Numerical Solution of One Dimensional Nonlinear Longitudinal Oscillations in a Class of Generalized Functions

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1 Proc. of te 8t WSEAS Int. Conf. on Matematical Metods and Computational Tecniques in Electrical Engineering, Bucarest, October 16-17, Numerical Solution of One Dimensional Nonlinear Longitudinal Oscillations in a Class of Generalized Functions Rafik Rasulov Te Seki Filial of te Istitute of Teacers, Seki, Az5, Republic of Azerbaycan, Haluk Kul Beykent University, Department of Computer Engineering, Istanbul 34900, Turkey July 12, 2006 Abstract In tis paper a new numerical metod for solution of initial value problem for te second order nonlinear wave equation wic describes te standing vibrations of a finite continuous and nonlinear string. Special auxiliary problem as been suggested and using te advantages of te suggested auxiliary problem, a finite difference sceme is constructed for te numerical solution of te main problem. K eywords: Longitudinal Oscillation. Numerical modeling. In a class of discontinuous function. Nonlinear wave propagation. 1 Introduction. Te problems associated wit wave propagation and standing oscillation in nonlinear fields ave been important for pysics and engineering, and 1

2 Proc. of te 8t WSEAS Int. Conf. on Matematical Metods and Computational Tecniques in Electrical Engineering, Bucarest, October 16-17, today it is still an actual topic. One of te nonlinear wave equations as been studied by Zabusky N.J in[2] for te first time and te exact solution as been acieved. In [1],[9] special metods for finding te exact solution of some nonlinear wave equations ave been developed. However, it is not always possible to obtain te exact solutions of tese problems. It is well known tat, numerical solutions play an important role, were obtaining exact solutions falls sort. For tis reason in tis paper an original numerical metod for finding te numerical solution of te problem for te second order nonlinear partial wave equation given below 2 u t 2 = x K ( ) u, (1.1) x u(x, 0) = u 0 (x), (1.2) u(x, 0) = u 1 (x) (1.3) t is suggested. Here te functions u 0 (x) and u 1 (x) are given functions (may be discontinuous as well) and te function K(s) satisfy te following conditions: (i) K(s) is continuous and differentiable, (ii) K(0) 0 for s 0, (iii) K (s) canges of sign. It is obvious tat te equation (1.1) degenerates twice, in deed since u K( ) = x x K ( u) 2 u and K (s) = 0, K(0) = 0 te equation (1.1) degenerates to first order differential equation. If K(s) = s te equation(1.1) x x 2 describes te little vibration of te string, but if K(s) = s3 + s te equation 3 (1.1) describes a larger vibration of te string. In addition, since te equation (1.1) is nonlinear in general, it is impossible to find te exact solution of te Caucy problem of te equation (1.1). As it is seen from te pysical structure of te problem K( u) u, it must x x be a continuous function, owever te second order derivatives of te solution may not exist. Because te classical solution of te problem (1.1)-(1.3) does not exist, te weak solution of tis problem is defined as Definition 1. A function u(x, t) wic satisfy te conditions (1.2),(1.3) is called a weak solution of te problem (1.1)-(1.3), if te following integral relation { ( ) } u f u f D t t K dxdt x x 2

3 Proc. of te 8t WSEAS Int. Conf. on Matematical Metods and Computational Tecniques in Electrical Engineering, Bucarest, October 16-17, f(0, t) + u 0 (x) dx t u 1 (x)f(0, t)dx = 0 (1.4) olds for every test function f(x, t) W 2 1,1( o R 2 l ) and f(x, T ) = 0, ere, W 2 1,1( o R 2 l ) is a Sobolev s space on o R 2 l. In literature tere are many familiar omogenous finite differences scemes in wic te differentiability property of te solution is neglected [3],[4],[7],[8]. In order to find te numerical solution of te problem (1.1)-(1.3) in accordance to [5],[6] we introduce te following auxiliary problem. 2 Auxiliary problem. Te problem 2 v t 2 = K ( 2 v x 2 ) 2 v x 2, (2.1) v(x, 0) = v 0 (x), (2.2) v(x, 0) = v 1 (x). (2.3) t is called an auxiliary problem. Here, te functions v 0 (x) and v 1 (x) are any continuous solutions of te equations dv 0 (x) dx = u 0 (x), dv 1 (x) dx = u 1 (x). (2.4) Teorem 1. If v ( x, t) is te classical solution of te auxiliary problem (2.1)-(2.3), ten te function defined by v(x, t) u(x, t) = (2.5) x is weak solution of te main problem (1.1)-(1.3). Wit reference to (2.5) te auxiliary equation can be written as 2 v t 2 = K ( ) u u x x. (2.6) 3

