Explicit Solution of Hyperbolic Partial Differential Equations by an Iterative Decomposition Method

Explicit Solution of Hyperbolic Partial Differential Equations by an Iterative Decomposition Method Adekunle, Y.A. 1* , Kadiri, K.O. 2 , Odetunde, O.S. 3 1 Department of Computer Science and Mathematics, Babcock University, Ilisan-Remo Ogun State, Nigeria [email protected] 2 . Department Electrical/Electronics Engineering, Federal Polytechnic, Offa, Kwara State, Nigeria [email protected] 3 . Department of Mathematical Science, Olabisi Onabanjo University, Ago-Iwoye, Ogun State,Nigeria [email protected] Abstract- In this paper, an iterative decomposition method is applied to solve partial differential equations. The solution of a partial differential equation of the hyperbolic form is obtained by the stated method in the form of an infinite series of easily computable terms. Some examples are given and the solutions obtained by the method are found to compare favourably with the known exact solutions. Keywords: Hyperbolic partial differential equation, iterative decomposition method, analytical solution . I. INTRODUCTION An equation involving two or more partial derivatives of an unknown function of two or more independent variables is called a Partial Differential equation. The most general second order linear partial differential equation in two independent variables x and y in this paper can be expressed generally in the form + + + + e (1) Or + + = H (x, y, u, ) (2) where a, b and c are funct ion of x and y and H is a linear function of u, , . Thus equation (2) is called linear, and is a form of (1). The second order derivatives occur only to the first degree .If in equation (2), b 2_ 4ac >0, the equation is called Hyperbolic. Since the governing equation in many phenomena of engineering, science, and mathematical physics lead to hyperbolic equations, such as the wave equations, much attention is given to the study of this class of partial differential equation by mathematicia ns. Various numerical methods have been applied to solve hyperbolic partial differential equations. See Ref[4 ]. However, many of these methods, commonly used as characteristic methods require large size of computational work, and round-off error causes loss of accuracy [2]. The present paper attempts to present a method for the expilict solution of hyperbolic partial differential equation. The results obtained are compared with known analytical solution. For problems whose analytical solutions are unknown, the method will be appropriate. The paper is organised as follows: section 2 presents the iterative decomposition method, section 3 considers the method of section 2 for hyperbolic equations. In section 4, we present examples to illustrate the simplicity, efficiency and accuracy of the method. A conclusion is drawn in section 5 II THE ITERATIVE DECOMPOSITION METHOD The idea of the iterative decomposition method can be conveyed considering equation (1) as an equation in the form (4) Where k is a constant and N(u) is the nonlinear term. We may find the solution of (4) in a series form such as (5) We can decompose the nonlinear operator N as (6) (IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 6, June 2011 307 http://sites.google.com/site/ijcsis/ ISSN 1947-5500

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Explicit Solution of Hyperbolic Partial Differential Equations by an Iterative Decomposition Method