The Approximate Solution of High-Order Nonlinear Ordinary Differential Equations by Improved Collocation Method with Terms of Shifted Chebyshev Polynomials

Abstract

In this paper, we present a direct computational method for solving the higher-order nonlinear differential equations by using collocation method. This method transforms the nonlinear differential equation into the system of nonlinear algebraic equations with unknown shifted Chebyshev coefficients, via Chebyshev–Gauss collocation points. The solution of this system yields the Chebyshev coefficients of the solution function. The method is valid for both initial-value and boundary-value problems. Several examples are presented to illustrate the accuracy and effectiveness of the method by the approximate solutions of very important equations of applied mathematics such as Lane–Emden equation, Riccati equation, Van der Pol equation. The approximate solutions can be very easily calculated using computer program Maple 13.