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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 741278, 12 pages
http://dx.doi.org/10.1155/2013/741278
Research Article

Fast Spectral Collocation Method for Solving Nonlinear Time-Delayed Burgers-Type Equations with Positive Power Terms

1Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
2Department of Mathematics, Faculty of Science, Beni-Suef University, Beni Suef 62511, Egypt
3Department of Mathematics, Faculty of Science, Um-Al-Qurah University, Makkah 21955, Saudi Arabia

Received 20 January 2013; Revised 6 June 2013; Accepted 6 June 2013

Academic Editor: Mustafa Bayram

Copyright © 2013 A. H. Bhrawy et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Since the collocation method approximates ordinary differential equations, partial differential equations, and integral equations in physical space, it is very easy to implement and adapt to various problems, including variable coefficient and nonlinear differential equations. In this paper, we derive a Jacobi-Gauss-Lobatto collocation method (J-GL-C) to solve numerically nonlinear time-delayed Burgers-type equations. The proposed technique is implemented in two successive steps. In the first one, we apply nodes of the Jacobi-Gauss-Lobatto quadrature which depend upon the two general parameters , and the resulting equations together with the two-point boundary conditions constitute a system of ordinary differential equations (ODEs) in time. In the second step, the implicit Runge-Kutta method of fourth order is applied to solve a system of ODEs of second order in time. We present numerical results which illustrate the accuracy and flexibility of these algorithms.