Shooting method

The shooting method solves a boundary-value problem by guessing unknown initial conditions, integrating an initial-value problem, and adjusting the guesses until the required boundary conditions are met.

Initial-value problem: A differential equation problem specified by conditions on the solution and its derivatives at one point. Each trial in the shooting method is an initial-value problem launched from one boundary.
Ordinary differential equation: An equation relating a function of one variable to its derivatives. Classical shooting is formulated for ordinary differential equation boundary problems.
Bratu problem: A nonlinear boundary-value problem involving an exponential source term and conditions at both endpoints. Its multiple solution branches make initial-slope guesses central to successful shooting.
Finite difference method: A numerical method that approximates derivatives with differences on a grid. It solves for values across the domain directly instead of integrating from a guessed endpoint slope.
Boundary-value problem: A differential equation problem whose solution must satisfy conditions at multiple points or boundaries. The shooting method replaces its separated boundary conditions with adjustable initial data.
Dirichlet boundary condition: A boundary condition that specifies the value of a function at a boundary point. Such prescribed endpoint values define the target that a trial trajectory must hit.
Sturm–Liouville theory: A theory of linear differential operators and eigenvalue problems with boundary conditions. Shooting can estimate eigenvalues by varying a parameter until the terminal condition is satisfied.
Finite element method: A numerical method that approximates a solution using local basis functions over a mesh. Its global discretization avoids relying on one trajectory remaining stable across the full interval.
Numerical root-finding: Algorithms that approximate zeros of functions using evaluations or local models. The endpoint error is treated as a function of the guessed initial condition and driven toward zero.
Neumann boundary condition: A boundary condition that specifies a derivative of a function at a boundary point. A boundary-value problem may prescribe endpoint slopes as well as function values.

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Numerical methods for ordinary differential equationsRelated: It shows how an initial-value solver can serve as a component in a different problem formulation.