A Pseudospectral Algorithm for Solving Multipantograph Delay Systems on a Semi-Infinite Interval Using Legendre Rational Functions

E H Doha,A H Bhrawy,D Baleanu,R M Hafez

doi:10.1155/2014/816473

Abstract

A new Legendre rational pseudospectral scheme is proposed and developed for solving numerically systems of linear and nonlinear multipantograph equations on a semi-infinite interval. A Legendre rational collocation method based on Legendre rational-Gauss quadrature points is utilized to reduce the solution of such systems to systems of linear and nonlinear algebraic equations. In addition, accurate approximations are achieved by selecting few Legendre rational-Gauss collocation points. The numerical results obtained by this method have been compared with various exact solutions in order to demonstrate the accuracy and efficiency of the proposed method. Indeed, for relatively limited nodes used, the absolute error in our numerical solutions is sufficiently small.

Highlights

Over the last three decades, the scientists have paid much attention to spectral methods due to their high accuracy
We present two numerical examples in order to show the accuracy of Legendre rational collocation method for solving multipantograph delay system
A collocation Legendre rational method has been proposed to obtain the approximate solutions of systems of multipantograph delay equations

Summary

Introduction

Over the last three decades, the scientists have paid much attention to spectral methods due to their high accuracy (see, for instance, [1–6] and the references therein). Spectral methods, in the context of numerical schemes for differential equations, generically belong to the family of weighted residual methods (WRMs) (cf Finlayson [7]). WRMs represent a particular group of approximation techniques, in which the residuals (or errors) are minimized in a certain way and thereby leading to specific methods including Galerkin, Petrov-Galerkin, collocation, and tau formulations. WRMs are traditionally regarded as the foundation and cornerstone of the finite element, spectral, finite volume, boundary element, and some other methods. The use of Jacobi rational functions has the advantage of obtaining the solutions in terms of the Jacobi rational parameters (see, e.g., [13–16]). The authors of [17, 18] proposed an efficient collocation schemes based on the operational matrices of rational Legendre and Chebyshev functions for solving problems in the half line

Objectives

Results

Conclusion