Solution of Differential Equations of Lane-Emden Type by Combining Integral Transform and Variational Iteration Method

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1 Nonlinear Analysis and Differential Equations, Vol. 4, 2016, no. 3, HIKARI Ltd, Solution of Differential Equations of Lane-Emden Type by Combining Integral Transform and Variational Iteration Method N. H. Zomot and O. Y. Ababneh Department of Mathematics, Faculty of Science Zarqa University, Zarqa, Jordan Copyright c 2016 N. H. Zomot and O. Y. Ababneh. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we present a reliable combined the modified Sumudu transform and the new modified variational iteration method to solve some nonlinear differential equations of Lane-Emden type. The results of these equations have been find in terms of convergent series with computable components. The nonlinear terms can be handled by using the variational iteration method. This method is efficient and easy to handle such nonlinear differential equations. Keywords: equation modified Sumudu transform, Lane-Emden type, nonlinear 1 Introduction Many problems in the literature of mathematical physics can be formulated as equations of Lane-Emden type [1, 2] defined in the form y + 2 x y + f(y) = 0, (1) where f(y) is some given function of y. A pivotal amount of work has been done on this type of problems for various of f(y). Many methods have been used to solve Lane-Emden initial value problem. For instance, Ramos [4] presented

2 144 N. H. Zomot and O. Y. Ababneh a series approach to the Lane-Emden equation and made comparisons with homotopy perturbation method. Dehghan and Shakeri [5] apply exponential transformation to the Lane-Emden equation in order to address the difficulty of a singular point at x = 0 and solve the resulting nonsingular problem using the variational iteration method. Momoniat and Harley [6] get an approximate implicit solution by reducing the Lane-Emden equation to a first-order differential equation using Lie group analysis and get a power series solution of the reduced equation. Approximate solutions of Lane-Emden type equations using the Adomian decomposition method [7, 8, 9]. Recently, Yang and Hou [10] proposed an approximation algorithm for the solution of a Lane-Emden type equation based on hybrid functions and the collocation method. In this work, we will use the new modified form of the variational iteration method, Elzaki variational iteration method, and this method provides an effective way of solving Lane-Emden equations. We present various forms of f(y) that have attracted the attention of the scientific community, due to their significant applications. The Modification of Sumudu Transform and variational iteration method The basic definition of the modification of Sumudu Transform is given as below: the transform of the function f(t) is E[f(t)] = v f(t)e t/v dt, t > 0. (2) 0 Tarig M. Elzaki and Sailh M. Elzaki in [15, 16, 17], showed the modified of Sumudu transform [19] or Elzaki transform was applied to partial differential equations, ordinary differential equations, system of ordinary and partial differential equations and integral equations. More details can be found in [11]. let us consider the following general differential equation L[u(x, t)] + N[u(x, t)] = g(x, t), u(x, 0) = h(x), (3) where L is a linear operator of the first order,n is a nonlinear operator and g(x; t) is inhomogeneous term. According to the variational iteration method, we can construct a correction functional as u n+1 = u n + t 0 λ [Lu n (x, s) + Nũ n (x, s) g(x, s)]ds, (4) where λ is a Lagrange multiplier(λ ), the subscript n denotes the n- the approximation,ũ n is considered as a restricted variation, i.e. δũ n = 0.

3 Solution of differential equations of Lane-Emden type 145 The successive approximation u n+1 of the solution u will be readily obtained upon using the determined Lagrange multiplier and any selective function u 0. Consequently, the solution is given by u = lim u n. In this paper, we assume n that L is an operator of the first order in the equation 3. Let us take t the modified Sumudu transform of both sides and apply the differentiation property of new transform. Then we get and E [Lu(x, t)] + E [Nu(x, t)] = E[g(x, t)] (5) E [u(x, t)] = ve [g(x, t)] + vh(x) ve[nu(x, t)]. (6) Applying the inverse of modified Sumudu transform of both sides of the equation, we have u(x, t) = G(x, t) E 1 {ve [Nu(x, t)]}, (7) where G(x; t) represents the terms arising from the source term and the prescribed initial condition. Taking the first partial derivative with respect to t, we have Or alternately u t (x, t) t G(x, t) + t E 1 {ve [Nu(x, t)]} = 0. (8) u n+1 (x, t) = G(x, t) E 1 {ve [Nu n (x, t)]}. (9) Thus, we can obtain the solution u by Illustrative examples u(x, t) = lim n u n (x, t). (10) In this section, we solve some examples of nonlinear differential equations of Lane-Emden type by using the modified Sumudu transform variational iteration method. Example 1: (The isothermal gas spheres equation) The isothermal gas spheres equation is y + 2 x y + e y = 0 (11) subject to the boundary conditions y(0) = 0, y (0) = 0, This model can be used to view the isothermal gas spheres, where the temperature remains constant.

4 146 N. H. Zomot and O. Y. Ababneh For a thorough discussion of the formulation of Eq.11, see [2]. A series solution of Eq.(27) is obtained by Wazwaz [18] as y(x) = 1 6 x x x x8... (12) Now, taking the transform on the given equation, we have E[xy ] + E[2y ] + E[xe y ] = 0 (13) E[xy ] = u 2 d [ [ ] T T (0)] du u y(0) 2 uy u u y(0) 2 uy (0) (14) = u 2 [ T u 2 2uT u 4 ] T u = T 3T u. E[2y ] = 2T u (15) Subtitute (14) and (15) in (13) to get T T u + E[xey ] = 0. (16) Multiplying equation (16) by u 2 and taking the inverse to obtain E [ 1 u 2 T ut ] + E [ 1 u 2 E[xe y ] ] = 0 xy(x) + E 1 [ u 2 E[xe y ] ] = 0 y(x) x E 1 [ u 2 E[xe y ] ]. According to the equation (9), the correction function is given by y n+1 x E 1 [ u 2 E[xe yn ] ]. Now let us apply the modified Sumudu transform variational iteration method, the solution in series is given by y 0 = 0, y 1 x E 1 [u 2 E[x]] x E 1 [u 2 u 3 ] x x 3 3! 6 x2.

5 Solution of differential equations of Lane-Emden type 147 y 2 [ x E 1 u 2 E[xe y 1 ] ] [ ] x E 1 u 2 E[xe 1 6 x2 ] [ ( x E 1 u 2 E[x x2 + 1 ) ] 72 x4 x ] x [u5 u u u ] = x2 3! + x4 5! 5x6 3 7! 5 7x ! +... Continue this process, and we can obtain the solution y(x) = 1 6 x x x x8... (17) which is the exact solution. Gupta [12], Rafig et al. [13], Yildirim et al. [14] obtained the same result by the variation iteration method and the homotopy perturbation method. Example 2: The Emden-Fowler type equations y + 2 x y + sin(y) = 0, (x > 0) (18) subject to the boundary conditions y(0) = 1, y (0) = 0. The series solution of Eq.(18) is obtained by Wazwaz [18] by using ADM is: y(x) = 1 k 1 6 x2 + k ( ) ( ) 1k 2 k x4 + k k2 2 x 6 + k 1 k 2 113k k2 2 x (19) Now, taking the transform on the given equation, we have Therefore, we get E[xy ] + E[2y ] + E[sin(y)] = 0. (20) T T u + E[xsin(y)] = 0. (21) Multiplying equation (21) by u 2 and taking the inverse to obtain xy(x) + E 1 [ u 2 E[xsin(y)] ] = 0 y(x) x E 1 [ u 2 E[xsin(y)] ].

6 148 N. H. Zomot and O. Y. Ababneh According to the equation (9), the correction function is given by y n+1 x E 1 [ u 2 E[xsin(y n )] ]. Now let us apply the modified Sumudu transform variational iteration method, the solution in series is given by y 0 = 1, y 1 y 2 x E 1 [u 2 E[x sin(1)] = k 1 6 x2 where k 1 = sin(1). x E 1 [u 2 E[x sin ( k 1 6 x2) ]]. Continue this process, we obtain the solution y(x) = 1 k 1 6 x2 + k ( ) ( ) 1k 2 k x4 + k k2 2 x 6 + k 1 k 2 113k k2 2 x (22) which present the exact solution. Example 3: The Emden-Fowler type equations y + 2 x y + sinh(y) = 0, (x > 0) (23) subject to the boundary conditions y(0) = 1, y (0) = 0. The Series solution of Eq.(23) is obtained by Wazwaz [18] by y(x) = 1 e e x2 + e e 2 x4 2e6 + 3e 2 + 3e e 3 x Now, taking the transform on the given equation, we have E[xy ] + E[2y ] + E[sinh(y)] = 0 (24) then we follow the same procedure in previous examples to get y n+1 x E 1 [ u 2 E[xsinh(y n )] ]. and apply the modified Sumudu transform variational iteration method, to get y 0 = 1, y 1 y 2 x E 1 [u 2 E[x sinh(1)] = k 1 6 x2 = e2 1 12e x2 = k 1 x 2. x E 1 [u 2 E[x sinh (k 1 x 2 )]]. Continue this process, we obtain the solution y(x) = 1 e e x2 + e e 2 x4 2e6 + 3e 2 + 3e e 3 x which present the exact solution.

7 Solution of differential equations of Lane-Emden type 149 Conclusion In this study, the modified Sumudu transform variational iteration method is successfully applied to obtain analytical solutions of nonlinear singular initial value problems of Emden-Fowler Type equations. We can conclude that method is efficient in finding the analytical solution for a wide class of Emden- Fowler Type problems. Acknowledgements. This research is funded by the Deanship of Research in Zarqa University, Jordan. References [1] H.T. Davis, Introduction to Nonlinear Dierential and Integral Equations, Dover, New York, [2] S. Chandrasekhar, Introduction to the Study of Stellar Structure, Dover, New York, [3] O.W. Richardson, The Emission of Electricity from Hot Bodies, Longmans, Green and Co., London, [4] J.I. Ramos, Series approach to the Lane-Emden equation and comparison with the homotopy perturbation method, Chaos Solitons Fractals, 38 (2008), no. 2, [5] M. Dehghan, F. Shakeri, Approximate solution of a differential equation arising in astrophysics using the variational iteration method, New Astron., 13 (2008), no. 1, [6] E. Momoniat, C. Harley, Approximate implicit solution of a Lane-Emden equation, New Astron., 11 (2006), [7] N.T. Shawagfeh, Nonperturbative approximate solution for Lane-Emden equation, J. Math. Phys., 34 (1993), no. 9, [8] A.M. Wazwaz, Analytical solution for the time-dependent Emden-Fowler type of equations by Adomian decomposition method, Appl. Math. Comput., 166 (2005),

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