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Finite differences
Finite differences turn the differential equation and boundary conditions into algebraic equations.
Collocation
Collocation enforces the differential equation at selected nodes using an approximate solution space.
Boundary residual
Boundary residuals are as important as differential-equation residuals.
Finite elements
Finite elements use weak formulations and local basis functions to assemble a global system.
BVP
A boundary-value problem imposes conditions at multiple boundary points, so the solution must be found globally.
Shooting
Shooting converts missing initial data into parameters and solves a nonlinear matching problem.