Alexandria Engineering Journal (2015) 54, 645–651 H O S T E D BY Alexandria University Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com ORIGINAL ARTICLE Analytical solutions of convection–diffusion problems by combining Laplace transform method and homotopy perturbation method Sumit Gupta a b c a,* , Devendra Kumar b, Jagdev Singh c Department of Mathematics, Jagan Nath Gupta Institute of Engineering and Technology, Jaipur 302022, Rajasthan, India Department of Mathematics, JECRC University, Jaipur 303905, Rajasthan, India Department of Mathematics, Jagan Nath University, Jaipur 303901, Rajasthan, India Received 25 February 2014; revised May 2015; accepted 12 May 2015 Available online June 2015 KEYWORDS Homotopy perturbation method; Laplace transform method; Linear and nonlinear convection–diffusion problems; He’s polynomials Abstract The aim of this paper was to present a user friendly numerical algorithm based on homotopy perturbation transform method for solving various linear and nonlinear convection-diffusion problems arising in physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to two processes: diffusion and convection The homotopy perturbation transform method is a combined form of the homotopy perturbation method and Laplace transform method The nonlinear terms can be easily obtained by the use of He’s polynomials The technique presents an accurate methodology to solve many types of partial differential equations The approximate solutions obtained by proposed scheme in a wide range of the problem’s domain were compared with those results obtained from the actual solutions The comparison shows a precise agreement between the results ª 2015 Faculty of Engineering, Alexandria University Production and hosting by Elsevier B.V This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Introduction The convection–diffusion equation is a combination of the diffusion and convection equations, and describes physical phenomena where particles, energy, or other physical quantities * Corresponding author Tel.: +91 9929764461 E-mail addresses: guptasumit.edu@gmail.com (S Gupta), devendra maths@gmail.com (D Kumar), jagdevsinghrathore@gmail.com (J Singh) Peer review under responsibility of Faculty of Engineering, Alexandria University are transferred inside a physical system due to two processes: diffusion and convection In general form the convection–diffusion equation is given as follows: @u ¼ r Á D ruị r ~ tuị ỵ R; @t ð1Þ where u is the variable of interest, D is the diffusivity, such as mass diffusivity for particle motion or thermal diffusivity for heat transport, ~ t is the average velocity that the quantity is moving For example, in advection, u might be the concentrat would be the velocity of the tion of salt in a river, and then ~ water flow As another example, u might be the concentration http://dx.doi.org/10.1016/j.aej.2015.05.004 1110-0168 ª 2015 Faculty of Engineering, Alexandria University Production and hosting by Elsevier B.V This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) 646 S Gupta et al t would be the averof small bubbles in a calm lake, and then ~ age velocity of bubbles rising towards the surface by buoyancy R describes ‘‘sources’’ or ‘‘sinks’’ of the quantity u For example, for a chemical species, R > means that a chemical reaction is creating more of the species, and R < means that a chemical reaction is destroying the species For heat transport, R > might occur if thermal energy is being generated by friction r represents gradient and rÁ represents divergence Previously various methods have been used to handle these problems such as variational iteration method (VIM) [1], Adomian’s decomposition method (ADM) [2], homotopy perturbation method (HPM) [3] and Bessel collocation method [4] Most of these methods have their inbuilt deficiencies like the calculation of Adomian’s polynomials, the Lagrange multiplier, divergent results and huge computational work In recent years numerical methods have also been applied various physical problems such as Volterra’s population growth model with fractional order [5], nonlinear Lane-Emden type equations [6], Hantavirus infection model [7], continuous population models for single and interacting species [8] The homotopy perturbation method (HPM), first proposed by He [9–15] for solving various linear and nonlinear initial and boundary value problems The HPM was also studied by many authors to handle nonlinear equations arising in science and engineering [16–23] The Laplace transform is totally incapable of handling nonlinear equations because of the