Řešení okrajových úloh pro obyčejné diferenciální rovnice metodou střelby

Abstract

This bachelor's thesis focuses on the numerical solution of boundary value problems for second-order ordinary differential equations. The introductory part describes basic numerical methods for solving initial value problems such as the Euler method, Heun’s method, and the fourth-order Runge-Kutta method. The main part of the thesis is dedicated to boundary value problems, their mathematical formulation, and the conditions for the existence and uniqueness of solutions. For the numerical solution, the shooting method was implemented, which transforms a boundary value problem into an initial value problem. Its combination with Newton’s method speeds up the solution of each problem. Moreover, Newton’s method was implemented using automatic differentiation provided by the JAX library, which simplifies the implementation of more complex models. The proposed methods were applied to several examples, including the equation of damped harmonic oscillations.