The combined Laplace-homotopy analysis method for partial differential equations

Transcription

1 Available online at wwwir-publicationcom/jmc J Math Computer Sci 6 (26), 88 2 Reearch Article The combined Laplace-homotopy analyi method for partial differential equation Javad Vahidi Department of Mathematic, Iran Univerity of Science and Technology, Tehran, Iran Abtract In thi paper, the Laplace tranform homotopy analyi method (LHAM) i employed to obtain approximate analytical olution of the linear and nonlinear differential equation Thi method i a combined form of the Laplace tranform method and the homotopy analyi method The propoed cheme find the olution without any dicretization or retrictive aumption and i free from round-off error and therefore, reduce the numerical computation to a great extent Some illutrative example are preented and the numerical reult how that the olution of the LHAM are in good agreement with thoe obtained by exact olution c 26 All right reerved Keyword: Homotopy analyi method, Laplace tranform method, partial differential equation 2 MSC: 54A4, 26E5 Introduction Nonlinear phenomena have important effect on applied mathematic, phyic and iue related to engineering; many uch phyical phenomena are modeled in term of nonlinear partial differential equation The importance of obtaining the exact or approximate olution of linear and nonlinear partial differential equation in phyic and mathematic i till a ignificant problem that need new method to dicover exact or approximate olution Mot new linear and nonlinear equation do not have a precie analytic olution; o, numerical method have largely been ued to handle thee equation There are alo analytic technique for nonlinear equation Some of the claic analytic method are Lyapunov artificial mall parameter method [], perturbation technique [3, 4], δ- expanion method [5] and Hirota bilinear method [6, 7] In recent year many author have paid attention to tudying the olution of linear and nonlinear partial differential equation by uing variou method Among thee are the Adomian decompoition method (ADM) [8], He emi-invere addre: jvahidi@iutacir (Javad Vahidi) Received

2 J Vahidi, J Math Computer Sci 6 (26), method [9], the tanh method, the homotopy perturbation method (HPM), the inhcoh method, the differential tranform method and the variational iteration method (VIM) [,, 2, 3, 4, 5, 6, 7, 8, 9] Mot of thee method have their inbuilt deficiencie like the calculation of Adomian polynomial, the Lagrange multiplier, divergent reult and huge computational work In 992, Liao [2] employed the baic idea of the homotopy in topology to propoe a general analytic method for nonlinear problem, namely the Homotopy analyi method (HAM), [6, 9, 2, 22] In recent year, homotopy analyi method ha been ued in obtaining approximate olution of a wide cla of differential, integral and integro-differential equation The method provide the olution in a rapidly convergent erie with component that are elegantly computed The Laplace tranform i totally incapable of handling nonlinear equation becaue of the difficultie that are caued by the nonlinear term Variou way have been propoed recently to deal with thee nonlinearitie uch a the Adomian decompoition method [23] and the Laplace decompoition algorithm [24, 25, 26, 27, 29] Inpired and motivated by the ongoing reearch in thi area, we ue the homotopy analyi method coupled with the Laplace tranformation for olving the linear and nonlinear equation in thi paper It i worth mentioning that the propoed method i an elegant combination of the Laplace tranformation and the homotopy analyi method Thi paper conider the effectivene of the Laplace homotopy analyi method in olving linear and nonlinear partial differential equation, both homogeneou and non-homogeneou 2 Baic idea of LHAM To illutrate the baic idea of thi method, we conider the general form of two-dimenional non-homogeneou partial differential equation with a variable coefficient of the form ubject to the boundary condition u t = u µ(x) 2 + ϕ(x, t), (2) x2 and the initial condition u(, t) = g (t), u(, t) = g (t), (22) u(x, ) = f(x) (23) The methodology conit of applying a Laplace tranform on both ide of Eq (2), (22) and in view of the initial condition, we get 2 u x u L[φ(x, t)] + f(x) + =, 2 µx µx u(, t) = g (t), u(, t) = g (t), (24) which i a econd-order boundary value problem By mean of generalizing the traditional homotopy method, Liao [22] contructed the o-called zero-order deformation equation ( p)l[φ(x, ; p) u (x, )] = p H(x, )N [φ(x, ; p)], (25) p [, ] i the embedding parameter, h i a non-zero auxiliary parameter, H(x, ) i an auxiliary function, u (x, ) i an initial gue of u(x, ) and φ(x, ; p) i an unknown function and L an auxiliary linear operator with the property L[φ(x, ; p)] = 2 φ(x, ; p) x 2 (26)

