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Numerical simulation of the generalized Huxley equation by He’s homotopy perturbation method
Applied Mathematics and Computation 192 (2007) 157–161 www.elsevier.com/locate/amc
Numerical simulation of the generalized Huxley equation by He’s homotopy perturbation method S.H. Hashemi, H.R. Mohammadi Daniali *, D.D. Ganji Mazandaran University, Department of Mechanical Engineering, P.O. Box 484, Babol, Iran
Abstract In this paper, the solution of the generalized Huxley equation is obtained by means of homotopy perturbation method and Adomian decomposition method. The comparison reveals that the former method is more effective than the later. In this method, a homotopy is constructed for the equation.The initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions. Moreover, we will show that He’s homotopy perturbation method overcome the difficulties arising in calculating Adomian polynomials. It is predicted that the HPM can be found wide application in engineering problems. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Nonlinear PDE; Huxley equation; Homotopy perturbation method; Adomian decomposition method; Lagrange multiplier
1. Introduction Nonlinear phenomena are of fundamental importance in various fields of science and engineering. Most nonlinear models of real-life problems are still very difficult to solve, either numerically or theoretically. Many assumptions have to be made unnecessarily to make nonlinear models solvable. The generalized Huxley equation ut uxx ¼ buð1 ud Þðud cÞ;
0 6 x 6 1; t P 0;
with the initial condition of hc c i1d uðx; 0Þ ¼ þ tanhðrcxÞ 2 2
ð1Þ
ð2Þ
describes nerve pulse propagation in nerve fibres and wall motion in liquid crystals. The exact solution of this equation was derived by Wang et al. [1], using nonlinear transformations and is given by
*
Corresponding author. E-mail addresses: [email protected] (S.H. Hashemi), [email protected] (H.R. Mohammadi Daniali), ddg_davood @yahoo.com (D.D. Ganji). 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.02.128
158
S.H. Hashemi et al. / Applied Mathematics and Computation 192 (2007) 157–161
1d c c ð1 þ d cÞq þ tanh rc x þ ; ð3Þ t 2 2 2ð1 þ dÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r ¼ dq=4ð1 þ dÞ andq ¼ 4bð1 þ dÞ. In order to solve Eq. (1) numerically, many researchers have used various numerical methods. Hashim et al. investigated the generalized Huxley equation, using Adomian decomposition method (ADM) [2] and Wazwaz studied the generalized forms of Burgers, Burgers–KdV and Burgers–Huxley equations [3]. Wang studied the solitary wave solutions of the generalized Burger’s–Huxley equation [1], Hashem et al. studied the generalized Burger’s–Huxley equation [4], and Estevez investigated non-classical symmetries and the singular modified Burger’s and Burger’s–Huxley equation [5]. In this paper, we solve the generalized Huxley equation by He’s homotopy perturbation method (HPM) [6–17]. Moreover, we compare the numerical results with the exact solutions and those were obtained by the ADM [2]. uðx; tÞ ¼
2. Basic idea of He’s homotopy perturbation method To illustrate the basic ideas of HPM, we consider the following nonlinear differential equation [15]: AðuÞ f ðrÞ ¼ 0;
