Abstract and Applied Analysis
Volume 2013 (2013), Article ID 823098, 12 pages
Research Article

A Collocation Method Based on the Bernoulli Operational Matrix for Solving High-Order Linear Complex Differential Equations in a Rectangular Domain

1Department of Applied Mathematics, School of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran
2Center of Excellence on Modelling and Control Systems, Ferdowsi University of Mashhad, Mashhad, Iran
3Department of Mathematics, University of Venda, Private Bag X5050, Thohoyandou 0950, South Africa

Received 19 November 2012; Accepted 14 February 2013

Academic Editor: Douglas Anderson

Copyright © 2013 Faezeh Toutounian 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.


This paper contributes a new matrix method for the solution of high-order linear complex differential equations with variable coefficients in rectangular domains under the considered initial conditions. On the basis of the presented approach, the matrix forms of the Bernoulli polynomials and their derivatives are constructed, and then by substituting the collocation points into the matrix forms, the fundamental matrix equation is formed. This matrix equation corresponds to a system of linear algebraic equations. By solving this system, the unknown Bernoulli coefficients are determined and thus the approximate solutions are obtained. Also, an error analysis based on the use of the Bernoulli polynomials is provided under several mild conditions. To illustrate the efficiency of our method, some numerical examples are given.