Solving Singular BVPs Ordinary Differential Equations by Modified Homotopy Perturbation Method

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1 Journal of mathematics and computer Science 7 (23) Solving Singular BVPs Ordinary Differential Equations by Modified Homotopy Perturbation Method Article history: Received March 23 Accepted Apri 23 Available online April 23 Mostafa Mahmoudi, 2 Mohammad V. Kazemi 3 Higher Institute of Pouyandegandanesh, Chalous, Iran, math.mahmoudi@yahoo.com 2 payamnoor university,nowshahr,iran 3 Higher Institute of Pouyandegandanesh, Chalous, Iran, mohammad_v_kazemi@yahoo.com Abstract In this paper, we use modified homotopy perturbation method to solving singular boundary value problems (BVP) of higher order ordinary differential equations. The proposed method can be applied to linear and nonlinear problems. The results prove that the modified HPM is a powerful tool for the solution of singular BVPs. Keywords: Singular boundary value problems, homotopy perturbation method, ordinary differential equations.. Introduction The homotopy perturbation method (HPM) is a new and ingenious method for solving linear and nonlinear differential and integral equations of various kinds. homotopy perturbation method is an analytical method which can be applied to the solution of linear, nonlinear deterministic and stochastic operator equations. HPM deforms a difficult problem into an infinite set of problems which are easier to solve without any need to transform nonlinear terms. The applications of HPM in nonlinear problems have been demonstrated by many researchers. In recent years, much attention has been devoted to the application of the HPM, to the solutions of various scientific models. The purpose of this paper is to 38

2 Mostafa Mahmoudi, Mohammad V. Kazemi/ J. Math. Computer Sci. 7 (23) introduce a new reliable modification of HPM. For this reason, a new differential operator is defined which can be used for higher-order singular boundary value problems. Consider the singular boundary value problem of n+ order ordinary differential equation in the form y (n+) + m y(n) + Ny = g() () y() = a, y () = a,, y n () = a n, y(b) = c, Where N is nonlinear differential operator of order less than n, g() is given function anda, a... a n, b, c are given constants. We propose the new differential operator, as below dn L(. ) = d n +n m d d m n (. ) (2) Where m n-, n so, the problem () can be written as Ly = g() Ny (3) The inverse operator L - is therefore consider n+ fold integral operator, as below b L (. ) = n m m n. (. )d. d (4) According to HPM we can determine the component y n (), and the series solution of y() can be obtained. y() = y () + py () + p 2 y 2 () + Putting P= the approimate solution therefore can be readily obtained. For numerical purposes, the n-term approimate Can be used approimate the eact solution. y = lim p y() = y () + y () + y 2 () + n Ψ n = y n () n= 39

3 Mostafa Mahmoudi, Mohammad V. Kazemi/ J. Math. Computer Sci. 7 (23) Numerical eamples In this section, few eamples are presented to understand better the confusion HPM Eample. Consider the linear BVP y + b y = b cos (2 b) b sin y() =, y() = cos put L(. ) = d d 2 b d d +b (. ) We construct the following homotopy L (. ) = 2 b (. )dd y + b y + p( b cos + (2 b) b sin ) = Equating the terms with the identical powers of P, p y + b y =, y ()= y()+(y() y()) b = cos b p : y + b y + b cos + (2-b) b sin = Ly = b cos (2 b) b sin y ()=L ( b cos (2 b) b sin ) y () = b cos b cos( ) p 2 : y + b y = Ly 2 = y 2 () = n 3 y 3 () = y() = y () + y () + y 2 () + = b cos 4

4 Mostafa Mahmoudi, Mohammad V. Kazemi/ J. Math. Computer Sci. 7 (23) This is the eact solution. Eample 2. Consider the nonlinear BVP y 2 y y y 2 = g() We put y() = y () =, y() = e g()= 7 2 e +6e 6e 6 e 2 We construct the following homotopy d2 L(. ) = d 2 5 d d 4 (.) L (. )= 4 5 (. ) ddd y 2 y +P ( y y2 g())= we use Taylor series of g() with order 4 Equating the terms with the identical powers of P g() P )y 2 y = y () = y() + (y() y()) n m y () = e 4 P )y 2 y y y 2 g() = y 2 y e4 e = Ly = e 4 + e 2 8 y () = L ( e 4 + e 2 8 ) 4

5 Mostafa Mahmoudi, Mohammad V. Kazemi/ J. Math. Computer Sci. 7 (23) y () = 4 5 ( e 4 + e 2 8 ) y () = y () + y () = P 2 ) y 2 2 y 2 y 2y y = y 2 2 y 2 = ( y 2 () = L ( ) y 2 () = 4 5 ( )ddd y 2 () = The eact solution isy() = 3 e 4. Discussion and Conclusion y () + y () + y 2 () = In this paper, we use modified homotopy perturbation method to solving singular boundary value problems (BVP) of higher order ordinary differential equations. The MHPM proposed in this investigation is simple and effective for solving higher order of BVP and can provide an accuracy approimate solution or eact solution. Mathematical has been used for computations in this paper. References [] G. Adomain, Solving frontier problems of physics: the decomposition method, Kluwer. Boston, MA. MR (95e:26). (994) Zbl [2] G. Adomian, Nonlinear Stochastic Operator Equations, Academic press, San Diego, CA (986). 42

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