r 2 X;
ð4Þ
with the boundary conditions of ou B u; ¼ 0; r 2 C; on
ð5Þ
where A, B, f(r) and C are a general differential operator, a boundary operator, a known analytical function and the boundary of the domain X, respectively. Generally speaking the operator A can be divided into a linear part L and a nonlinear part N(u). Eq. (4) can therefore, be rewritten as LðuÞ þ N ðuÞ f ðrÞ ¼ 0;
ð6Þ
By the homotopy technique, we construct a homotopy vðr; pÞ : X ½0; 1 ! R; which satisfies H ðv; pÞ ¼ ð1 pÞ½LðvÞ Lðu0 Þ þ p½AðvÞ f ðrÞ ¼ 0;
p 2 ½0; 1; r 2 X
ð7Þ
or H ðv; pÞ ¼ LðvÞ Lðu0 Þ þ pLðu0 Þ þ p½N ðvÞ f ðrÞ ¼ 0;
ð8Þ
where p 2 ½0; 1 is an embedding parameter, while u0 is an initial approximation of Eq. (4), which satisfies the boundary conditions. Obviously, from Eqs. (7) and (8) we will have H ðv; 0Þ ¼ LðvÞ Lðu0 Þ ¼ 0; H ðv; 1Þ ¼ AðvÞ f ðrÞ ¼ 0;
ð9Þ ð10Þ
The changing process of p from zero to unity is just that of v(r, p) from u0(r) to u(r). In topology, this is called deformation, while LðvÞ Lðu0 Þ and AðvÞ f ðrÞ are called homotopy. According to the HPM, we can first use the embedding parameter p as a ‘‘small parameter’’, and assume that the solution of Eqs. (7) and (8) can be written as a power series in p: v ¼ v0 þ pv1 þ
ð11Þ
Setting p = 1 yields in the approximate solution of Eq. (4) to u ¼ lim v ¼ v0 þ v1 þ p!1
ð12Þ
The combination of the perturbation method and the homotopy method is called the HPM, which eliminates the drawbacks of the traditional perturbation methods while keeping all its advantage. The series (12) is convergent for most cases. However, the convergent rate depends on the nonlinear operator A(v). Moreover, He made the following suggestions [15]:
S.H. Hashemi et al. / Applied Mathematics and Computation 192 (2007) 157–161
159
(1) The second derivative of N(v) with respect to vmust be small because the parameter may be relatively large, i.e. p ! 1. (2) The norm of L1 oN must be smaller than one so that the series converges [16]. ov 3. Applications To solve Eq. (1) with an initial condition (2) by means of HPM, we consider the following Process: 3.1. He’s HPM After separating the linear and nonlinear parts of the equation, we apply homotopy perturbation to Eq. (7). A homotopy can be constructed as follows: 0 1
o2 o vðx; tÞ vðx; tÞ bvðx; tÞ o o ox2 A¼0 ð1 pÞ vðx; tÞ u0 ðx; tÞ þ p@ ot ð13Þ ot ot d d ð1 vðx; tÞ Þðvðx; tÞ cÞ Substituting Eq. (11) into Eq. (13) and rearranging the resultant equation based on powers of p-terms, one has o v0 ðx; tÞ ¼ 0; ð14Þ ot 2 o o d 2d d v1 ðx; tÞ bv0 ðx; tÞv0 ðx; tÞ v ðx; tÞ þ bv ðx; tÞv ðx; tÞ bv ðx; tÞv ðx; tÞ c þ bv ðx; tÞc ¼ 0; 0 0 0 0 0 0 ot ox2 ð15Þ with the following conditions: hc c i1d d u0 ðx; 0Þ ¼ 0; u0 ðx; 0Þ ¼ þ tanhðrcxÞ ; 2 2 dt d ui ðx; 0Þ ¼ 0; i ¼ 1; 2; . . . : ui ðx; 0Þ ¼ 0; dt
ð16Þ
