Numerical Solution of Higher-Order Linear and Nonlinear Ordinary Differential Equations with Orthogonal Rational Legendre Functions

In this paper, we describe a method for the solution of linear and nonlinear ordinary differential equations ODE's of arbitrary order with initial or boundary conditions. In this direction we first investigate some properties of orthogonal rational Legendre functions, and then we give the least square method based on these basis functions for the solution of such equations. In this method the solution of an ODE is reduced to a minimization problem, which is then numerically solved via Maple 16. Finally results of this method which are obtained in the form of continuous functions, will be compared with the numerical results in other references.