3 J Vahidi, J Math Computer Sci 6 (26), We define a nonlinear operator N a N [φ(x, ; p)] = 2 φ(x, ; p) φ(x, ; p) + x 2 µ(x) L[ϕ(x, t)] + f(x) (27) µ(x) It i important, that one ha great freedom to chooe auxiliary thing in LHAM Obviouly, when p = and p =, it hold φ(x, ; ) = u (x, ), φ(x, ; ) = u(x, ), (28) repectively Thu, a p increae from to, the olution φ(x, ; p) varie from the initial gue u (x, ) to the olution u(x, ) Expanding φ(x, ; p) in Taylor erie with repect to p, we have φ(x, ; p) = u (x, ) + + m= u m (x, )p m, (29) [ ] m φ(x, ; p) u m (x, ) = (2) m! p m p= If the auxiliary linear operator, the initial gue, the auxiliary parameter, and the auxiliary function are o properly choen, the erie (2) converge at p =, then we have u(x, ) = u (x, ) + + m= u m (x, ), (2) which mut be one of olution of original nonlinear equation, a proved by [22] A = and H(x, ) =, Eq (25) become ( p)l[φ(x, ; p) u (x, )] + p N [φ(x, ; p)] =, (22) which i ued motly in the homotopy perturbation method [3], a the olution obtained directly, without uing Taylor erie Define the vector u n = {u (x, ), u (x, ),, u n (x, )} Differentiating equation (25) m time with repect to the embedding parameter p and then etting p = and finally dividing them by m!, we have the o-called mth-order deformation equation and L[u m (x, ) χ m u m (x, )] = H(x, )R m ( u m ), (23) R m ( [ u m ) = (m )! χ m = ] m N [φ(x, ; p)], (24) p m p= {, m,, m > (25) It hould be emphaized that u m (x, ) for m i governed by the linear equation (23) under the linear boundary condition that come from original problem, which can be eaily olved by ymbolic computation oftware uch a Matlab Taking the invere Laplace tranform from u m (x, ) we obtain component u m (x, )of erie olution of Eq (2) 3 Application In order to ae the advantage and the accuracy of Laplace-homotopy analyi method for olving partial differential equation, we will conider the following three example

4 J Vahidi, J Math Computer Sci 6 (26), Example 3 Conider the following homogeneou problem u t + u u x =, u(x, ) = x (3) LHAM approach By applying a Laplace tranform on both ide of Eq (3) we get u(x, ) = x [ L u u ] (32) x Now, we olve Eq (32) by mean of HAM Therefore, we chooe the linear oprator L[φ(x, ; p)] = φ(x, ; p) (33) We define a nonlinear operator a N [φ(x, ; p)] = φ(x, ; p) + x + [ ] L φ(x, t; p) φ(x, t; p) (34) x Uing the definition (33) and (34), we contruct the zeroth-order deformation equation ( p)l[φ(x, ; p) u (x, )] = p H(x, )N [φ(x, ; p)] Thu, we obtain the mth-order (m ) deformation equation L[u m (x, ) χ m u m (x, )] = H(x, )R m ( u m ), (35) [ R m ( u m ) = u m m ] L u m i u i x i= ( x ) + ( χ m ) Now, the olution of the mth-order (m ) deformation equation (35) i a follow u m (x, ) = χ m u m (x, ) + H(ξ, )R m ( u m (ξ, )) The initial approximation u (x, ) can be freely choen, here we et u (x, ) = x By chooing H(x, ) =, the component of the LHAM olution for Eq (32) are derived a follow u (x, ) =! 2 x, u 2 (x, ) = 2! 3 x, u 3 (x, ) = 3! 3 x, u 4 (x, ) = 4! 3 x, In the ame manner, the ret of component were obtained uing the Matlab package