With the effective initial approximation for v0 from the conditions (16) and Solution of Eqs. (14) and (15) may be written as follows: hc c i1d ð17Þ v0 ðx; tÞ ¼ þ tanhðrcxÞ 2 2 1 1þ2d 1 1 2 v1 ðx; tÞ ¼ 2 ð1 þ tanhðrcxÞÞd 2ðdÞ cð d Þ r2 tanhðrcxÞ d 1þ2d 1þd 1þ2d 1þ2d 1 1 2 þ 2ðdÞ cð d Þ r2 tanhðrcxÞ d 2ð d Þ cð d Þ r2 tanhðrcxÞ þ 2ð d Þ cð d Þ r2 1 1 1 d 1þd 1þd 1 1 þ 2ð d Þ cð d Þ b 2ð d Þ cðdÞ ð1 þ tanhðrcxÞÞd d2 2ð d Þ cð d Þ bd2 1 1 1 d 1þ2d 1 1 1 2ð d Þ cð d Þ r2 d þ 2ð d Þ cðdÞ 2ð d Þ cðdÞ ð1 þ tanhðrcxÞÞd bd2 1 1 1 2d 1 1 2ð d Þ cðdÞ 2ð d Þ cðdÞ ð1 þ tanhðrcxÞÞd d2 t : ð18Þ In the same manner, the rest of components were obtained using the Maple Package. According to the HPM, we can conclude that uðx; tÞ ¼ limp!1 vðx; tÞ ¼ v0 ðx; tÞ þ v1 ðx; tÞ þ Therefore, substituting the values of v0(x, t) and v1(x, t) from Eqs. (17), (18) into. Eq. (19) yields:
ð19Þ
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uðx; tÞ ¼
1 1þ2d 1 1þ2d 1 1 ð1 þ tanhðrcxÞÞd 2ðdÞ cð d Þ r2 tanhðrcxÞ2 þ 2ðdÞ cð d Þ r2 tanhðrcxÞ2 d 2 d 1 1 1 d 1þd 1þ2d 1þ2d 1þd 1 1 2ð d Þ cð d Þ r2 tanhðrcxÞ þ 2ð d Þ cð d Þ r2 þ 2ð d Þ cð d Þ b 2ð d Þ cðdÞ ð1 þ tanhðrcxÞÞd d2 1 1 1 d 1þd 1þ2d 1 1 1 1 2ð d Þ cð d Þ bd2 2ð d Þ cð d Þ r2 d þ 2ð d Þ cðdÞ 2ð d Þ cðdÞ ð1 þ tanhðrcxÞÞd bd2 i1d 1 1 hc c 1 2d 1 1 2ð d Þ cðdÞ 2ð d Þ cðdÞ ð1 þ tanhðrcxÞÞd d2 t þ þ tanhðrcxÞ : 2 2
ð20Þ
Table 1 Numerical solutions for b = 1, c = 0.001 and d = 1 x
t
Exact
ADM [2]
HPM
0.1
0.05 0.1 1
5.00030171E04 5.00042665E04 5.00267553E04
5.00005184E04 4.99992690E04 4.99767803E04
5.00005184E04 4.99992690E04 4.99767803E04
0.5
0.05 0.1 1
5.00100882E04 5.00113376E04 5.00338263E04
5.00075895E04 5.00063401E04 4.99838513E04
5.00075895E04 5.00063401E04 4.99838513E04
0.9
0.05 0.1 1
5.00171593E04 5.00184087E04 5.00408974E04
5.00146605E04 5.00134111E04 4.99909224E04
5.00146605E04 5.00134111E04 4.99909224E04
Table 2 Numerical solutions for b = 1, c = 0.001 and d = 2 x
t
Exact
ADM [2]
HPM
0.1
0.05 0.1 1
2.23618841E02 2.23624429E02 2.23724988E02
2.23607664E02 2.23602076E02 2.23501462E02
2.23607664E02 2.23602076E02 2.23501462E02
0.5
0.05 0.1 1
2.23644658E02 2.23650245E02 2.23750792E02
2.23633483E02 2.23627895E02 2.23527292E02
2.23633483E02 2.23627895E02 2.23527292E02
0.9
0.05 0.1 1
2.23670472E02 2.23676058E02 2.23776594E02
2.23659298E02 2.23653711E02 2.23553120E02
2.23659298E02 2.23653711E02 2.23553120E02
Table 3 Numerical solutions for b = 1, c = 0.001 and d = 3 x
t
Exact
ADM [2]
HPM
0.1
0.05 0.1 1
7.93740204E02 7.93760039E02 7.94116901E02
7.93700531E02 7.93680693E02 7.93323439E02
7.93700531E02 7.93680693E02 7.93323439E02
0.5
0.05 0.1 1
7.93819558E02 7.93839389E02 7.94196179E02
7.93779893E02 7.93760059E02 7.93402876E02
7.93779893E02 7.93760059E02 7.93402876E02
0.9
0.05 0.1 1
7.93898897E02 7.93918724E02 7.94275442E02
7.93859239E02 7.93839409E02 7.93482298E02
7.93859239E02 7.93839409E02 7.93482298E02
S.H. Hashemi et al. / Applied Mathematics and Computation 192 (2007) 157–161
161
4. Numerical results and discussion We illustrate the accuracy and efficiency of HPM by applying the method to Eq. (1) and compare the results with to the ADM [2] applied to same equation. For this purpose, we consider the same parameter values for the generalized Huxley equation (1) as considered in [2], namely; b = 1, c = 0.001 and d ¼ 1; 2; 3. The numerical results of the exact solution, five-terms approximate of the ADM and two-terms approximate of HPM, for d ¼ 1; 2; 3, are given in Tables 1–3. The results clearly show that HPM is more efficient than the ADM. Moreover, the HPM avoids the needs for calculating the Adomian polynomials which can be difficult in some cases. 