Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 823098, 12 pages http://dx.doi.org/10.1155/2013/823098 Research Article A Collocation Method Based on the Bernoulli Operational Matrix for Solving High-Order Linear Complex Differential Equations in a Rectangular Domain Faezeh Toutounian, 1,2 Emran Tohidi, 1 and Stanford Shateyi 3 1 Department of Applied Mathematics, School of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran 2 Center of Excellence on Modelling and Control Systems, Ferdowsi University of Mashhad, Mashhad, Iran 3 Department of Mathematics, University of Venda, Private Bag X5050, Tohoyandou 0950, South Africa Correspondence should be addressed to Stanford Shateyi; stanford.shateyi@univen.ac.za Received 19 November 2012; Accepted 14 February 2013 Academic Editor: Douglas Anderson Copyright © 2013 Faezeh Toutounian et al. Tis 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. Tis paper contributes a new matrix method for the solution of high-order linear complex diferential equations with variable coefcients 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. Tis matrix equation corresponds to a system of linear algebraic equations. By solving this system, the unknown Bernoulli coefcients 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 efciency of our method, some numerical examples are given. 1. Introduction Complex diferential equations have a great popularity in science and engineering. In real world, physical events can be modeled by complex diferential equations usually. For instance, the vibrations of a one-mass system with two DOFs are mostly described using diferential equations with a complex dependent variable [1, 2]. Te various applications of diferential equations with complex dependent variables are introduced in [2]. Since a huge size of such equations cannot be solved explicitly, it is ofen necessary to resort to approximation and numerical techniques. In recent years, the studies on complex diferential equa- tions, such as a geometric approach based on meromorphic function in arbitrary domains [3], a topological description of solutions of some complex diferential equations with multivalued coefcients [4], the zero distribution [5], growth estimates [6] of linear complex diferential equations, and also the rational together with the polynomial approxima- tions of analytic functions in the complex plane [7, 8], were developed very rapidly and intensively. Since the beginning of 1994, the Laguerre, Chebyshev, Taylor, Legendre, Hermite, and Bessel (matrix and colloca- tion) methods have been used in the works in [9–19] to solve linear diferential, integral, and integrodiferential-diference equations and their systems. Also, the Bernoulli matrix method has been used to fnd the approximate solutions of diferential and integrodiferential equations [20–22]. In this paper, in the light of the above-mentioned methods and by means of the matrix relations between the Bernoulli polynomials and their derivatives, we develop a new method called the Bernoulli collocation method (BCM) for solving high-order linear complex diferential equation  () ()+ −1 ∑ =0   () () ()=(), ≥1,=+,∈[,],∈[,], (1) with the initial conditions  () (0)=  , =0,1,...,−1. (2)