,b;t), where I_r^C(a,b;t)=\int_a^b(t-s)^{-\alpha}(s-a)^r\,\dd s. The corresponding moments are evaluated exactly by incomplete beta functions.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.89
Problem formulation.
Design a method for a PDE with both Caputo time derivative and fractional Laplacian.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
For 0<\alpha<1, the Caputo derivative is {}^CD_{0+}^{\alpha}f(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}f'(s)\,\dd s. If f(t)=t^\beta, \beta>0, then f'(s)=\beta s^{\beta-1}. Thus {}^CD_{0+}^{\alpha}t^\beta = \frac{\beta}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}s^{\beta-1}\,\dd s. Set s=t\xi. Then {}^CD_{0+}^{\alpha}t^\beta = \frac{\beta t^{\beta-\alpha}}{\Gamma(1-\alpha)} B(\beta,1-\alpha) = \frac{\Gamma(\beta+1)}{\Gamma(\beta+1-\alpha)}t^{\beta-\alpha}. For f(t)=1, f'(t)=0, so {}^CD^\alpha1=0.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.90
Problem formulation.
Develop a solver for distributed-order diffusion and analyze the order quadrature.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 15.91
Problem formulation.
Propose a definition and numerical method for a variable-order Caputo–Katugampola derivative.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
For 0<\alpha<1, the Caputo–Katugampola derivative is {}^CD_{0+}^{\alpha,\rho}f(t) = \frac{\rho^\alpha}{\Gamma(1-\alpha)} \int_0^t(t^\rho-s^\rho)^{-\alpha}f'(s)\,\dd s. Let y=t^\rho and v(y)=f(y^{1/\rho}). Then {}^CD_{0+}^{\alpha,\rho}f(t) = \rho^\alpha\,{}^CD_{0+}^{\alpha}v(y). For f(t)=t^\beta, v(y)=y^{\beta/\rho}, so {}^CD_{0+}^{\alpha,\rho}t^\beta = \rho^\alpha \frac{\Gamma(\beta/\rho+1)} {\Gamma(\beta/\rho+1-\alpha)} t^{\beta-\alpha\rho}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 15.92
Problem formulation.
Design preconditioners for implicit fractional time-stepping systems.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Distinguish forward error from backward error; a small residual does not imply a small forward error when the problem is ill-conditioned.
Exercise 15.93
Problem formulation.
Develop a parallel-in-time history algorithm for fractional differential equations.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.94
Problem formulation.
Design a GPU implementation of fast fractional convolution.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.95
Problem formulation.
Create certified benchmark problems with exact Caputo and Caputo–Katugampola solutions.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
For 0<\alpha<1, the Caputo–Katugampola derivative is {}^CD_{0+}^{\alpha,\rho}f(t) = \frac{\rho^\alpha}{\Gamma(1-\alpha)} \int_0^t(t^\rho-s^\rho)^{-\alpha}f'(s)\,\dd s. Let y=t^\rho and v(y)=f(y^{1/\rho}). Then {}^CD_{0+}^{\alpha,\rho}f(t) = \rho^\alpha\,{}^CD_{0+}^{\alpha}v(y). For f(t)=t^\beta, v(y)=y^{\beta/\rho}, so {}^CD_{0+}^{\alpha,\rho}t^\beta = \rho^\alpha \frac{\Gamma(\beta/\rho+1)} {\Gamma(\beta/\rho+1-\alpha)} t^{\beta-\alpha\rho}.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 15.96
Problem formulation.
Design reproducible experiments comparing all methods in this chapter, including accuracy, runtime, memory, and residual norms.
Method.
Use fractional integral definitions, Caputo/Caputo–Katugampola kernels, Volterra forms, convolution weights, beta-function moments, and residual minimization.
Detailed solution and justification.
Fractional equations are nonlocal. The Caputo derivative is {}^CD_{0+}^{\alpha}u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t(t-s)^{-\alpha}u'(s)\,\dd s. Numerical schemes approximate this history integral by convolution weights, panel interpolation, or Volterra quadrature. For residual minimization, define \mathcal R[u](x,t)=0 and minimize J(\theta)=\int_0^T\int_\Omega |\mathcal R[u_\theta](x,t)|^2\,\dd x\,\dd t. Accurate memory evaluation is essential because an inaccurate fractional derivative contaminates the residual.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Interactive tools
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