Applied Mathematical Sciences, Vol. 6, 2012, no. 8, 357 – 367
Numerical Solution of Deformation Equations
in Homotopy Analysis Method
J. Izadian and M. MohammadzadeAttar
Department of Mathematics, Faculty of Sciences, Mashhad Branch, Islamic Azad University, Mashhad, Iran
M. Jalili
Department of Mathematics, Neyshabur Branch, Islamic Azad University, Neyshabur, Iran [email protected]
Abstract
In this paper, we consider the homtopy analysis method (HAM) for solving nonlinear ordinary differential equation with boundary conditions. The partial sum of solution series is determined using finite difference method and spectral method. These methods are used for solving deformation equation. For improving the rate of convergence, we apply the methods of Marinca and Niu-Wang. The results are compared with finite difference method and spectral method. Unlike HAM, these methods don’t need convergence control parameter. Numerical experiments show efficiency and performance of proposed methods.
Keywords: Homotopy analysis method, Deformation equation, Spectral method, Finite difference method
1 Introduction
Ordinary differential equations with boundary conditions (BVP) are very important in various domains of sciences and engineering. In most cases, it is very complicated to achieve analytic solutions of these equations. Numerical finite difference methods and spectral methods can be used for solving certain types of these problems with applying nonlinear equation solvers as Newton's methods 358 J. Izadian, M. MohammadzadeAttar and M. Jalili
[7,8], but these techniques are valid only for weakly nonlinear problems. Recently, homotopy analysis method (HAM) has been successfully applied to various types of these problems. In 2003, Liao published the book [1] in which he summarized the basic ideas of the homotopy analysis method and gave the details of his approach both in the theory and on a large number of practical examples and in a series of papers [2-4,9], he developed HAM. Homotopy series solution is depended to convergence control parameter. For instance Liao has used the functional series that their coefficients are the control parameters. These parameters can be used for accelerating the convergence of solutions series, or finding a homotopy series that is convergent for certain value of homotopy parameters. Some authors like Marinca et al. [5], Yabushita et al. [10], and Niu et al. [6] have considered the optimization convergence control parameters that can be obtained by minimizing the square residual error. In this paper, the known methods like finite difference method and spectral method are applied to solving deformation equation, next, Marinca, and Niu-Wang methods are used to minimize the task of determining the solution, and these two methods are compared by presenting some numerical experiments. This paper is organized as follows. In section 2 the basic idea of HAM is presented. In the section 3 the application of finite difference and Chebyshev collocation method for solving deformation equation is described. In section 4 Marinca and Niu-Wang methods are combined with numerical methods of section 3, next in the section 5 the numerical results are presented. The results of Marinca and Niu- Wang methods are compared for two boundary value problems. Finally, we discuss and analyse the results.
2 The basic idea of HAM for solving BVP
We consider the following boundary value problems: