Applications of Homotopy Perturbation Transform Method for Solving Newell-Whitehead-Segel Equation

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1 General Letters in Mathematics Vol, No, Aug 207, pp5-4 e-issn , p-issn Available online at wwwrefaadcom Applications of Homotopy Perturbation Transform Method for Solving Newell-Whitehead-Segel Equation Mohamed Zellal, Kacem Belghaba2 Hassiba Benbouali University of Chlef BP 5, Hay Essalem, 02000, Algeria 2 Labratory of Mathematic and Its Applications LAMAP, University of Oran, Ahmed Ben Bella, PO Box 524, Oran, 000, Algeria mzellal@yahoofr, 2 belghaba@yahoofr Abstract In the present work, we apply an analytical method known as Homotopy Perturbation Transform Method, is used for solving Newell-Whitehead-Segel Equation Five tests are examined to validate the accuracy of this algorithm Numerical results show that the proposed method is a more reliable, efficient and convenient one for solving different cases of NewellWhitehead-Segel equation Keywords: He s polynomials, Homotopy Perturbation Transform Method, Laplace transform method, WhiteheadSegel equation 200 MSC No: 5K5, 5A22, 47j0, 49M27 Introduction The Newell-Whitehead-Segel equations have wide applicability in mechanical and chemical engineering, ecology, biology and bio-engineering Newell-Whitehead-Segel NWSequation is solved by using the Adomian decomposition and multi-quadric quasi-interpolation methods [2], an approximate solution to the Newell-Whitehead-Segel equation by the Adomian decomposition method [8], homotopy perturbation method [9], new iterative method [0], He s variational iteration method [], Laplace Adomian Decomposition Method [2] and a comparison between the reduced differential transform method and the Adomian decomposition method for the Newell-Whitehead-Segel equation [] In this article, different types of nonlinear Newell- Whitehead-Segel equations are solved by the Homotopy Perturbation Transform Method HPTM which is introduced by Khan and Wu [7] by combining the Homotopy perturbation method []-[5]-[] and Laplace transform method for solving various types of linear and nonlinear partial differential equations The Newell-Whitehead-Segel equation models written as: u 2u = k 2 x, t + aux, t bum x, t, t x where ux, t is a function of the spatial variable x and the temporal variable t with x R and t 0 The function ux, t may be thought of as the nonlinear distribution of temperature in an infinitely thin and long rod or as the flow velocity of a fluid in an infinitely long pipe with small diameter a, b and k are real numbers with k > 0 and m is a positive integer Corresponding authormohamed Zellal mzellal@yahoofr 200 Mathematics Subject Classification:5K5, 5A22, 47j0, 49M27

2 Mohamed Zellal et al 2 Analysis of Homotopy perturbation transform method HPTM To illustrate the basic idea of HPTM [7], we consider a general nonlinear partial differential equation with the initial conditions of the form Dux, t + Rux, t + Nux, t = gx, t, ux, 0 = hx, u t x, 0 = fx, 2 where D is the second order linear differential operator D = 2 / t 2, R is the linear differential operator of less order than D, N represents the general non-linear differential operator and gx, t is the source term Taking the Laplace transform denoted throughout this paper by L on both sides of Eq 2: L[Dux, t] + L[Rux, t] + L[Nux, t] = L[gx, t] Using the differentiation property of the Laplace transform, we have L[ux, t] = hx + fx s s 2 s 2 L[Rux, t] + s 2 L[gx, t] L[Nux, t] 4 s2 Operating with the Laplace inverse on both sides of Eq 4 gives ux, t = Gx, t L s 2 L [Rux, t + Nux, t], 5 where Gx, trepresents the term arising from the source term and the prescribed initial conditions Now, we apply the homotopy perturbation method and the nonlinear term can be decomposed as ux, t = Nux, t = for some He s polynomials H n u see[]-[4] that are given by [ H n u 0,, u n = n + ] n! p n N p i u i i=0 p n u n x, t, p n H n u, 7 p=0, n = 0,, 2,, Substituting Eqs and 7 in Eq 5 we get [ [ + p n u n x, t = Gx, t p L s 2 L R p n u n x, t + p n H n u 8 ]], 9 which is the coupling of the Laplace transform and the homotopy perturbation method using He s polynomials Comparing the coefficient of like powers of p, the following approximations are obtained p 0 : u 0 x, t = Gx, t, 0 p : u x, t = L s 2 L[Ru 0x, t + H 0 u], p 2 : u 2 x, t = L s 2 L[Ru x, t + H u], p : u x, t = L s 2 L[Ru 2x, t + H 2 u],

