Comparative numerical laboratory
Fractional Methods: Memory Weights and Short-Memory Truncation — Chapter 15
Plot Caputo L1 weights and compare full-memory versus truncated-memory accumulation.
\[
\omega_k=(k+1)^{1-\alpha}-k^{1-\alpha}, 0<\alpha<1.
\]
Methods compared
| Method | Convenience | Inconvenience / warning |
|---|---|---|
| Full memory | Faithful to the fractional convolution. | Cost grows with the number of time steps. |
| Short memory | Reduces cost. | Introduces truncation error that must be quantified. |
| Fast convolution viewpoint | Can reduce cost while preserving long memory. | Implementation is more complex. |
Illustrative plot / animation
Numerical diagnostics
What the student must compare
- weight decay
- memory tail
- truncation fraction
- cost versus accuracy