5 J Vahidi, J Math Computer Sci 6 (26), Taking the invere Laplace of component yield u (x, ) = x, u (x, ) = xt, u 2 (x, ) = xt 2, u 3 (x, ) = xt 3, u 4 (x, ) = xt 4, o that the olution u(x, t) i given by u(x, t) = x( + t + t 2 + t 3 + ) (36) in erie form, and u(x, t) = x t i in cloed form That i the exact olution of Eq (3) (37) Example 32 Let u conider the one dimenional non-homogeneou problem ubject to boundary condition u t + u x + u 2 e x ( + 2t) =, (38) u(, t) =, u x (, t) = e t t, and the initial condition u(x, ) = x that i eaily een to have the exact olution u(x, t) = te x +xe t HAM approach To olve Eq (38) by mean of HAM, we chooe the linear operator L[φ(x, t; p)] = φ(x, t; p) t Furthermore, Eq (38) ugget to define the nonlinear operator N [φ(x, t; p)] = φ(x, t; p) t + 2 φ(x, t; p) x 2 + φ(x, t; p) e x ( + 2t) (39) Uing above definition, we contruct the zeroth-order deformation equation ( p)l[φ(x, t; p) u (x, t)] = p H(x, t)n [φ(x, t; p)] (3) According to Eq (39), (3), we gain the mth-order (m ) deformation equation L[u m (x, t) χ m u m (x, t)] = H(x, t)r m ( u m ), R m ( u m ) = u m t + 2 u m x 2 + u m ( χ m )e x ( + 2t)

6 J Vahidi, J Math Computer Sci 6 (26), Now, the component of olution of Eq (38) for (m ) become u m (x, t) = χ m u m (x, t) + t H(x, ξ)r m ( u m (x, ξ))dξ The initial approximation u (x, t) can be freely choen, here we et u (x, t) = x Then, the olution of Eq (38) in erie form i given by u(x, t) = u (x, t) + u (x, t) + (3) LHAM approach By applying a Laplace tranform on both ide of Eq (38), (39) and in view of the initial condition, we get ( 2 u + ( + )u x e x x2 + ) =, (32) 2 u(, ) = 2, u x(, ) = (33) Now, we olve Eq (32) by mean of HAM Therefore, we chooe the linear oprator We define a nonlinear operator a N [φ(x, ; p)] = 2 φ(x, ; p) x 2 L[φ(x, ; p)] = 2 φ(x, ; p) x 2 (34) ( + ( + )φ(x, ; p) x e x + ) (35) 2 Uing the definition (34) and (35), we contruct the zeroth-order deformation equation ( p)l[φ(x, ; p) u (x, )] = p H(x, )N [φ(x, ; p)] Thu, we obtain the mth-order (m ) deformation equation L[u m (x, ) χ m u m (x, )] = H(x, )R m ( u m ), (36) ( R m ( u m ) = 2 u m + ( + )u x 2 m ( χ m ) (x + e x + )) 2 Now, the olution of the mth-order (m ) deformation equation (36) i a follow u m (x, ) = χ m u m (x, ) + x x The initial approximation u (x, ) can be freely choen, here we et u (x, ) = x 2 + x +, H(ξ, )R m ( u m (ξ, ))dξdξ

7 J Vahidi, J Math Computer Sci 6 (26), which atifie the boundary condition (33) By chooing H(x, ) =, the component of the LHAM olution for Eq (32) are derived a follow u (x, ) = ( ) x + 2x 3x 2 3x 2 + x 3 + x 3 + 6e x + 2e x, 6 ( 2 ) u 2 (x, ) = 2 x 5 5x 4 + 6x 3 8x x e x 2 2 ( ) 2x 3 6x x e x 2 2 ( ) 2x 3 6x x e x 2 ( 2 ) 2x 5 x 4 + 8x 3 24x x e x 2 2 ( ) 2x 3 6x x e x 2 2 ( x 5 5x 4 + 2x 3 6x 2 + 2x 2 + 2e x) 2 In the ame manner, the ret of component were obtained uing the Matlab package Taking the invere Laplace of component yield u (x, t) = xe t + ( x)t, u 2 (x, t) = h 6 ( x3 + 3x 2 2x + 2 2e x ) + h 6 ( x3 + 3x 2 6x + 6 6e x ) Finally, the M th-order approximation olution of Eq (38) i a follow S M = M u k (x, t), k= Note The erie (3) contain the auxiliary parameter The validity of the method i baed on uch an aumption that the erie (2) converge at p = It i the auxiliary parameter which enure that thi aumption can be atified A pointed out by Liao [2], in general, by mean of the o-called -curve, it i traightforward to chooe a proper value of which enure that the olution erie i convergent In thi way, we chooe everal valid in Table Table decribe the analytic 3rd-order approximation olution of Eq (38) by the LHAM with = and the analytic 8th-order approximation olution of Eq (38) by the HAM with different value of when x = and t 7 Alo, Table how that, unlike the LHAM the abolute error of the HAM i dramatically increaed a the time value t increae, o the HAM olution validity range i retricted to a hort region (t < 2) But the validity range can be increaed by increaing the term of the olution to more than 8 term On the other hand, reult of the LHAM are in good agreement with