3 Applications of HPTM for Solving Newell-Whitehead-Segel Equation 7 Then the solution is ux, t = lim p i=0 p i u i x, t = u 0 x, t + u x, t + u 2 x, t + For later numerical computation, the expression of ψ n for n below denotes the n - term approximation to ux, t n ψ n x, t = u i x, t 2 Absolute error = ux, t ψ n x, t Remark: Consider the following linear operators and their invers operators i=0 L t = t, L xx = x 2, [ n! L s L tn = L L [ s L ] Numerical results = t 0 s n+2 ds 2 u n x 2 = u n xx, ] = tn+, for n = 0,, 2,, n + To illustrate the accuracy and the effectiveness of the present method, several test examples are considered in this section Example : Consider NWS equation if k = 5, a = 2, b = and m = 2 : subject to the initial condition ux, t t = 5 2 ux, t x 2 + 2ux, t + u 2 x, t, ux, 0 = α, α R 4 Taking Laplace transform both of sides, subject to the initial condition we have L[ux, t] = α s + s L[5u xx + 2u + u 2 ] 5 Taking Inverse Laplace transform we get ux, t = α + pl s L[5u xx + 2u + u 2 ] By homotopy perturbation method, we get ux, t = Substitute Eq 7 in Eq, we get p n u n x, t 7 p n u n x, t = α + pl s L 5 pn u n x, t xx +2 pn u n x, t + pn H n u 8

4 8 Mohamed Zellal et al Where H n are He s polynomials that represents the nonlinear terms The first few components of He s polynomials are given by H 0 u = u 2 0 H u = 2u 0 u H 2 u = 2u 0 u 2 + u 2 H u = 2u 0 u + 2u u 2 By comparing the coefficient of like powers of p, we have: p 0 : u 0 x, t = α, p : u x, t = L s L [5 u 0 xx + 2u 0 + H 0 u] = α α + 2 t, p 2 : u 2 x, t = L s L [5 u xx + 2u + H u] = αα + 2α + t 2, p : u x, t = L s L [5 u 2 xx + 2u 2 + H 2 u] = 2αα + 2α + 2 t, p 4 : u 4 x, t = L s L [5 u xx + 2u + H u] = αα + 2α + 5α 2 + 0α + 2 t4 the approximate solution of equation -4 is given as: ux, t = α α + 2 t + αα + 2α + t 2 + 2αα + 2α + 2 t +αα + 2α + 5α 2 + 0α + 2 t4 + the closed form solution will be as follows: ux, t = 2αe 2t 2 + α e 2t 9 which is an exact solution and is same as obtained by [0] Example 2: Consider NWS equation if k =, a = 2, b = and m = 2: with the initial condition ux, t t = 2 ux, t x 2 + 2ux, t u 2 x, t, 20 ux, 0 = β, β R 2 As before, taking Inverse Laplace transform we find ux, t = β + pl s L[u xx + 2u u 2 ] 22

5 Applications of HPTM for Solving Newell-Whitehead-Segel Equation 9 By applying homotopy perturbation transform method, subject to initial condition we have p n u n x, t = β + pl s L pn u n x, t xx +2 pn u n x, t pn u n x, t 2 By comparing the coefficient of like powers of p, we get: p 0 : u 0 x, t = β, [ p : u x, t = L s L [ u 0 xx + 2u 0 u 2 ] ] 0 = β 2 β t, p 2 : u 2 x, t = L s L [u xx + 2u u 0 u ] = 2β 2 β β t2 2, [ p : u x, t = L s L [ u 2 xx + 2u 2 u 0 u 2 u 2 ] ] = 2β 2 β 27β 2 8β + 2 t!, Then the series solution is given by ux, t = β + β 2 β t + 2β 2 β β t2 2 +2β 2 β 27β 2 8β + 2 t! + = 2βe 2t 2 + β e 2t which is an exact solution and is the same as obtained by [9] and [] Example : Consider NWS equation if k =, a =, b = and m = 2 : ux, t t = 2 ux, t x 2 + ux, t u 2 x, t, 2 subject to the initial condition ux, 0 = + e x 2 24 By applying aforesaid method subject to initial condition we have p n u n x, t = + e x 2 + pl s L pn u n x, t xx + pn u n x, t pn u n x, t 2

