Solving Two Emden Fowler Type Equations of Third Order by the Variational Iteration Method

Transcription

1 Appl. Math. Inf. Sci. 9, No. 5, Applied Mathematics & Information Sciences An International Journal Solving Two Emden Fowler Type Equations of Third Order the Variational Iteration Method Abdul-Majid Wazwaz Department of Mathematics, Saint Xavier University, Chicago, IL 6655, U.S.A Received: 27 Jan. 215, Revised: 28 Apr. 215, Accepted: 29 Apr. 215 Published online: 1 Sep. 215 Abstract: In this paper, we establish two kinds of Emden Fowler type equations of third order. We investigate the linear and the nonlinear third-order equations, with specified initial conditions, using the systematic variational iteration method. We corroborate this study investigating several Emden Fowler type eamples with initial value conditions. Keywords: Emden Fowler equation; variational iteration method; Lagrange multipliers. 1 Introduction Many scientific applications in the literature of mathematical physics and fluid mechanics can be distinctively described the Emden Fowler equation y + k y + fgy=,y=y,y =, 1 where f and gy are some given functions of and y, respectively, and k is called the shape factor. The Emden Fowler equation 1 describes a variety of phenomena in fluid mechanics, relativistic mechanics, pattern formation, population evolution and in chemically reacting systems. For f = 1 and gy = y m, Eq. 1 becomes the standard Lane Emden equation of the first order and inde m, given y + k y + y m =,y= y,y =, 2 The Lane Emden equation 2 models the thermal behavior of a spherical cloud of gas acting under the mutual attraction of its molecules [1 12] and subject to the classical laws of thermodynamics. Moreover, the Lane Emden equation of first order is a useful equation in astrophysics for computing the structure of interiors of polytropic stars. On the other side, for f = 1 and gy = e y, Eq. 1 becomes the standard Lane Emden equation of the second order that models the non-dimensional density distribution y in an isothermal gas sphere [9 22] Moreover, the Lane Emden equation 2 describes the temperature variation of a spherical gas cloud under the mutual attraction of its molecules and subject to the laws of thermodynamics. In addition, the Lane-Emden equation of the first kind appears also in other contet such as in the case of radiatively cooling, self-gravitating gas clouds, in the mean-field treatment of a phase transition in critical adsorption and in the modeling of clusters of galaies [17 19]. The Lane-Emden equation was first studied astrophysicists Jonathan Homer Lane and Robert Emden, where they considered the thermal behavior of a spherical cloud of gas acting under the mutual attraction of its molecules and subject to the classical laws of thermodynamics. The well-known Lane-Emden equation has been used to model several phenomena in mathematical physics and astrophysics such as the theory of stellar structure, the thermal behavior of a spherical cloud of gas, isothermal gas spheres, the theory of thermionic currents, and in the modeling of clusters of galaies. The Emden Fowler equation was studied Fowler [2] to describe a variety of phenomena in fluid mechanics and relativistic mechanics among others. The singular behavior that occurs at = is the main difficulty of Equations 1 and 2. Corresponding author wazwaz@su.edu c 215 NSP