difficulties that are caused by the nonlinear terms Various ways have been proposed recently to deal such nonlinearities such as the Laplace decomposition algorithm [24–27] and the homotopy perturbation transform method (HPTM) [28–30] to produce highly effective techniques for solving many nonlinear problems The basic motivation of this paper is to apply an effective modification of HPM to overcome the deficiency We implement the homotopy perturbation transform method (HPTM) for solving the convection-diffusion equations Using this method, all conditions can be satisfied Also very accurate results are obtained in a wide range via one or two iteration steps The suggested HPTM provides the solution in a rapid convergent series which may lead the solution in closed form The use of He’s polynomials in nonlinear terms first proposed by Ghorbani [31, 32] Several examples are given to verify the reliability and efficiency of the HPTM Analysis of the method Using the differentiation property of the Laplace transform, we have hðxÞ fðxÞ 1 ỵ LẵRux; tị ỵ Lẵgx; tị s s s s LẵN ux; tị: s L ẵux; tị ẳ 5ị Operating with the Laplace inverse on both sides of Eq (5) gives ! 6ị ux; tị ẳ Gx; tị L1 L ẵRux; tị ỵ Nux; tị : s where Gðx; tÞ represents the term arising from the source term and the prescribed initial conditions Now we apply the HPM ux; tị ẳ X pn un x; tị 7ị n¼0 and the nonlinear term can be decomposed as N ux; tị ẳ X pn Hn uị 8ị nẳ0 for some He’s polynomials Hn ðuÞ that are given by " !# X @n i Hn ðu0 ; u1 ; ;un ị ẳ N p ui ; n ¼ 0;1; 2; n! @pn i¼0 pẳ0 9ị Substituting Eqs (7) and (8) in Eq (6), we get X pn un x;tị ẳ Gx;tị nẳ0 " À1 Àp L " ##! 1 X X n n L R p un x;tị ỵ p Hn uị : s2 nẳ0 nẳ0 10ị which is the coupling of the Laplace transform and the HPM using He’s polynomials Comparing the coefficient of like powers of p, the following approximations are obtained p0 : u0 x; tị ẳ Gx; tị p1 : u1 x; tị ẳ L1 s12 LẵRu0 x; tị ỵ H0 uị ; p2 : u2 x; tị ẳ L1 s12 LẵRu1 x; tị ỵ H1 uị ; p3 : u3 x; tị ẳ L1 s12 LẵRu2 x; tị ỵ H2 ðuÞ ; ; ð11Þ The HPTM is a combined form of the HPM and Laplace transform method We apply the HPTM to the following general nonlinear partial differential equation with the initial conditions of the form and so on Dux; tị ỵ Rux; tị ỵ Nux; tị ẳ gx; tị; 2ị ux; 0ị ẳ hxị; 3ị In this section, we discuss the implementation of our numerical method and investigate its accuracy and stability by applying it to numerical examples on the convection-diffusion equations ut x; 0ị ẳ fxị: where D is the second order linear differential operator D ¼ @ =@t2 , R is the linear differential operator of less order than D; N represents the general nonlinear differential operator and gðx; tÞ is the source term Taking the Laplace transform (denoted in this paper by L) on both sides of Eq (2): LẵD ux; tị ỵ LẵR ux; tị ỵ LẵN ux; tị ẳ L ẵgx; tÞ: ð4Þ Numerical examples and error estimation Example 3.1 Let us consider the following diffusionconvection problem @u @ u ¼ À u; @t @x2 with the initial condition ux; 0ị ẳ x ỵ ex 12ị Analytical solutions of convection–diffusion problems 647 The surface shows the solution uðx; tÞ for Eq (12): (a) exact solution; (b) approximate solution (18); (c) juex À uapp j Figure Taking Laplace transform on both the sides, subject to the initial condition, we get Lẵux; tị ẳ p0 : x ỵ ex ỵ L ẵuxx u: s s 13ị Taking inverse Laplace transform, we get ! uðx; tÞ ẳ x ỵ ex ị ỵ pL1 L ẵuxx u : s ð14Þ By homotopy perturbation method, we have ux; tị ẳ Comparing the coefcients of various powers of p, we get X pn un ðx; tÞ: ð15Þ u0 x; tị ẳ x ỵ ex ; p : p2 : u1 x; tị ẳ x t; u2 x; tị ẳ x t2! ; p3 : u3 x; tị ẳ x t3! ; 17ị and so on Therefore the series solution is given by t2 t3 x ux; tị ẳ e ỵ x t ỵ ỵ ; 2! 3! 18ị nẳ0 Using (15) in (14), we have X pn un x;tị ẳ x ỵ ex ị nẳ0 þ pLÀ1 " " ! !## 1 X X L pn un ðx; tÞ ÀL pn un ðx; tị : s nẳ0 nẳ0 xx 16ị which converge very rapidly to the exact solution ux; tị ẳ ex ỵ xeÀt The numerical results of uðx; tÞ for the approximate solution (18) obtained by using HPTM, the exact solution and the absolute error E7 uị ẳ juex uapp j for various values of t and x are shown by Fig 1(a)–(c) It is observed from Fig 1(a) and (b) that uðx; tÞ increases with the increase in x and decrease in t Fig 1(a)–(c) clearly show that the 648 S Gupta et al Figure The surface shows the solution uðx; tÞ for Eq (19): (a) exact solution; (b) approximate solution (22); (c) juex À uapp j approximate solution (18) obtained by the present method is very near to the exact solution It is to be noted that only the seventh order term of the HPTM was used in evaluating the approximate solutions for Fig It is evident that the efficiency of the present method can be dramatically enhanced by computing further terms of uðx; tÞ when the HPTM is used Comparing the coefficients of various powers of p, we get p0 : p1 : p2 : p : Example 3.2 Let us consider the following diffusionconvection problem @u @ u ẳ ỵ ỵ cos x À sin2 xÞ u; @t @x2 ð19Þ with the initial condition ux; 0ị ẳ 101 ecos x11 By applying aforesaid method subject to the initial condition, we have " ! 