Application variational iteration method with studying the convergence to nonlinear PDEs

Abstract

This article is devoted to implement variational iteration method (VIM) for solving nonlinear partial differential equations. This method is based on the use of Lagrange multiplier for identification of optimal value of a parameter in a functional. This procedure is a powerful tool for solving large amount of problems. Using VIM, it is possible to find a sequence of functions which converges to the exact solution or an approximate solution of the problem. Our emphasis will be on study the convergence of the proposed method. Convergence analysis is reliable enough to estimate the maximum absolute error of the solution given by VIM.

Keywords

• Variational iteration method

References

1. Abassy, T. A., Magdy El-Tawil, A. and El Zoheiry, H., Solving nonlinear partial differential equations using the modified variational iteration Pad´e technique, Journal of Computational and Applied Mathematics, 207, 73-91, (2007).
2. Abdou, M. A. and Soliman, A. A., New applications of variational iteration method, Physica D. 211, 1-8, (2005).
3. Biazar, J. and Ghazvini, H., He’s variational iteration method for solving hyperbolic differential equations, International Journal of Nonlinear Sciences and Numerical Simulation, 8:(3), 311-314, (2007).
4. Eid, A. and Khader, M. M. Numerical studies using FDM for viscous dissipation and thermal radiation effects on the slip flow and heat transfer due to a stretching sheet embedded in a porous medium with variable thickness and variable thermal conductivity, New Trends in Mathematical Sciences, 4(1), 38-50, (2016).
5. Elsgolts, L. Differential Equations and the Calculus of Variations, translated from the Russian by G. Yankovsky, Mir, Moscow, (1977).
6. Evans, L. C., Partial Differential Equations, American Mathematical Society, Providence, RI, (1998).
7. Ganji, D. D., Afrouzi, G. A. and Talarposhti, R. A., Application of He’s variational iteration method for solving the reaction diffusion equation with ecological parameters, Computers and Mathematics with Applications, 54, 1010-1017, (2007).
8. He, J. H., Variational iteration method for autonomous ordinary differential systems, Applied Mathematics and Computation, 114:(2-3), 115-123, (2000).

9. He, J. H., Variation iteration method-a kind of non-linear analytical technique: some examples, International Journal of Non-Linear Mechanics, 34, 699-708, (1999).
10. He, J. H. and Xu-Hong Wu, Construction of solitary solution and compacton-like solution by variational iteration method, Chaos, Solitons and Fractals, 29, 108-113, (2006).
11. Khader, M. M. and Sweilam, N. H., Numerical and analytical study for integro-differential equations using spectral collocation method, New Trends in Mathematical Sciences, 3(4), 144-153, (2015).
12. Sadighi, A. and Ganji, D. D., Solution of the generalized nonlinear Boussinesq equation using homotopy perturbation and variational iteration methods, International Journal of Nonlinear Sciences and Numerical Simulation, 8:(3), 435-445, (2007).
13. Soliman, A. A., Numerical simulation of the generalized regularized long wave equation by He’s variational iteration method, Mathematics and Computers in Simulation, 70, 119-124, (2005).
14. Sweilam, N. H. and Khader, M. M., Variational iteration method for one dimensional nonlinear thermo-elasticity, Chaos, Solitons and Fractals, 32, 145-149, (2007).
15. Sweilam, N. H., Khader, M. M. and Al-Bar, R. F., Numerical studies for a multi-order fractional differential equation, Physics Letters A, 371, 26-33, (2007).
16. Sweilam, N. H. and Khader, M. M., On the convergence of VIM for nonlinear coupled system of partial differential equations, Int. J. of Computer Maths., 87(5), 1120-1130, (2010).
17. Tari, H., Ganji, D. D. and Rostamian, M., Approximate solutions of K(2,2), KdV and modified KdV equations by variational iteration method, homotopy perturbation method and homotopy analysis method, International Journal of Nonlinear Sciences and Numerical Simulation, 8:(2), 203-210, (2007).
18. Tatari, M. and Dehghan, M., On the convergence of He’s variational iteration method, Journal of Computational and Applied Mathematics, 207, 121-128, (2007).