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Numerical Solution of Deformation Equations in Homotopy Analysis 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: ňƳËʚÎʛƷ Ɣ ̉ , Î ∈ ʞ·, ¸ʟ , ʚ ̊ʛ ̋ where is a nonlinear Ëdifferentialʚ·ʛ Ɣ ë , Ë operatorʚ¸ʛ Ɣof ì , Ësecond order, ∈ ∁ ʞ·,is ¸ʟindependent variableň and is an unknown function. For this problem the homotopy Î equation can be written as follows : Ë where ʚ̊ Ǝ ÇʛņƳĈʚÎ, Çʛ ƎË is̉ ʚanÎʛ embeddingƷ Ɣ ÇňƳĈ ʚparameter,Î, ÇʛƷ , is an initialʚ̋ ʛguess of isͬ linear ∈ ʞ·, differential ¸ʟ, ͥ ∈ ʞ̉, operator ̊ʟ of second order, and Ë̉ is the homotopy series Ë, Ė ĈʚÎ, Çʛ ∞ à ĈʚÎ, Çʛ ƔË̉ʚÎʛ ƍ ȕ ËÃʚÎʛÇ ÃͰ̊ Numerical solution of deformation equations 359 That is assumed to be convergent on [0,1]. It easily deduces that à (3) ̊ Ą à When , itË holdsÃʚÎʛ Ɣ Ã! ĄÇ ĈʚÎ, Çʛ|ÇͰ̉ . ͥ Ɣ 1 ∞ , (4) ĈʚÎ, ̊ʛ ƔË̉ʚÎʛ ƍ ∑ÃͰ̊ ËÃʚÎʛ Ɣ ËʚÎʛ that is a solution of (2), and consequently the solution of (1), see [1]. The initial guess satisfies two boundary conditions of (1), see [1,2,4]. Referring the homotopy literature [1-4], one can find the deformation equations of order m is ͩͤʚͬʛ found as follows: m ≥1, (5) where ņʞËÃʚÎʛ Ǝ āÃËÃͯ̊ʚÎʛʟ Ɣ ®Ãͯ̊, Á ̊ Ą ®Á Ɣ Á ňƳĈʚÎ, ÇʛƷ|ÇͰ̉ , Á Ɣ ̊,̋,…,ÃƎ̊, and Á! ĄÇ , ̉ à ƙ ̊ āà Ɣ ʤ Then, the equation (5) is a second order̊ à linear ODEs Ƙ 1 with boundary conditions, By accepting isË aà solutionʚ·ʛ ƔË ofà ʚ ¸ʛ Ɣ ̉ , à ƚ ̊ Ë̉ʚÎʛ ņƳËʚÎʛƷ Ɣ ̉, Ëʚ·ʛ Ɣ ë , , Ëʚ¸ʛ Ɣ ì, The homotopy series solution (4) can be determined step by step by solving following linear problem : ņʞËÃʚÎʛ Ǝ āÃËÃͯ̊ʚÎʛʟ Ɣ ® Ãͯ̊ , à ƚ ̊ (6) ËÃʚ·ʛ ƔËÃʚ¸ʛ Ɣ ̉ In fact, one can only computes partial sum of series solution. However, in general these equations are solved analytically or with pade’ homotopy method [1]. In these methods the control parameters are used to guarantee convergence of homotopy series, for more information, the reader is refered the appropriate literature. This partial sum can be determined numerically by solving (6), with a proper numerical method. Here the finite difference method and the spectral method are applied to solve deformation equation. 360 J. Izadian, M. MohammadzadeAttar and M. Jalili 3 Numerical solution of deformation equation 3.1 Finite difference method Consider a partition of [a,b] , as follows: ·ƔÎ̉ ƗÎ̊ ƗÎ̋ Ɨ ⋯ Ɨ ÎÄ Ɣ ¸ οͮ̊ Ǝο Ɣ ¾ , ¿ Ɣ ̉,̊,̋,...,ÄƎ̊, and take ̋ (for simplicity), by omitting the truncation error, equation (6) º ̋ yields : ņ Ɣ ºÎ ʚÃʛ ʚÃʛ ʚÃʛ ʚÃ̊ʛ ʚÃ̊ʛ ʚÃ̊ʛ (7) Ì¿~̊ͮÌ¿̊ͯ̋Ì¿ Ì¿~̊ ͮÌ¿̊ ͯ̋Ì¿ ̋ ̋ where ¾ Ǝ āà ¾ Ɣ ®Ãͯ̊ʚοʛ, ¿Ɣ̊,̋,…,ÄƎ̊, , ʚÃʛ Ì¿ ≅ ËÃʚοʛ , ¿ Ɣ ̉,̊,̋,…,Ä , à ƚ ̉ Now by applying boundary conditions, we have a linear system of equations that can be solved by one of suited methods. Then we have an approximation of on [a,b] . By repeating M-times this procedure, finally we obtain an approximation of partial sum of series solution: ËÃʚÎʛ © ËȭʚÎʛ ƔËȮ̉ʚÎʛ ƍ ȕ ËȮ͉ʚÎʛ ≅ ËʚÎʛ, where , .¿Ͱ̊ ËȮÃʚÎʛ ≅ ËÃʚÎʛ ̉ ƙ à ƙ © 3.2 Spectral method Another method that can be used for solving deformation equations, is spectral method with Chebyshev points. Here, we consider an approximation of , that is solution of deformation equation (5) as given: ËÃʚÎʛ (8 ) Ä where is Lagrange Ëȭpolynomials,ÃʚÎʛ Ɣ ∑¿Ͱ̉ Âand¿ʚÎ ʛËÃʚοʛ, are Chebyshev points [8].