Applications of the galerkin methods for ordinary differential equations

In this study, after the basic concepts of ordinary differential equations are given, information about the commonly used methods in numerical solutions of these equations is given. Differential equations defined by initial and boundary values are used to model and solve real-world problems. However, exact solutions of these equations are often complex and rarely found. Therefore, approximation techniques are used for numerical solution of differential equations and supported by mathematical analysis. In order to find solutions to second-order boundary value problems, it is necessary to express the boundary value problem in variational form. Variational methods provide solutions to boundary value problems by using trial functions satisfying certain boundary conditions and using integral formulation to minimize the error. With methods such as Galerkin's method, we can obtain functions and differential equations satisfying the boundary conditions.