Shooting Method: Numerical Solution for Boundary Value Problems

The shooting method is a classical numerical approach for solving boundary value problems, particularly effective for ordinary differential equations with boundary conditions. Its core concept involves transforming boundary value problems into initial value problems by iteratively adjusting initial conditions until the solution satisfies the given boundary constraints.

MATLAB implementation of the shooting method typically involves these key steps:

Problem Transformation: Convert boundary value problem to initial value problem with guessed initial conditions (typically unknown derivative values). This requires defining the ODE system using function handles.

Numerical Integration: Employ ODE solvers like `ode45` or `ode15s` to integrate the differential equations. These solvers use adaptive step-size control for accuracy, where `ode45` is suitable for non-stiff problems while `ode15s` handles stiff systems better.

Boundary Condition Matching: Evaluate whether the solution meets specified boundary conditions at the endpoint. This involves creating a residual function that calculates the difference between computed and target boundary values.

Iterative Optimization: Apply root-finding algorithms like Newton-Raphson method or bisection method to refine initial guesses. The optimization process can be implemented using `fzero` or custom iteration loops with convergence tolerance checks.

The shooting method's advantage lies in its simplicity and intuitive approach, making it particularly effective for linear or weakly nonlinear problems. However, for strongly nonlinear systems or cases with multiple solutions, advanced strategies like multiple shooting method may be necessary to ensure convergence. MATLAB's flexibility and comprehensive numerical computation toolbox make it an ideal platform for implementing shooting method algorithms, allowing efficient handling of various ODE types and boundary conditions.