¿ʚ Ifʛ we denote , for i=0,1,2,…,n$ and , then (8)  ΠʚÃʛ ͬ , ͝ Ɣ 0,1,2,…,͢ can be written as: Ì¿ ƔËÃʚοʛ à ƚ ̉ Ä ʚÃʛ ËȭÃʚÎʛ Ɣ ȕ ¿ʚÎʛÌ¿ , ʚ̒ʛ By twice differentiation the equation¿Ͱ̉ (9) with respect to x, we have ′ Ä ′ ʚÃʛ ËȲÃʚÎʛ Ɣ ȕ  ¿ʚÎʛÌ¿ , ¿Ͱ̉ Numerical solution of deformation equations 361 ′′ ′′ . Ä =ʚÃʛ Or by using the spectral differentiationËȲ ÃʚÎʛ Ɣ ∑ matrix¿Ͱ̉  ¿ʚ ÎDʛÌ¿( d ij ) n+1, n + 1 see[7,8], we have ′ Ä ʚÃʛ Ëȱ ÃƳÎÀƷ Ɣ ȕ º¿À Ì¿ , ¿ Ɣ ̉,̊,̋,…,ÄƎ̊, ¿Ͱ̉ ′′ Ä Ä ʚÃʛ ËȲ ÃʚÎʛ Ɣ ȕ ȕ º¿À ºÁ¿ Ì¿ . By substituting these derivatives in deformation¿Ͱ̉ ÁͰ̉ equation, a linear system of (n+1) equations with (n-1) unknowns , is obtained: ʚ̭ʛ Ì̩ ,¿Ɣ̊,̋,…,Ä (10) Ä Ä ʚÃʛ Ä Ä ʚÃͯ̊ʛ ∑¿Ͱ̊ ∑ÁͰ̊ º¿À ºÁ¿ ÌÁ Ǝ āà ∑¿Ͱ̊ ∑ÃͰ̊ º¿À ºÁ¿ ÌÁ Ɣ ®Ãͯ̊ƳÎÀƷ, ÀƔ̊,̋,…,ÄƎ̊ The initial solution is considered as following : Ë̉ʚÎʛ ì Ǝ ë ¸ë Ǝ ìë Ë̉ʚÎʛ Ɣ Î ƍ , Î ∈ ʞ·, ¸ʟ, that can be easily transformed¸ Ǝ · to [-1,1]¸ Ǝ· by a linear transformation to use Chebyshev points. For , the boundary conditions are used. Then (7) is reduced to a linear system with n-1 unknowns that can be easily ƚ ̊ ËÃʚÎ̉ʛ ƔËÃʚÎÄʛ Ɣ ̉ solved. One knows that the coefficient matrix of this system is not sparse, however regarding the spectral precision of method this matrix can be of an acceptable order [8]. 4 Finding optimal control parameters To have a solution series that is convergent for q=1, sometimes we need to apply convergence control parameters. In such cases using the so called h-curve can be useful for determining the convergence control parameter [4]. However there are the methods that allow us to find the control parameter by a minimization process. Here we apply two methods to compute computing optimal control parameters due to Marinca and Niu-Wang [6]. For this purpose one chooses the following zero-th order deformation equation (or homotopy equation): where ʚ̊ Ǝ ÇʛņƳĈʚÎ, Çʛ ƎË̉ʚÎʛƷ Ɣ łʚÇʛňƳĈʚÎ, ÇʛƷ, ∞ . Á łʚÇʛ Ɣ ∑ÃͰ̊ ¾ÁÇ 362 J. Izadian, M. MohammadzadeAttar and M. Jalili Then the homotopy solution is the function of , we can determine such that Ĉ can be approximated¾Á ,by Á ∈a ŋpartial sum of small ¾orderÁ , Á ( ∈ ŋ ĈʚÎ, ¾, Ç .ʛ For determining͜ Ɣ ʚͥ͜,͜ ͦmore, … , ͜ (precise, … ʛʛ approximate solution one note that for the exact solution u(x) ,the following equations is satisfied: Then to find of a goodňƳË approximateʚοʛƷƔ̉ , solution ¿Ɣ̉,̊,̋,…,Ä, , each must become very small. Thus we define as [6] ,the following∗ function ∗ Ë ʚÎʛ |ňʚË ʚοʛʛ| ̋ (11) Ä ̊ ̋ © ¿ Where ¢isʚ ¾M-th, ¾ partial, … , ¾ sumʛ Ɣ ∑of¿Ͱ̉ seriesʠňƳË ȭsolution.ʚÎ ʛƷʡ ,In method of Marinca, one minimizeËȭʚ Î(11)ʛ to find an optimal approximation of at end of M-th step of i=1,2,…,M computing in method of Niu-wangËȭʚÎʛ in every step one minimizes a function of one variable that is defined by Ë¿ʚÎʛ, ̋ Ä ¿ Ô¿ʚ¾¿ʛ Ɣ ȕ ƶň ʬȕ ËÀʚÎÁʛʭƺ , ¿ Ɣ ̊,̋,…,ª , ÁͰ̊ ÀͰ̉ where have been determined, in previous step.
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