5. Conclusion In this paper, the homotopy perturbation method has been successfully used to study the generalized Huxley equation. The solution obtained by means of the homotopy perturbation method is an infinite power series for appropriate initial condition, which can be, in turn, expressed in a closed form. The results obtained here were compared with the exact solutions and the results reported by using Adomian decomposition method. The results revealed that the homotopy perturbation method is a powerful mathematical tool for the exact and numerical solutions of nonlinear equations in terms of accuracy and efficiency. References [1] X.Y. Wang, Z.S. Zhu, Y.K. Lu, Solitary wave solutions of the generalized Burgers–Huxley equation, Phys. Lett. A 23 (1990) 271–274. [2] I. Hashim, M.S.M. Noorani, B. Batiha, A note on the Adomian decomposition method for the generalized Huxley equation, Appl. Math. Comput, in press. doi:10.1016/j.amc.2006.03.011. [3] A.M. Wazwaz, Travelling wave solutions of generalized forms of Burgers, Burgers–KdV and Burgers–Huxley equations, Appl. Math. Comput. 169 (2005) 639–656. [4] I. Hashim, M.S.M. Noorani, M.R. Said Al-Hadidi, Solving the generalized Burgers–Huxley equation using the Adomian decomposition method, Math. Comput. Modell. 43 (2006) 1404–1411. [5] P.G. Estevez, Non-classical symmetries and the singular modified Burger’s and Burger’s–Huxley equation, Phys. Lett. A 27 (1994) 2113–2127. [6] J.H. He, Homotopy perturbation method: a new nonlinear analytical technique, Appl. Math. Comput. 135 (2003) 73–79. [7] J.H. He, A coupling method of a homotopy technique and a perturbation technique for non-linear problems, Int. J. Non-linear Mech. 35 (2000) 37–43. [8] J.H. He, A new approach to nonlinear partial differential equations, Commun. Nonlin. Sci. Numer. Simul. 2 (4) (1997) 230–235. [9] J.H. He, Non-linear oscillation with fractional derivative and its approximation, in: Int. Conf. on Vibration Engineering 98, Dalian, China, 1988. [10] J.H. He, Approximate solution of nonlinear differential equations with Convolution product nonlinearities, Comput. Methods Appl. Mech. Eng. 167 (1998) 69–73. [11] D.D. Ganji, The application of He’s homotopy perturbation method to nonlinear equation arising in heat transfer, Phys. Lett. A 355 (2006). [12] D.D. Ganji, A. Rajabi, Assessment of homotopy-perturbation and perturbation methods in heat radiation equations, Int. Commun. Heat Mass Transfer 33 (2006) 391–400. [13] D.D. Ganji, M. Rafei, Explicit solutions of Helmholtz equation and fifth-order KDV equation using homotopy perturbation method, Int. J. Non-linear Sci. Numer. Simul. 7 (3) (2006) 321–329. [14] M. Rafei, D.D. Ganji, H.R. Mohammadi Daniali, H. Pashaei, Application of homotopy perturbation method to the RLW and generalized modified Boussinesq equations, Phys. Lett. A 364 (2007) 1–6. [15] J.H. He, J. Comput. Methods Appl. Mech. Eng. 178 (3–4) (1999) 257–262. [16] J.H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Eng. 178 (1999) 257–262. [17] D.D. Ganji, S.H. Hashemi, Explicit solution for fourth-order equations with variable coefficients in higher-dimensional spaces using He’s homotopy perturbation method, Far East J. Appl. Math. 26 (3) (2007) 305–320.