6 40 Mohamed Zellal et al By comparing the coefficient of like powers of p, we have: p 0 : u 0 x, t = + e x 2, [ p : u x, t = L s L [ u 0 xx + u 0 u 2 ] ] 0 Proceeding in a similar manner, we obtain e x = 5 + e x t, p 2 : u 2 x, t = L s L [u xx + u 2u 0 u ] = 25 e x 2e x + e x 4 t 2, [ p : u x, t = L s L [ u 2 xx + u 2 2u 0 u 2 u 2 ] ] = e x 4e 2x 7e x + + e x 5 t, Then the series solution of equation 2-24 in closed form is given by ux, t = = e x + e x e x t e x 2e x + e x 4 t 2 e x 4e 2x 7e x + + e x 5 t + + e x 5t 2 As presented by [9] and []

7 Applications of HPTM for Solving Newell-Whitehead-Segel Equation 4 Table : Comparison of the exact solution and ψ 4 when t = 00 in Example x Exact value ψ 4 u ψ E E E E- 997E E E E E E E E E E-2 808E E E E E E E-2 Example 4: Consider linear Newell-Whitehead-Segel equation if k =, a =, b = and m = 4 : ux, t t = 2 ux, t x 2 + ux, t u 4 x, t, 25 subject to the initial condition ux, 0 = + e x By applying the present method HPTM subject to initial condition we have p n u n x, t = + e x 2 + pl s L 0 pn u n x, t xx + pn u n x, t pn H n u Where p n H n u = u 4, the first few components of He s polynomials are given by H 0 u = u 4 0 H u = 4u 0u H 2 u = 2u 2 02u 0 u 2 + u 2 + 4u 2 0u 2

8 42 Mohamed Zellal et al By comparing the coefficient of like powers of p, p 0 : u 0 x, t = + e x 2, 0 p : u x, t = L s L [u 0 xx + u 0 H 0 u] = 7 5 e x 0 + e x 5 0 t, p 2 : u 2 x, t = L s L [u xx + u H u] = e x 0 2e x 0 + e x 8 0 p : u x, t = L s L [u 2 xx + u 2 H 2 u] = 4 e x 0 4e x t 2, 27e x + e x 0 t, The other components of the HPTM can be determined in a similar way The 4-term approximate solution of equation 25-2 in is given as: ψ 4 x, t = + e x e x 0 2e x 0 + e x 8 0 e x 0 + e x e x 0 4e x x 0 27e e x t 0 t 2 t Hence the exact solution is given as; ux, t = { 2 tanh 2 0 x } 2 7t +, 0 2 which is in full agreement with [9] and []

9 Applications of HPTM for Solving Newell-Whitehead-Segel Equation 4 Table 2: Absolute Error u ψ 4 in Example 4 x u ψ 4 u ψ 4 u ψ 4 t = 0 t = 0 t = E E- 2455E E-4 452E- 4272E E E E E E- 4E E-4 52E E E E- 4409E-2 Example 5: Consider NWS equation if k =, a =, b = 4 and m = : ux, t t = 2 ux, t x 2 + ux, t 4u x, t, 27 subject to the initial condition e x ux, 0 = 4 e x + e 28 2 x According to Homotopy Perturbation Transform Method procedures, we have p n e x u n x, t = 4 e x + e + 2 x pl s L pn u n x, t xx + pn u n x, t 4 pn H n u Where p n H n u = u,

