Modifed Finite Integration Method by Using Legendre Polynomials for Solving Linear Ordinary Differential Equations

In this paper, we construct a numerical procedure which is called the finite integration method by using the Legendre polynomial. This numerical procedure is for solving the linear ordinary differential equations. That is, we define the solution as a linear combination of the Legendre polynomials and we use the zeros of Legendre polynomial as computational grid points. We implement this procedure with several numerical examples to demonstrate the accuracy of our method comparing to the finite difference method, the traditional finite integration methods and their analytical solutions.