Numerical Solution of Bratu-type Equations by the Variational Iteration Method

Yıl 2010, Cilt: 39 Sayı: 1, 23 - 29, 01.01.2010

B. Batiha

Öz

Numerical Solution of Bratu-type Equations by the Variational Iteration Method

Öz

In this paper, the variational iteration method (VIM) is applied to obtain approximate analytical solution of Bratu-type equations without any discretization. Comparisons with the exact solutions reveal that VIM is very effective and convenient.

Anahtar Kelimeler

Variational iteration method, Bratu’s problem, Boundary value problems, Initial value problems, 2000 AMS Classification: 65 L 05

Kaynakça

• Abbasbandy, S. A new application of He’s variational iteration method for quadratic Riccati differential equation by using Adomian’s polynomials, J. Comput. Appl. Math. doi:10.1016/j.cam.2006.07.012.
• Abulwafa, E. M., Abdou, M. A. and Mahmoud, A. A. The solution of nonlinear coagulation problem with mass loss, Chaos Solitons Fractals 29 (2), 313–330, 2006.
• Aregbesola, Y. Numerical solution of Bratu problem using the method of weighted residual, Electronic Journal of Southern African Mathematical Sciences Association 3 (01), 1–7, 2003. [4] Ascher, U. M., Matheij, R. and Russell, R. D. Numerical solution of boundary value problems for ordinary differential equations(SIAM, Philadelphia, PA, 1995).
• Batiha, B., Noorani, M. S. M. and Hashim, I. Application of variational iteration method to the generalized Burgers-Huxley equation, Chaos Solitons and Fractals 36 (3), 660-ˆu663, 2008.
• Batiha, B., Noorani, M. S. M. and Hashim, I. Numerical simulation of the generalized Huxley equation by He’s variational iteration method, Appl. Math. Comput. 186, 1322–1325, 2007. [7] Bildik, N. and Konuralp, A. The use of variational iteration method, differential transform method and Adomian decomposition method for solving different types of nonlinear partial differential equations, Int. J. Nonlinear. Sci. Numer. Simul. 7, 65–70, 2006.
• Boyd, J. P. An analytical and numerical study of the two-dimensional Bratu equation, J.Scien. Computing 1 (2), 183–206, 1986.
• Boyd, J. P. Chebyshev polynomial expansions for simultaneous approximation of two branches of a function with application to the one-dimensional Bratu equation, Appl. Math. Comput. 142, 189–200, 2003. [10] Chang, S. H. and Chang, transform nonlinear Appl. differential
• doi:10.1016/j.amc.2007.05.026. functions, Math. Comput.
• , [11] Dehghan, M. and Tatari, M. The use of He’s variational iteration method for solving a Fokker-Planck equation, Phys. Scr. 74, 310–316, 2006.
• El-Zoheiry, E., Abassy, T. A. and El-Tawil, M. A. Toward a modified variational iteration method, J. Comput. Appl. Math. doi:10.1016/j.cam.2006.07.019.
• He, J. H. A new approach to nonlinear partial differential equations, Commun. Nonlin. Sci. Numer. Simul. 2, 230-ˆu235, 1997.
• He, J. H. A simple perturbation approach to Blasius equation, Appl. Math. Comput. 140, 217–222, 2003.
• He, J. H. Approximate analytical solution of Blasius’ equation, Commun. Nonlin. Sci. Nu- mer. Simul. 3, 260–263, 1998.
• He, J. H. Non-perturbative methods for strongly nonlinear problems (Berlin: dissertation.de- Verlag im Internet GmbH, 2006).
• He, J. H. Some asymptotic methods for strongly nonlinear equations, Int. J. Modern Phys. B 20, 1141–1199, 2006.
• He, J. H. Variational iteration method–a kind of non-linear analytical technique: some ex- amples, Int. J. Non-Linear Mech. 34, 699–708, 1999.
• He, J. H. Variational iteration method for autonomous ordinary differential systems, Appl. Math. Comput. 114, 115ˆu-123, 2000.
• Inokuti, M., Sekine, H. and Mura, T. General use of the Lagrange multiplier in nonlinear mathematical physics. In: Nemat-Nassed S, ed. Variational method in the mechanics of solids(Pergamon Press, 1978).
• Jeffrey, A. and Mohamad, M. N. B. Exact solutions to the KdV-Burgers equation, Wave Motion 14, 369–375, 1991.
• Khuri, S. A. A new approach to Bratu’s problem, Appl. Math. Comput. 147, 131–136, 2004. [23] Odibat, Z. M. and Momani, S. Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. Nonlinear Sci. Numer. Simul. 7, 27–34, 2006.
• Soliman, A. A. A numerical simulation and explicit solutions of KdV-Burgers’ and Lax’s seventh-order KdV equations, Chaos Solitons Fractals 29 (2), 294–302, 2006.
• Syam, M. I. and Hamdan, A. An efficient method for solving Bratu equations, Appl. Math. Comput. 176, 704–713, 2006.
• Wadati, M. The exact solution of the modified Korteweg-de Vries equation, J. Phys. Soc. Jpn. 32, 1681–1687, 1972.
• Wang, X. Y. Exact and explicit solitary wave solutions for the generalized Fisher equation, Phys Lett A 131, 277¡-279, 1988.
• Wazwaz, A. M. Adomian decomposition method for a reliable treatment of the Bratu-type equations, Applied Mathematics and Computation 166, 652–663, 2005.
• Wazwaz, A. M. A comparison between the variational iteration method and Adomian de- composition method, J. Comput. Appl. Math. doi:10.1016/j.cam.2006.07.018.