10 44 Mohamed Zellal et al Table : Comparison of the exact solution and ψ 4 for different time when x = in Example 5 t Exact value ψ 4 u ψ E- 9489E E E E E E- 2028E E E E E E E E E E-2 By comparing the coefficient of like powers of p, we find p 0 e x : u 0 x, t = 4 e x + e, 2 [ x p : u x, t = L s L [ u 0 xx + u 0 4u ] ] 0 = e x e 2 x e x 2 t, + e 2 x p 2 : u 2 x, t = L [ s L [ u xx + u 2u 2 0u ] ] = e x e 2 e x x + e e x t 2, + e 2 x 2 x [ p : u x, t = L s L [ u 2 xx + u 2 4u 0 2u 0 u 2 + u 2 8u 0 u 2 4u 2 u 2 ] ] 0 = e x e 4 Thus, the solution in series form is given by: 2 4e x x + e 2x + e x e x 4 t, + e 2 x ux, t = = e x 4 e x + e x e x e e x e 4 4 e x + e 4 e x e 2 x e x 2 t + e 2 x 2 e x x + e e x t 2 + e 2 x 2 x 2 4e x x + e 2x + e x e x 4 t + + e 2 x e x 2 x 9 2 t As presented by [9]

11 Applications of HPTM for Solving Newell-Whitehead-Segel Equation 45 It can be observed through tables - and figures -9 that this method is efficient and accurate for different values of time and place 4 Conclusion In this paper, the homotopy perturbation transforms method HPTM is successfully applied to solve Newell- Whitehead-Segel equations It is worth mentioning that the nonlinear terms can be easily handled by the use of Hes Polynomials which reduces the computational work as compared to other methods, without using discretization, linearization, or restrictive assumptions, while still maintaining the fast convergence and accuracy of this method Finally, we conclude that the HPTM is considered as a best and accurate method to finding analytic as well as numerical solutions for wider classes of linear and nonlinear differential equations ACKNOWLEDGEMENTS We thank the editor and the referee for their comments References [] J Biazar, H Ghazvini, Convergence of the homotopy perturbation method for partial differential equations, Nonlinear Analysis: Real World Applications, 0, 2009, pp2 240 [2] R Ezzati, K Shakibi, Using adomian s decomposition and multi-quadric quasi-interpolation methods for solving Newell Whitehead equation Procedia Computer Science,, 20, pp [] A Ghorbani, J Saberi-Nadja, perturbation method for calculating adomian polynomials, Inter- national Journal of Nonlinear Sciences and Numerical Simulation, 82007, pp [4] A Ghorbani adomians polynomials: He polynomials, Chaos Solitons Fractals, 92009, pp [5] J H He Homotopy perturbation technique Computer Methods in Applied Mechanics and Engineering, 78, 999, pp [] JH He Homotopy perturbation method: a new nonlinear analytical technique, App Math Comp, 5200, pp7 79 [7] Y Khan and Q Wu Homotopy perturbation transform method for nonlinear equations using He s polynomials Computer and Mathematics with Applications, Vol, No8, 20, pp9 97 [8] S A Manaa An Approximate solution to the Newell-Whitehead-Segel equation by the Adomian decomposition method Raf J Comp Math, Vol8, No, 20, pp7 80 [9] S S Nourazar, M Soori, A Nazari-Golshan On the exact solution of Newell-Whitehead-Segel equation using the homotopy perturbation method Journal of Applied Sciences Research, Vol5, No8, 20, pp400 4 [0] J Patade, S Bhalekar Approximate analytical solutions of Newell-Whitehead-Segel equation using a new iterative method World Journal of Modelling and Simulation, Vol, No2, 205, pp94 0 [] A Prakash, M Kumar He s Variational Iteration Method for the Solution of Nonlinear Newell-Whitehead-Segel Equation Journal of Applied Analysis and Computation, Vol, No, 20, pp [2] P Pue-on, Laplace Adomian Decomposition Method for Solving Newell- Whitehead-Segel Equation Applied Mathematical Sciences, Vol 7, No 2,

12 4 Mohamed Zellal et al [] A Saravanan, N Magesh A comparison between the reduced differential transform method and the Adomian decomposition method for the Newell-Whitehead-Segel equation Egyptian Math Soc, 2, 20, pp259 25