Approximate Solution of Convection- Diffusion Equation by the Homotopy Perturbation Method

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1 Gen. Math. Notes, Vol. 1, No., December 1, pp ISSN ; Copyright ICSRS Pblication, 1 Available free online at Approximate Soltion of Convection- Diffsion Eqation by the Homotopy Pertrbation Method Mehdi Gholami Porshokohi 1,, Behzad Ghanbari 1, Mohammad Gholami and Majid Rashidi 1 Department of Mathematics, Faclty of science, Islamic Azad University, Takestan Branch, Iran m_gholami_p@yahoo.com, b.ghanbary@yahoo.com Department of Agricltral Machinery, Faclty of Agricltre, Islamic Azad University, Takestan Branch, Iran gholamihassan@yahoo.com, majidrashidi81@yahoo.com Abstract In recent years, a new difference scheme with high accracy has been applied for solving convection-diffsion eqation [1]. In this letter, we solve this eqation by homotopy pertrbation method (HPM) [-4]. To illstrate the ability and reliability of the method some examples are provided. The reslts reveal that the method is very effective and simple Keywords: Homotopy pertrbation method; Convection-diffsion eqation MSC No: 65Q, 65Q1. 1 Introdction Consider the convection-diffsion eqation [1] Corresponding athor

2 19 Approximate soltion of convection-diffsion eqation by the HPM ( 1) + ε = γ x 1, t. Sbject to the initial condition, ( ) ( ) conditions (, t), t. ( 1, t), t. x, = g x, x 1 and bondary = = where the parameter γ is the viscosity coefficient and ε is the phase speed and both are assmed to be positive. g is a given fnction of sfficient smoothness. To illstrate the basic concepts of homotopy pertrbation method, consider the following non-linear fnctional eqation: A( ) = f ( r), r Ω, ( ) With the following bondary conditions: ( ) B,, r. n = Γ Where A is a fnctional operator, B is a bondary operator, f ( r ) is a known analytic fnction, and Γ is the bondary of the domain Ω. Generally speaking, the operator A can be decomposed into two parts L and N, where L is a linear and N is a non-linear operator. Therefore Eq. ( ) can be rewritten as the following: L( ) + N ( ) f ( r) =. ( 3 ) We constrct a homotopy v r p [ ] (, ) : Ω,1 R, which satisfies: Or ( ) ( ) ( ) ( ) ( ) ( ) [ ] H v, p = 1 p L v L + p A v f r =, p,1, r Ω. ( ) ( ) ( ) ( ) ( ) ( ) [ ] H v, p = L v L + pl + p N v f r =, p,1, r Ω., Where is an initial approximation to the soltion of Eq.( ). In this method, homotopy pertrbation parameter p is sed to expand the soltion, as a power series, say; v = v + pv + p v +, 1

3 Mehdi Gholami Porshokohi et al. 11 Usally an approximation to the soltion, will be obtained by taking the limit, as p tends to 1, = lim v = v + v + v, p 1 1 For solving Eq. ( 1 ), by homotopy pertrbation method, we constrct the following homotopy: v v v v ( 1 p) p ε γ, t t + + = Or v v v + p ε γ +, = t t x x t 4 to be in the following form Sppose that the soltion of Eq. ( ) v = v + pv1 + p v + ( 5 ) Sbstitting Eq. ( 5 ) into Eq.( 4 ), and eqating the coefficients of the terms with the identical powers of p, ( 4 ) v p : =, t t 1 v1 v v p : + + ε γ, = v1 ( x,) = t v v1 v1 p : + ε γ =, v ( x,) = 3 v3 v v p : + ε γ =, v 3 ( x,) = v j v j j 1 v j 1 p : + ε γ =, v j ( x,) = For simplicity we take (, ) = (, ) = (,) v x t x t x Having this assmption we get the following iterative eqation t v j 1 v j 1 v j = γ ε dt, j = 1,,3,... x x

4 111 Approximate soltion of convection-diffsion eqation by the HPM Therefore, the approximated soltions of Eq.( 1 ) can be obtained, by setting p = 1 = lim v = v + v + v + v +... p nmerical examples In this section, we present examples of convection-diffsion eqation and reslts will be compared with the exact soltions. Example1. Let s consider the convection-diffsion eqation +.1 =.1 x 1, t. With the following initial condition ( ) 5 The exact soltion is ( ) x x, = e sinπ 5x (.5.1π ) t x, t = e sinπ Approximation to the soltion of example 1 can be readily obtained by = vi i= The reslts corresponding absolte errors are presented in Fig.1. Fig.1. The absolte error between exact and nmerical soltions in Example 1.

5 Mehdi Gholami Porshokohi et al. 11 Example. Consider the following the convection-diffsion eqation with. x bondary conditions ( x,) = e sinπ The exact soltion is ( ). x (.4+.5π ) t x, t = e sinπ +. =.5 x 1, t. Approximation to the soltion of example can be readily obtained by = vi i= The reslts corresponding absolte errors are presented in Fig.. Fig.. The absolte error between exact and nmerical soltions in Example. Example3. We consider the convection-diffsion eqation with bondary conditions ( ).5 x x, = e sinπ The exact soltion is ( ).5x (.15+.π ) t x, t = e sinπ +.1 =. x 1, t.

6 113 Approximate soltion of convection-diffsion eqation by the HPM Approximation to the soltion of example 3 can be readily obtained by = vi i= The reslts corresponding absolte errors are presented in Fig.3. Fig.3. The absolte error between exact and nmerical soltions in Example 3. 4 Conclsion In this paper, we proposed the homotopy pertrbation method for solving the convection-diffsion eqations. The obtained soltions, in comparison with exact soltions admit a remarkable accracy. The comptations associated with the examples in this paper were performed sing maple 1. References [1] H.Ding and Y.Zhang, A new difference scheme with high accracy and absolte stability solving convection-diffsion eqations, J. Compt. Appl. Math. 3(9), [] J-H. He, Homotopy pertrbation techniqe. Compt Methods Appl Mech Eng 1999;178(3/4), 57 6.

7 Mehdi Gholami Porshokohi et al. 114 [3] J-H. He, Homotopy pertrbation method: a new nonlinear analytical techniqe, Appl Math Compt, 135( 3), [4] J-H. He, A copling method of homotopy techniqe and pertrbation techniqe for nonlinear problems, Int J Nonlinear Mech, 35(1) (),