Numerical simulation of the generalized Huxley equation by He’s homotopy perturbation method S.H. Hashemi, H.R. Mohammadi Daniali *, D.D. Ganji Mazandaran University, Department of Mechanical Engineering, P.O. Box 484, Babol, Iran
Abstract In this paper, the solution of the generalized Huxley equation is obtained by means of homotopy perturbation method and Adomian decomposition method. The comparison reveals that the former method is more effective than the later. In this method, a homotopy is constructed for the equation.The initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions. Moreover, we will show that He’s homotopy perturbation method overcome the difficulties arising in calculating Adomian polynomials. It is predicted that the HPM can be found wide application in engineering problems. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Nonlinear PDE; Huxley equation; Homotopy perturbation method; Adomian decomposition method; Lagrange multiplier
1. Introduction Nonlinear phenomena are of fundamental importance in various fields of science and engineering. Most nonlinear models of real-life problems are still very difficult to solve, either numerically or theoretically. Many assumptions have to be made unnecessarily to make nonlinear models solvable. The generalized Huxley equation ut uxx ¼ buð1 ud Þðud cÞ;
0 6 x 6 1; t P 0;
with the initial condition of hc c i1d uðx; 0Þ ¼ þ tanhðrcxÞ 2 2
ð1Þ
ð2Þ
describes nerve pulse propagation in nerve fibres and wall motion in liquid crystals. The exact solution of this equation was derived by Wang et al. [1], using nonlinear transformations and is given by
*
Corresponding author. E-mail addresses: [email protected] (S.H. Hashemi), [email protected] (H.R. Mohammadi Daniali), ddg_davood @yahoo.com (D.D. Ganji). 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.02.128
158
S.H. Hashemi et al. / Applied Mathematics and Computation 192 (2007) 157–161
1d c c ð1 þ d cÞq þ tanh rc x þ ; ð3Þ t 2 2 2ð1 þ dÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r ¼ dq=4ð1 þ dÞ andq ¼ 4bð1 þ dÞ. In order to solve Eq. (1) numerically, many researchers have used various numerical methods. Hashim et al. investigated the generalized Huxley equation, using Adomian decomposition method (ADM) [2] and Wazwaz studied the generalized forms of Burgers, Burgers–KdV and Burgers–Huxley equations [3]. Wang studied the solitary wave solutions of the generalized Burger’s–Huxley equation [1], Hashem et al. studied the generalized Burger’s–Huxley equation [4], and Estevez investigated non-classical symmetries and the singular modified Burger’s and Burger’s–Huxley equation [5]. In this paper, we solve the generalized Huxley equation by He’s homotopy perturbation method (HPM) [6–17]. Moreover, we compare the numerical results with the exact solutions and those were obtained by the ADM [2]. uðx; tÞ ¼
2. Basic idea of He’s homotopy perturbation method To illustrate the basic ideas of HPM, we consider the following nonlinear differential equation [15]: AðuÞ f ðrÞ ¼ 0;
r 2 X;
ð4Þ
with the boundary conditions of ou B u; ¼ 0; r 2 C; on
ð5Þ
