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(IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 6, June 2011 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 adekunleya@gmail.com 2 . Department Electrical/Electronics Engineering, Federal Polytechnic, Offa, Kwara State, Nigeria kadiritoyin2007@yahoo.com 3 . Department of Mathematical Science, Olabisi Onabanjo University, Ago-Iwoye, Ogun State,Nigeria todetunde@yahoo.co.uk Abstract- In this paper, an iterative decomposition method such as the wave equations, much attention is given is applied to solve partial differential equations. The to the study of this class of partial differential solution of a partial differential equation of the equation by mathematicians. hyperbolic form is obtained by the stated method in the Various numerical methods have been applied to form of an infinite series of easily computable terms. Some examples are given and the solutions obtained by solve hyperbolic partial differential equations. See the method are found to compare favourably with the Ref[4 ]. However, many of these methods, commonly known exact solutions. used as characteristic methods require large size of computational work, and round-off error causes loss Keywords: Hyperbolic partial differential equation, of accuracy [2]. The present paper attempts to present iterative decomposition method, analytical solution. a method for the expilict solution of hyperbolic partial differential equation. The results obtained are compared with known analytical solution. For I. INTRODUCTION problems whose analytical solutions are unknown, An equation involving two or more partial the method will be appropriate. derivatives of an unknown function of two or more The paper is organised as follows: section 2 independent variables is called a Partial Differential presents the iterative decomposition method, section equation. 3 considers the method of section 2 for hyperbolic The most general second order linear partial equations. In section 4, we present examples to differential equation in two independent variables x illustrate the simplicity, efficiency and accuracy of and y in this paper can be expressed generally in the the method. A conclusion is drawn in section 5 form + + + +e (1) II THE ITERATIVE DECOMPOSITION METHOD The idea of the iterative decomposition method Or can be conveyed considering equation (1) as an equation in the form + + = H (x, y, u, ) (2) (4) Where k is a constant and N(u) is the nonlinear term. where a, b and c are function of x and y and H is a We may find the solution of (4) in a series form such linear function of u, , . as Thus equation (2) is called linear, and is a form of (5) (1). The second order derivatives occur only to the We can decompose the nonlinear operator N as first degree .If in equation (2), b2_ 4ac >0, the equation is called Hyperbolic. Since the governing equation in many phenomena of engineering, science, and (6) mathematical physics lead to hyperbolic equations, 307 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 6, June 2011 From equations (5) and (6), equation (4) is then equivalent to = {- } (15) (7) From equation (7), we define the following Then, the solution of equation two will be given as iterative scheme (x, y) = N ( (8) We can determine the component un as far as our desired accuracy of the approximation. The n-terms can be used to approximate the m=1, 2, 3... solution. Thus, (9) III Application of the iterative Decomposition Method. III Numerical Experiments. Consider the hyperbolic equation (2) with the following initial conditions; We shall now consider the application of the iterative decomposition method. (10) (x,0) =g(x) (11) Example 4. [ ] Consider the hyperbolic partial differential equation Using , we can put (16) , with the initial condition (12) (17) in operation form as u = H(x,y,u, , )- The exact solution of this problem is (13) The inverse operator of denoted is Putting equation (16) in operator form, we have defined as (.) = - (18) Then L= u(x,y) = u(x,0) + (x,0) + H Applying the inverse operator L1 to both sides of (18) (x,y,u, , )- } (14) where and 308 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 6, June 2011 Then . Then we have the exact solution -( } (19) By the iterative decomposition method, after only two terms of the solution series. -( IV Conclusion } dy dy =0 (20) Our aim in this work has been to derive an approximation algorithm for the explicit solution of hyperbolic partial differential equations. This we have been able to do via a method which does not require elaborate mathematical computation and Then the solution is which is assumptions. This method is a significant the same as the given exact solution. improvement over some other decomposition methods, notably the well-known Adomian Example 4.2 [ ] consider the equation Decomposition Method. REFERENCES (21) [1] G. Adomian, Nonlinear Stochastic Systems and Application to Physics, Kluwer Academic Press. 1989 With the associated initial condition [2] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Press. 1994. u(x,0) =x (22) [3] J. Biazar, and H. Ebrahimi, An approximation to the solution of hyperbolic equations by Adomian decomposition method and comparison with characteristic method, Appl. Math. Comput. 163, 633-638. 2005. The exact solution of the problem is [4] G. Evens, J. Blackledge, and P. Yardley, Numerical Method for Partial Differential Equations, Springer. 2000. The equation may be written in operator form as [5] O. Taiwo, S. Odetunde, and .Y Adekunle, Numerical Approximation of One Dimensional Biharmonic Equations by an LU= Iterative Decomposition Method, J. Math. Sc. 20 (1) 37-44. 2008 Applying the inverse operator ,we have From (14) and (15) 309 http://sites.google.com/site/ijcsis/ ISSN 1947-5500