AN APPROXIMATE SOLUTION FOR DIFFERENT TYPES OF WAVE PROBLEMS

Year 2009, Volume: 4 Issue: 2, 189 - 198, 01.12.2009

Morteza Mıansarı Amin Bararı Hessam Mırgolbabaeı Mohammad Vahdatırad

Abstract

In this article, He’s variational iteration method (VIM) is implemented to solve the non-homogeneous dissipative wave, Helmholtz and some nonlinear fifth-order Korteweg-de Vries (FKdV) partial differential equations with specified initial conditions. The initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary or initial conditions after few iterations. Comparison of the results with those obtained by exact solution and Adomian’s decomposition method reveals that VIM is very effective, convenient and quite accurate to both linear and nonlinear problems. It is predicted that VIM can be widely applied in engineering

Keywords

Variational iteration method, Helmholtz equation, FKdV equation, nonlinear partial differential equations

DALGA PROBLEMLERİNİN FARKLI TİPLERİ İÇİN BİR YAKLAŞIK ÇÖZÜM

Abstract

Bu makalede, He’nin varyasyonel iterasyon yöntemi (VIM), belli başlangıç koşulları ile homojen olmayan dissipative dalga, Helmholtz ve bazı lineer olmayan beşinci mertebeden Korteweg-de Vries (FKdV) kısmi diferansiyel denklemlerini çözmek için uygulanmıştır. Başlangıç yaklaşımları, birkaç iterasyon sonra başlangıç ve sınır koşullarının uygulanmasıyla belirlenebilen mümkün bilinmeyen sabitler ile keyfi olarak seçilebilir. Analitik çözüm ve Adomian'ın ayrıştırma yöntemi ile elde edilen sonuçların karşılaştırılması, VIM’in çok etkili, uygun ve hem lineer hem de lineer olmayan problemler için oldukça hatasız olduğunu ortaya koymaktadır. VIM’in mühendislikte yaygın olarak uygulanabildiği tahmin edilmektedir.

Keywords

Varyasyonel iterasyon yöntemi, Helmholtz denklemi, FKdV denklemi, lineer olmayan kısmi diferansiyel denklemler

References

• ABDOU MA, SOLIMAN AA, 2005. Variational iteration method for solving Burger’s and Coupled Burger’s equations, Journal of Computational and Applied Mathematics, 181 (2), 245–251.
• AKYLAS TR, YANG TS, 1995. On short-scale oscillatory tails of long-wave disturbances, Studies in Applied Mathematics, 94, 1-20.
• BEALE JT, 1991. Exact solitary waves with capillary ripples at infinity, Communications Pure Applied Mathematics, 44, 211-247.
• BODY JP, 1991. Weak Non-Local Solitons for Capillary-Gravity Waves: Fifth-Order Korteweg-de Vries Waves, Physica D, 48, 129-146.
• BURDEN RL, FAIRES JD, 1993. Numerical Analysis, Boston: PWS Publishing Company, USA, pp. 729.
• DRAZIN PG, JOHNSON RS, 1989. Solitons: An Introduction, Cambridge University Press, UK, pp. 226.
• EL-SAYED SM, KAYA D, 2004. Comparing numerical methods for Helmholtz equation model problem, Applied Mathematics and Computation, 150, 763-773.
• GANJI DD, RAJABI A, 2006. Assessment of homotopy-perturbation and perturbation methods in heat radiation equations, International Communications in Heat and Mass Transfer, 33 (3), 391- 400.
• HE JH, 1998a. Approximate analytical solution for seepage flow with fractional derivatives in porous media, Computer Methods in Applied Mechanics and Engineering. 167, 57–68.
• HE JH, 1998b. Approximate solution for nonlinear differential equations with convolution product nonlinearitiesü, Computer Methods in Applied Mechanics and Engineering, 167, 69–73.
• HE JH, 1999a. Variational iteration method: A kind of nonlinear analytical technique: some examples, International Journal of Nonlinear Mechanics, 344, 699–708.
• HE JH, 1999b. Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering, 178, 257–262.
• HE JH, 2000. A coupling method of a homotopy technique and a perturbation technique for non-linear problems, International Journal of Nonlinear Mechanics, 35, 37– 43.
• HE JH, 2004. Comparison of homotopy perturbation method and homotopy analysis method, Applied Mathematics and Computation, 156, 527-539.
• HE JH, 2005. Homotopy perturbation method for bifurcation of nonlinear problems, International Journal of Nonlinear Sciences and Numerical Simulation, 6 (2), 207–208.
• HE JH, 2006. Some asymptotic methods for strongly nonlinear equations, International Journal of Modern Physics B, 20, 1141-1199.
• HE JH, Wu XH, 2007a. Variational iteration method: New development and applications, Computers & Mathematics with Applications, 54, 881-894.
• HE JH, 2007b. Variational iteration method - Some recent results and new interpretations, Journal of Computational and Applied Mathematics, 207, 3-17.
• HE JH, 2008. An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering, International Journal of Modern Physics B, 22, 3487-3578.
• HUNTER JK, SCHEURLE J, 1988. Existence of petrubed solitay wave solutions to a model equation for water waves, Physic D, 32, 253-568.
• INC M, CHERRUAULT Y, ABBAOUI K, 2004. A computational approach to the wave equations, Kybernetes, 33, 80-97.
• KAYA D, 2003. An explicit and numerical solutions of some fifth-order kdv equation by decomposition method, Applied Mathematics and Computation, 144, 353- 363.
• MIANSARI Mo, GANJI DD, MIANSARI ME, 2008. Application of He’s variational iteration method to nonlinear heat transfer equations, Physics Letters A, 372, 779–785.
• RAFEI M, GANJI DD, 2006. Explicit solutions of Helmholtz equation and fifth-order Kdv equation using homotopy perturbation method, International Journal of Nonlinear Sciences and Numerical Simulation, 7 (3), 321-328.
• WAZWAZ AM, 2002. Partial Differential Equations: Methods and Applications, Balkema, Rotterdam, pp. 76.