Residual Correction Procedure with Bernstein Polynomials for Solving Important Systems of Ordinary Differential Equations

摘要

One of the most attractive subjects in applied sciences is to obtain exact or approximate solutions for different types of linear and nonlinear systems. Systems of ordinary differential equations like systems of second-order boundary value problems (BVPs), Brusselator system and stiff system are significant in science and engineering. One of the most challenge problems in applied science is to construct methods to approximate solutions of such systems of differential equations which pose great challenges for numerical simulations. Bernstein polynomials method with residual correction procedure is used to treat those challenges. The aim of this paper is to present a technique to approximate solutions of such differential equations in optimal way. In it, we introduce a method called residual correction procedure, to correct some previous approximate solutions for such systems. We study the error analysis of our given method. We first introduce a new result to approximate the absolute solution by using the residual correction procedure. Second, we introduce a new result to get appropriate bound for the absolute error. The collocation method is used and the collocation points can be found by applying Chebyshev roots. Both techniques are explained briefly with illustrative examples to demonstrate the applicability, efficiency and accuracy of the techniques. By using a small number of Bernstein polynomials and correction procedure we achieve some significant results. We present some examples to show the efficiency of our method by comparing the solution of such problems obtained by our method with the solution obtained by Runge-Kutta method, continuous genetic algorithm, rational homotopy perturbation method and adomian decomposition method.