where A, B, f(r) and C are a general differential operator, a boundary operator, a known analytical function and the boundary of the domain X, respectively. Generally speaking the operator A can be divided into a linear part L and a nonlinear part N(u). Eq. (4) can therefore, be rewritten as LðuÞ þ N ðuÞ f ðrÞ ¼ 0;
ð6Þ
By the homotopy technique, we construct a homotopy vðr; pÞ : X ½0; 1 ! R; which satisfies H ðv; pÞ ¼ ð1 pÞ½LðvÞ Lðu0 Þ þ p½AðvÞ f ðrÞ ¼ 0;
p 2 ½0; 1; r 2 X
ð7Þ
or H ðv; pÞ ¼ LðvÞ Lðu0 Þ þ pLðu0 Þ þ p½N ðvÞ f ðrÞ ¼ 0;
ð8Þ
where p 2 ½0; 1 is an embedding parameter, while u0 is an initial approximation of Eq. (4), which satisfies the boundary conditions. Obviously, from Eqs. (7) and (8) we will have H ðv; 0Þ ¼ LðvÞ Lðu0 Þ ¼ 0; H ðv; 1Þ ¼ AðvÞ f ðrÞ ¼ 0;
ð9Þ ð10Þ
The changing process of p from zero to unity is just that of v(r, p) from u0(r) to u(r). In topology, this is called deformation, while LðvÞ Lðu0 Þ and AðvÞ f ðrÞ are called homotopy. According to the HPM, we can first use the embedding parameter p as a ‘‘small parameter’’, and assume that the solution of Eqs. (7) and (8) can be written as a power series in p: v ¼ v0 þ pv1 þ
ð11Þ
Setting p = 1 yields in the approximate solution of Eq. (4) to u ¼ lim v ¼ v0 þ v1 þ p!1
ð12Þ
The combination of the perturbation method and the homotopy method is called the HPM, which eliminates the drawbacks of the traditional perturbation methods while keeping all its advantage. The series (12) is convergent for most cases. However, the convergent rate depends on the nonlinear operator A(v). Moreover, He made the following suggestions [15]:
S.H. Hashemi et al. / Applied Mathematics and Computation 192 (2007) 157–161
159
(1) The second derivative of N(v) with respect to vmust be small because the parameter may be relatively large, i.e. p ! 1. (2) The norm of L1 oN must be smaller than one so that the series converges [16]. ov 3. Applications To solve Eq. (1) with an initial condition (2) by means of HPM, we consider the following Process: 3.1. He’s HPM After separating the linear and nonlinear parts of the equation, we apply homotopy perturbation to Eq. (7). A homotopy can be constructed as follows: 0 1
o2 o vðx; tÞ vðx; tÞ bvðx; tÞ o o ox2 A¼0 ð1 pÞ vðx; tÞ u0 ðx; tÞ þ p@ ot ð13Þ ot ot d d ð1 vðx; tÞ Þðvðx; tÞ cÞ Substituting Eq. (11) into Eq. (13) and rearranging the resultant equation based on powers of p-terms, one has o v0 ðx; tÞ ¼ 0; ð14Þ ot 2 o o d 2d d v1 ðx; tÞ bv0 ðx; tÞv0 ðx; tÞ v ðx; tÞ þ bv ðx; tÞv ðx; tÞ bv ðx; tÞv ðx; tÞ c þ bv ðx; tÞc ¼ 0; 0 0 0 0 0 0 ot ox2 ð15Þ with the following conditions: hc c i1d d u0 ðx; 0Þ ¼ 0; u0 ðx; 0Þ ¼ þ tanhðrcxÞ ; 2 2 dt d ui ðx; 0Þ ¼ 0; i ¼ 1; 2; . . . : ui ðx; 0Þ ¼ 0; dt
ð16Þ
With the effective initial approximation for v0 from the conditions (16) and Solution of Eqs. (14) and (15) may be written as follows: hc c i1d ð17Þ v0 ðx; tÞ ¼ þ tanhðrcxÞ 2 2 1 1þ2d 1 1 2 v1 ðx; tÞ ¼ 2 ð1 þ tanhðrcxÞÞd 2ðdÞ cð d Þ r2 tanhðrcxÞ d 1þ2d 1þd 1þ2d 1þ2d 1 1 2 þ 2ðdÞ cð d Þ r2 tanhðrcxÞ d 2ð d Þ cð d Þ r2 tanhðrcxÞ þ 2ð d Þ cð d Þ r2 1 1 1 d 1þd 1þd 1 1 þ 2ð d Þ cð d Þ b 2ð d Þ cðdÞ ð1 þ tanhðrcxÞÞd d2 2ð d Þ cð d Þ bd2 1 1 1 d 1þ2d 1 1 1 2ð d Þ cð d Þ r2 d þ 2ð d Þ cðdÞ 2ð d Þ cðdÞ ð1 þ tanhðrcxÞÞd bd2 1 1 1 2d 1 1 2ð d Þ cðdÞ 2ð d Þ cðdÞ ð1 þ tanhðrcxÞÞd d2 t : ð18Þ In the same manner, the rest of components were obtained using the Maple Package. According to the HPM, we can conclude that uðx; tÞ ¼ limp!1 vðx; tÞ ¼ v0 ðx; tÞ þ v1 ðx; tÞ þ Therefore, substituting the values of v0(x, t) and v1(x, t) from Eqs. (17), (18) into. Eq. (19) yields:
ð19Þ
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uðx; tÞ ¼
1 1þ2d 1 1þ2d 1 1 ð1 þ tanhðrcxÞÞd 2ðdÞ cð d Þ r2 tanhðrcxÞ2 þ 2ðdÞ cð d Þ r2 tanhðrcxÞ2 d 2 d 1 1 1 d 1þd 1þ2d 1þ2d 1þd 1 1 2ð d Þ cð d Þ r2 tanhðrcxÞ þ 2ð d Þ cð d Þ r2 þ 2ð d Þ cð d Þ b 2ð d Þ cðdÞ ð1 þ tanhðrcxÞÞd d2 1 1 1 d 1þd 1þ2d 1 1 1 1 2ð d Þ cð d Þ bd2 2ð d Þ cð d Þ r2 d þ 2ð d Þ cðdÞ 2ð d Þ cðdÞ ð1 þ tanhðrcxÞÞd bd2 i1d 1 1 hc c 1 2d 1 1 2ð d Þ cðdÞ 2ð d Þ cðdÞ ð1 þ tanhðrcxÞÞd d2 t þ þ tanhðrcxÞ : 2 2
ð20Þ
Table 1 Numerical solutions for b = 1, c = 0.001 and d = 1 x
t
Exact
ADM [2]
HPM
0.1
0.05 0.1 1
5.00030171E04 5.00042665E04 5.00267553E04
5.00005184E04 4.99992690E04 4.99767803E04
5.00005184E04 4.99992690E04 4.99767803E04
0.5
0.05 0.1 1
5.00100882E04 5.00113376E04 5.00338263E04
5.00075895E04 5.00063401E04 4.99838513E04
5.00075895E04 5.00063401E04 4.99838513E04
0.9
0.05 0.1 1
5.00171593E04 5.00184087E04 5.00408974E04
5.00146605E04 5.00134111E04 4.99909224E04
5.00146605E04 5.00134111E04 4.99909224E04
Table 2 Numerical solutions for b = 1, c = 0.001 and d = 2 x
t
Exact
ADM [2]
HPM
0.1
0.05 0.1 1
2.23618841E02 2.23624429E02 2.23724988E02
2.23607664E02 2.23602076E02 2.23501462E02
2.23607664E02 2.23602076E02 2.23501462E02
0.5
0.05 0.1 1
2.23644658E02 2.23650245E02 2.23750792E02
2.23633483E02 2.23627895E02 2.23527292E02
2.23633483E02 2.23627895E02 2.23527292E02
0.9
0.05 0.1 1
2.23670472E02 2.23676058E02 2.23776594E02
2.23659298E02 2.23653711E02 2.23553120E02
2.23659298E02 2.23653711E02 2.23553120E02
Table 3 Numerical solutions for b = 1, c = 0.001 and d = 3 x
t
Exact
ADM [2]
HPM
0.1
0.05 0.1 1
7.93740204E02 7.93760039E02 7.94116901E02
7.93700531E02 7.93680693E02 7.93323439E02
7.93700531E02 7.93680693E02 7.93323439E02
0.5
0.05 0.1 1
7.93819558E02 7.93839389E02 7.94196179E02
7.93779893E02 7.93760059E02 7.93402876E02
7.93779893E02 7.93760059E02 7.93402876E02
0.9
0.05 0.1 1
7.93898897E02 7.93918724E02 7.94275442E02
7.93859239E02 7.93839409E02 7.93482298E02
7.93859239E02 7.93839409E02 7.93482298E02
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4. Numerical results and discussion We illustrate the accuracy and efficiency of HPM by applying the method to Eq. (1) and compare the results with to the ADM [2] applied to same equation. For this purpose, we consider the same parameter values for the generalized Huxley equation (1) as considered in [2], namely; b = 1, c = 0.001 and d ¼ 1; 2; 3. The numerical results of the exact solution, five-terms approximate of the ADM and two-terms approximate of HPM, for d ¼ 1; 2; 3, are given in Tables 1–3. The results clearly show that HPM is more efficient than the ADM. Moreover, the HPM avoids the needs for calculating the Adomian polynomials which can be difficult in some cases. 