1 X X 1 pn un x; tị ẳ ecosx11 ỵ pL1 L pn un x; tị 10 s nẳ0 nẳ0 xx X ! n ỵ ỵ cosx sin xịL p un x;y;tị : s 20ị u0 x; tị ẳ 101 ecos x11 ; u1 x; tị ẳ 101 ecos x11 tị; u2 x; tị ẳ 101 ecos x11 t2! ; u3 x; tị ẳ 101 ecos xÀ11 À t3! ; ð21Þ and so on Therefore the approximate solution is given by uðx; tÞ ¼ cos xÀ11 t2 t3 e t ỵ ỵ ; 10 2! 3! ð22Þ which converge very rapidly to the exact solution uðx; tị ẳ 101 ecos x11t The numerical results of uðx; tÞ for the approximate solution (22) obtained with the help of HPTM, the exact solution and the absolute error E7 uị ẳ juex uapp j for various values of t and x are shown by Fig 2(a)–(c) From Fig 2(a)–(c), we Analytical solutions of convection–diffusion problems Figure 649 The surface shows the solution uðx; tÞ for Eq (23): (a) exact solution; (b) approximate solution (26); (c) juex À uapp j observed that the approximate solution (22) obtained by the proposed method is very near to the exact solution Example 3.3 Let us consider the following diffusionconvection problem @u @ u ¼ À u; @t @x2 x; t R ð23Þ with the initial condition ux; 0ị ẳ 12 x ỵ ex=2 By applying aforesaid method subject to the initial condition, we have " ! 1 x X X À1 n x=2 n ỵ pL p un x; tị ẳ ỵe p un x; tị L s nẳ0 nẳ0 xx !# X pn un ðx; tÞ : ð24Þ À L 4s n¼0 Comparing p, we get the coefficients of various powers of p : u0 x; tị ẳ x2 ỵ ex=2 ; u1 x; tị ẳ x2 4t ; p2 : u2 x; tị ẳ x2 p0 : p : u3 ðx; tị ẳ t=4ị2 2! ; x t=4ị 3! ð25Þ ; and so on Therefore the series solution is given by x=2 ux; tị ẳ e ! x t t=4ị2 t=4ị3 ỵ ỵ ỵ ; 2! 3! ð26Þ which converge very rapidly to the exact solution ux; tị ẳ ex=2 ỵ x2 et=4 The numerical results of uðx; tÞ for the approximate solution (26) derived with the application of HPTM, the exact solution and the absolute error E7 uị ẳ juex uapp j for various values of t and x are described by Fig 3(a)–(c) It is to be noted from Fig 3(a) and (b) that uðx; tÞ increases with the 650 S Gupta et al Figure The surface shows the solution uðx; tÞ for Eq (27): (a) exact solution; (b) approximate solution (30); (c) juex À uapp j increase in x and decrease in t Fig 3(a)–(c) clearly show that the approximate solution (26) obtained by the present approach is very near to the exact solution Comparing the coefficients of various powers of p, we get p0 : Example 3.4 Let us consider the following nonlinear diffusion-convection problem @u @ u @u ẳ ỵ u ux u2 ỵ u; @t @x2 @x x; t R 27ị with the initial equation ux; 0ị ẳ e x By applying aforesaid method subject to the initial condition, we have ! ! 1 X X X À1 n x n n L p un ðx; tị ẳ e ỵ pL p un x; tị L p un x; tị s nẳ0 nẳ0 nẳ0 xx x ( ! ! ) 1 X X ÀL pn un x; tị pn un x; tị nẳ0 L X nẳ0 nẳ0 !2 n p un x;tị ỵL x X nẳ0 !3 p un x; tị 5: n 28ị u0 x; tị ẳ e x ; p : p2 : u1 x; tị ẳ e x t; u2 x; tị ẳ e x t2! ; p3 : u3 x; tị ẳ e x 29ị t3 ; 3! and so on Therefore the series solution is given as t2 t3 ux; tị ẳ e x ỵ t ỵ ỵ ỵ ; 2! 3! ð30Þ which converge very rapidly to the exact solution ux; tị ẳ exỵt The numerical results of uðx; tÞ for the approximate solution (30) find by applying HPTM, the exact solution and the absolute error E7 ðuÞ ¼ juex À uapp j for various values of t and x are depicted by Fig 4(a)–(c) It is to be noted from Analytical solutions of convection–diffusion problems Fig 4(a) and (b) that uðx; tÞ increases with the increase in both x and t From Fig 4(a)–(c), we can see that the approximate solution (26) obtained by the present scheme is very near to the exact solution Conclusions In this paper, we have applied the homotopy perturbation transform method (HPTM) for solving convection-diffusion equations The proposed method is applied without using linearization, discretization or restrictive assumptions It may be concluding that the HPTM using He’s polynomials is very powerful and efficient in finding the analytic solutions for a wide class of problems The solution procedure using He’s polynomials is simple, but the calculation of Adomian’s polynomials is complex The method gives more realistic series solutions that converge 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