8 J Vahidi, J Math Computer Sci 6 (26), thoe obtained by exact olution The LHAM olution i almot valid for a large wide range of time that how the preent method can olve a non-homogeneou parabolic equation with a high degree of accuracy by the three firt term only Table : The abolute error of LHAM and HAM with x = t LHAM( = ) HAM( = 5) HAM( = ) Example 33 Let u conider the one dimenional non-homogeneou problem ubject to boundary condition 2 u t 2 u + u =, (37) 2 x2 u(, t) = coh t, u x (, t) =, (38) and the initial condition u(, t) = in x +, u t (x, ) =, that i eaily een to have the exact olution u(x, t) = in x + coh t HAM approach To olve Eq(37) by mean of HAM, we chooe the linear operator L[φ(x, t; p)] = 2 φ(x, t; p) t 2 Furthermore, Eq (37) ugget to define the nonlinear operator N [φ(x, t; p)] = 2 φ(x, t; p) t φ(x, t; p) x 2 + φ(x, t; p) (39) Uing above definition, we contruct the zeroth-order deformation equation ( p)l[φ(x, t; p) u (x, t)] = p H(x, t)n [φ(x, t; p)] (32) According to Eq (39), (32), we gain the mth-order (m ) deformation equation L[u m (x, t) χ m u m (x, t)] = H(x, t)r m ( u m ),

9 J Vahidi, J Math Computer Sci 6 (26), R m ( u m ) = 2 u m + 2 u m + u t 2 x 2 m Now, the component of olution of Eq (37) for (m ) become u m (x, t) = χ m u m (x, t) + t t The initial approximation u (x, t) can be freely choen, here we et u (x, t) = + in x Then, the olution of Eq (37) in erie form i given by H(x, ξ)r m ( u m (x, ξ))dξdξ u(x, t) = u (x, t) + u (x, t) + (32) LHAM approach By applying a Laplace tranform on both ide of Eq (37), (38) and in view of the initial condition, we get 2 u x + ( 2 2 )u + ( + in x) =, (322) u(, ) = 2, u x(, ) = (323) Now, we olve Eq (322) by mean of HAM Therefore, we chooe the linear oprator We define a nonlinear operator a L[φ(x, ; p)] = 2 φ(x, ; p) x 2 (324) N [φ(x, ; p)] = 2 φ(x, ; p) x 2 + ( 2 )φ(x, ; p) + ( + in x) (325) Uing the definition (324) and (325), we contruct the zeroth-order deformation equation ( p)l[φ(x, ; p) u (x, )] = p H(x, )N [φ(x, ; p)] Thu, we obtain the mth-order (m ) deformation equation L[u m (x, ) χ m u m (x, )] = H(x, )R m ( u m ), (326) R m ( u m ) = 2 u m + ( 2 )u x 2 m + ( χ m )(( + in x)) Now, the olution of the mth-order (m ) deformation equation (326) i a follow u m (x, ) = χ m u m (x, ) + x x The initial approximation u (x, ) can be freely choen, here we et u (x, ) = H(ξ, )R m ( u m (ξ, ))dξdξ 2 + x,