5. Conclusion In this paper, the homotopy perturbation method has been successfully used to study the generalized Huxley equation. The solution obtained by means of the homotopy perturbation method is an infinite power series for appropriate initial condition, which can be, in turn, expressed in a closed form. The results obtained here were compared with the exact solutions and the results reported by using Adomian decomposition method. The results revealed that the homotopy perturbation method is a powerful mathematical tool for the exact and numerical solutions of nonlinear equations in terms of accuracy and efficiency. References [1] X.Y. Wang, Z.S. Zhu, Y.K. Lu, Solitary wave solutions of the generalized Burgers–Huxley equation, Phys. Lett. A 23 (1990) 271–274. [2] I. Hashim, M.S.M. Noorani, B. Batiha, A note on the Adomian decomposition method for the generalized Huxley equation, Appl. Math. Comput, in press. doi:10.1016/j.amc.2006.03.011. [3] A.M. Wazwaz, Travelling wave solutions of generalized forms of Burgers, Burgers–KdV and Burgers–Huxley equations, Appl. Math. Comput. 169 (2005) 639–656. [4] I. Hashim, M.S.M. Noorani, M.R. Said Al-Hadidi, Solving the generalized Burgers–Huxley equation using the Adomian decomposition method, Math. Comput. Modell. 43 (2006) 1404–1411. [5] P.G. Estevez, Non-classical symmetries and the singular modified Burger’s and Burger’s–Huxley equation, Phys. Lett. A 27 (1994) 2113–2127. [6] J.H. He, Homotopy perturbation method: a new nonlinear analytical technique, Appl. Math. Comput. 135 (2003) 73–79. [7] J.H. He, A coupling method of a homotopy technique and a perturbation technique for non-linear problems, Int. J. Non-linear Mech. 35 (2000) 37–43. [8] J.H. He, A new approach to nonlinear partial differential equations, Commun. Nonlin. Sci. Numer. Simul. 2 (4) (1997) 230–235. [9] J.H. He, Non-linear oscillation with fractional derivative and its approximation, in: Int. Conf. on Vibration Engineering 98, Dalian, China, 1988. [10] J.H. He, Approximate solution of nonlinear differential equations with Convolution product nonlinearities, Comput. Methods Appl. Mech. Eng. 167 (1998) 69–73. [11] D.D. Ganji, The application of He’s homotopy perturbation method to nonlinear equation arising in heat transfer, Phys. Lett. A 355 (2006). [12] D.D. Ganji, A. Rajabi, Assessment of homotopy-perturbation and perturbation methods in heat radiation equations, Int. Commun. Heat Mass Transfer 33 (2006) 391–400. [13] D.D. Ganji, M. Rafei, Explicit solutions of Helmholtz equation and fifth-order KDV equation using homotopy perturbation method, Int. J. Non-linear Sci. Numer. Simul. 7 (3) (2006) 321–329. [14] M. Rafei, D.D. Ganji, H.R. Mohammadi Daniali, H. Pashaei, Application of homotopy perturbation method to the RLW and generalized modified Boussinesq equations, Phys. Lett. A 364 (2007) 1–6. [15] J.H. He, J. Comput. Methods Appl. Mech. Eng. 178 (3–4) (1999) 257–262. [16] J.H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Eng. 178 (1999) 257–262. [17] D.D. Ganji, S.H. Hashemi, Explicit solution for fourth-order equations with variable coefficients in higher-dimensional spaces using He’s homotopy perturbation method, Far East J. Appl. Math. 26 (3) (2007) 305–320.