4 Proc. of te 8t WSEAS Int. Conf. on Matematical Metods and Computational Tecniques in Electrical Engineering, Bucarest, October 16-17, Te auxiliary problem as te following advantages: Te second order derivatives of u(x, t)wit respect to x does not ave to exist and it is not necessary to use tis second order derivatives in te algoritm written to find te solution eiter. Te v(x, t) is absolutely continuous. Bot advantages of te auxiliary problem enables to construct te economical numerical algoritm for te problem (1.1)-(1.3). 3 Numerical algoritm in a class of discontinues functions. In order to develop a numerical algoritm for te problem (2.1)-(2.3), at first, we cover te region Rl 2 by te grid as ω,τ = {(x i, t k ) x i = i, t k = kτ, i = 0, ±1, ±2,..., k = 0, 1, 2,...; > 0, τ > 0}. Here, and τ are te steps of te grid ω,τ wit respect to x and t variables, respectively. We approximate te problem (2.1),(2.3) to finite differences as follows V i,k+1 2V i,k + V i,k 1 τ 2 ( ) ( ) Ui,k U i 1,k Ui,k U i 1,k = K (3.1) V i,0 = v 0 (x i ), (3.2) V i,1 V i,0 = u 1 (x i ) (3.3) τ Te following teorem olds; Teorem 2. If V i,k+1 is te numerical solutions of te auxiliary problem (2.1)-(2.3), ten te relations defined by V i,k+1 V i 1,k+1 is numerical solution of te problem = U i,k+1 (3.4) 4

5 Proc. of te 8t WSEAS Int. Conf. on Matematical Metods and Computational Tecniques in Electrical Engineering, Bucarest, October 16-17, U i,k+1 2U i,k + U i,k 1 τ 2 = 1 [ K ( ) ( )] Ui+1,k U i,k Ui,k U i 1,k K (3.5) U i,0 = u 0 (x i ), (3.6) U i,1 U i,0 = u 1 (x i ). (3.7) τ Te difference sceme (3.1) is te second order wit respect to τ, owever, te order of it can be made iger by applying, for example, te Runge-Kutta metod. As it can be seen from (3.1)-(3.3), te suggested algoritms are very effective and economic from a computational point of view. 4 Conclusion In tis study an original metod for obtaining te numerical solution of te Caucy problem for te second order nonlinear partial equation is suggested. We ave introduced an auxiliary problem wose differentiability property of solution is one order iger tan te differentiability property of te solution of te main problem. By tis auxiliary problem an economical and effective differences sceme as been developed. References [1] Ames, W. F. Nonlinear Partial Differential Equations in Engineering. Academic Press, New York, London, [2] Zabusky, N.,J. Exact Solution for te Vibrations of a Nonlinear Continuous Model String. Journal of Matematical Pysics, vol. 3, No 5, pp , [3] Godunov, S.K. Equations of Matematical Pysics. Nauka, Moskow, [4] Godunov, S.K., Ryabenkii, V.S. Finite Difference Scemes. Moskow, Nauka,

6 Proc. of te 8t WSEAS Int. Conf. on Matematical Metods and Computational Tecniques in Electrical Engineering, Bucarest, October 16-17, [5] Rasulov, M. A. On a Metod of Solving te Caucy Problem for a First Order Nonlinear Equation of Hyperbolic Type wit a Smoot Initial Condition. Soviet Mat. Dok. 43, No.1, [6] Rasulov, M. A. Finite Difference Sceme for Solving of Some Nonlinear Problems of Matematical Pysics in a Class of Discontinuous Functions. Baku, 1996, (in Russian). [7] Ricmyer, R. D., Morton, K. W. Difference Metods for Initial Value Problems. New York, Wiley, Int., [8] Smoller, J. A. Sock Wave and Reaction Diffusion Equations. Springer- Verlag, New York Inc., [9] Witam, G. B. Linear and Nonlinear Waves. Wiley Int., New York, pp ,