10 J Vahidi, J Math Computer Sci 6 (26), which atifie the boundary condition (323) By chooe H(x, ) =, the component of the LHAM olution for Eq (322) are derived a follow u (x, ) = ( ) 2 x x in x x 3, 6 u 2 (x, ) = 2 ( x 5 2x 3 + 2x 2 in x ) 2 + ( 2 x 5 + 2x 3 + 2x 2 in x ) 2 + ( ) x x 3 + 2x 3 2 In the ame manner, the ret of component were obtained uing the Matlab package Taking the invere Laplace of component yield u (x, t) = x + coh t, u 2 (x, t) = h ( x 3 + (6x 6 in x x 3 )δ(, t) ) 6 u 3 (x, t) = h2 ( x 5 + (2x 2 in x + x 5 2x 3 )δ(3, t) ) 2 + h ( x 3 ( + ) + (6x 6 in x x 3 x 5 )δ(, t) ) 6 Finally, the M th-order approximation olution of Eq (37) i a follow S M = M u k (x, t) k= LHAM Exact HAM u(,t) t Figure : Compariion between LHAM, HAM and exact olution for x = Table 2 decribe the analytic 4th-order approximation olution of Eq (37) by the LHAM with = and the analytic th-order approximation olution of Eq (37) by the HAM with different

11 J Vahidi, J Math Computer Sci 6 (26), value of when x = and t 7 Table 2 how that, unlike the LHPM, the abolute error of the HAM i dramatically increaed a the time value t increae, o the HAM olution validity range i retricted to a hort region (t < 2) But the validity range can be increaed by increaing the term of the olution to more than term On the other hand, reult of the LHAM are in good agreement with thoe obtained by exact olution The LHAM olution i almot valid for a large wide range of time that how that the preent method can olve a non-homogeneou parabolic equation with a high degree of accuracy by the four firt term only Table 2: The abolute error of LHAM and HAM with x = t LHAM( = ) HAM( = 9) HAM( = ) Example 34 Let u conider the one dimenional non-homogeneou problem ubject to boundary condition u t = 2 u x + 2 e x (co x in x), (327) u(, t) = in t, u x (, t) = in t, and the initial condition u(x, ) = x that i eaily een to have the exact olution u(x, t) = x+e x in t HAM approach To olve Eq (327) by mean of HAM, we chooe the linear operator L[φ(x, t; p)] = φ(x, t; p) t Furthermore, Eq (327) ugget to define the nonlinear operator N [φ(x, t; p)] = φ(x, t; p) t 2 φ(x, t; p) x 2 e x (co x in x) (328) Uing above definition, we contruct the zeroth-order deformation equation ( p)l[φ(x, t; p) u (x, t)] = p H(x, t)n [φ(x, t; p)] (329) According to Eq (328), (329), we gain the mth-order (m ) deformation equation L[u m (x, t) χ m u m (x, t)] = H(x, t)r m ( u m ), R m ( u m ) = u m t + 2 u m x 2 ( χ m )e x (co x in x)

12 J Vahidi, J Math Computer Sci 6 (26), Now, the component of olution of Eq (327) for (m ) become u m (x, t) = χ m u m (x, t) + t H(x, ξ)r m ( u m (x, ξ))dξ The initial approximation u (x, t) can be freely choen, here we et u (x, t) = x Then, the olution of Eq (327) in erie form i given by u(x, t) = u (x, t) + u (x, t) + (33) LHAM approach By applying a Laplace tranform on both ide of Eq (327), (328) and in view of the initial condition, we get ( ) 2 u u + x + e x =, (33) x2 2 + u(, ) = 2 +, u x(, ) = 2 + (332) Now, we olve Eq (33) by mean of HAM Therefore, we chooe the linear oprator We define a nonlinear operator a N [φ(x, ; p)] = 2 φ(x, ; p) x 2 L[φ(x, ; p)] = 2 φ(x, ; p) x 2 (333) ( ) φ(x, ; p) + x + e x (334) 2 + Uing the definition (333) and (334), we contruct the zeroth-order deformation equation ( p)l[φ(x, ; p) u (x, )] = p H(x, )N [φ(x, ; p)] Thu, we obtain the mth-order (m ) deformation equation L[u m (x, ) χ m u m (x, )] = H(x, )R m ( u m ), (335) ( ( )) R m ( u m ) = 2 u m u x 2 m + ( χ m ) x + e x 2 + Now, the olution of the mth-order (m ) deformation equation (326) i a follow u m (x, ) = χ m u m (x, ) + x x The initial approximation u (x, ) can be freely choen, here we et u (x, ) = x x, H(ξ, )R m ( u m (ξ, ))dξdξ

13 J Vahidi, J Math Computer Sci 6 (26), 88 2 which atifie the boundary condition (332) By chooing H(x, ) =, the component of the LHAM olution for Eq (33) are derived a follow u (x, ) = ( ) x 6x 3x 2 + x 3 + 6e x, In the ame manner, the ret of component were obtained uing the Matlab package Taking the invere Laplace of component yield u (x, t) = x (x ) in t, u (x, t) = h ( ( 6 + 6x 3x 2 + x 3 + 6e x ) co t ) 6 + h ( (6 6x 6e x ) in t ) 6 Finally, the M th-order approximation olution of Eq (327) i a follow S M = M u k (x, t) k= 5 5 LHAM Exact HAM u(,t) 5 u(3,t) t t 5 5 u(6,t) 5 u(9,t) t t Figure 2: Compariion between LHAM, HAM and exact olution for x =, x = 3, x = 6, x = 9 Table 3 decribe the analytic 5th-order approximation olution of Eq (327) by the LHAM with = and the analytic th-order approximation olution of Eq (327) by the HAM with different value of when x = and t 7

14 J Vahidi, J Math Computer Sci 6 (26), 88 2 Table 3: The abolute error of LHAM and HAM with x = t LHAM( = ) HAM( = ) HAM( = ) Concluion In thi paper, the LHAM and HAM were ued to obtain the analytic olution of linear and non-linear partial differential equation The comparion between the LHAM and HAM were made and it wa found that LHAM i more effective than HAM On the other hand, the LHAM olution i almot valid for a large wide range of time that how that the preent method can olve a nonhomogeneou parabolic equation with a high degree of accuracy Hence, it may be concluded that thi method i a powerful and an efficient technique in finding the analytic olution for wide clae of problem The computation aociated with the example in thi paper were performed uing Matlab 7 Reference [] G Adomian, Solving Frontier Problem of Phyic: The Decompoition Method, Fundamental Theorie of Phyic, 6 Kluwer Academic Publiher Group, Dordrecht, (994) [2] A K Alomari, A novel olution for fractional chaotic Chen ytem, J Nonlinear Sci Appl, 8 (25), [3] J Biazar, M Gholami Porhokuhi, B Ghanbari, Extracting a general iterative method from an adomian decompoition method and comparing it to the variational iteration method, Comput Math Appl, 59 (2), , 2 [4] C Chun, Fourier-erie-baed variational iteration method for a reliable treatment of heat equation with variable coefficient, Int J Nonlinear Sci Numer Simul, (29), [5] N Faraz, Y Khan, A Yildirim, Analytical approach to two-dimenional vicou flow with a hrinking heet via variational iteration algorithm-ii, J King Saud Univerity, (2), 77 8 [6] J H He, Homotopy perturbation technique, Comput Method Appl Mech Eng, 78 (999), [7] J H He, Variational iteration method-a kind of nonlinear analytical technique: ome example, Int J Nonlinear Mech, 34 (999), [8] J H He, X H Wu, Variational iteration method: new development and application, Comput Math Appl, 54 (27), [9] J H He, G C Wu, F Autin, The variational iteration method which hould be followed, Nonlinear Sci Lett A, (2), 3 [] E Heameddini, H Latifizadeh, Recontruction of variational iteration algorithm uing the Laplace tranform, Int J Nonlinear Sci Numer Simul, (29), [] R Hirota, Exact olution of the Korteweg-de Vrie equation for multiple colliion of oliton, Phy Rev Lett, 27 (97), [2] S Ilam, Y Khan, N Faraz, F Autin, Numerical olution of logitic differential equation by uing the Laplace decompoition method, World Appl Sci J, 8 (2), 5 [3] H Jafari, C Chun, S Seifi, M Saeidy, Analytical olution for nonlinear Ga Dynamic equation by Homotopy Analyi Method, Appl Appl Math, 4 (29), 49 54

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