2 243 A. M. Wazwaz: Solving Two Emden Fowler Type Equations of Third... We note that 1 was derived using the equation k d k d y+ fgy=,y=y,y =, d d 3 where k is called the shape factor. The Lane Emden equation and the Emden Fowler equation were subjected to a considerable size of investigation, both numerically and analytically. A variety of useful methods were used to obtain eact and approimate solutions as well. Eamples of the methods that were applied are the Adomian decomposition method [6, 7, 18], the variational iteration method [8, 1, 14, 23], the homotopy perturbation method [22], the rational Legendre pseudospectral approach [17], and other methods as well [19, 2]. We aim in this work to establish two kinds of Emden Fowler type equations of third-order. Our approach depends mainly on using different orders of the differential operators involved in the Emden Fowler sense given in 3. Our net goal of this work is to apply the variational iteration method to handle the developed Emden Fowler type equations of third-order. Several numerical eamples, with specified initial conditions, of each model will be eamined to handle the singular point that eists in each model. 2 Constructing Emden Fowler type equations of third-order To derive the Emden-Fowler type equations of third-order, we use the sense of 3 and set k dm k dn d m d n y+ fgy=. 4 To determine third-order equations, it is clear that we should select m+n= 3,m,n 1 5 that leads to the following two choices and Substituting m=2,n=1 in 4 gives m=2,n=1, 6 m=1,n=2. 7 k d2 d 2 k d y+ fgy=. 8 d This in turn gives the first Emden Fowler type equation of third order in the form d 3 y + 2k d 2 y d 3 + kk 1 dy d 2 2 d + fgy=,y=y,y =y =, 9 or equivalently + 2k y + kk 1 y + fgy=,y=y 2,y =y =. 1 Notice that the singular point = appears twice as and 2 with shape factors 2k and kk 1, respectively. Moreover, the third term vanishes for k = 1 and the shape factor in this case reduces to 2. For f=1, Eqs. 1 becomes the Lane Emden type equation of the third-order given + 2k y + kk 1 y + gy=,y=y 2,y =y =. 11 In the other case, we substitute m = 1,n=2 in 4 to obtain k d k d2 d d 2 y+ fgy=. 12 This in turn gives the second kind of Emden Fowler type equation of third order of the form d 3 y d 3 + k d 2 y d 2 + fgy=,y= y,y =y =, 13 or equivalently + k y + fgy=,y=y,y =y =. 14 Unlike the first kind, the singular point in the second case =appears once with shape factor k. Moreover, the firstorder term y term vanishes in the second kind. For f=1, Eqs. 14 becomes the Lane Emden type equations of the third order given + k y + gy=,y=y,y =y = Analysis of the method In this section, we will present the analysis for the use of the variational iteration method VIM for a reliable treatment of the two models of the Emden Fowler type equations of third-order. The main focus will be, as will be seen later, on the derivation of a variety of Lagrange multipliers for each models and investigate several eamples as well. 3.1 The first model: the VIM and the Lagrange multipliers In this section we will present the essential steps for using the variational iteration method and the determination of the Lagrange multipliers for various values of k. Consider the differential equation Ly+Ny=ht, 16 c 215 NSP

3 Appl. Math. Inf. Sci. 9, No. 5, / where L and N are linear and nonlinear operators respectively, and ht is the source term. To use the VIM, a correction functional for equation 16 should be used in the form y n+1 =y n + λly n ξ+nỹ n ξ dξ, 17 where λ is a general Lagrange s multiplier, which can be identified optimally via the variational theory, and ỹ n is a restricted variation, which means δỹ n =. For Eq. 1 the correction functional reads y n+1 =y n + λ,t n t+ 2k t y n kk 1 t+ y n t+ ft gy nt 18 where δ gy n t=. To determine the optimal value of λ,t, we take the variation for both sides with respect to y n to obtain δy n+1 =δy n +δ λ,t nt+ 2k or equivalently t y nt+ kk 1 δy n+1 =δy n +δ λ,t nt+ 2k y nt+ ft gy n t 19 t y nt+ kk 1 y nt 2 where we used δ gy n t=. For illustrative purposes, we evaluate the integral at the right side in steps. We first integrate I 1 = λ,t n tdt. 21 Integrating I 1 parts three times gives I 1 =λ,ty n λ,ty n+ λ,ty n t= λ,ty n tdt. 22 We net integrate the second term of the integral in 2, therefore we set I 2 = λ,k t y nt dt. 23 Integrating I 2 parts twice gives I 2 = λ,k + t y n 2ktλ,t 2kλ,t y n t t= 2k λ,t 4ktλ,t+4kλ,t y t 3 n tdt. 24 We finally integrate the third term of the integral in 2, therefore we set I 3 = λ,t kk 1 y ntdt. 25 Integrating I 3 parts once gives I 3 = λ,t kk 1 y n t= kk 1tλ,t 2kk 1λ,t y t 3 n tdt. 26 In view of 21 26, Eq. 2 becomes δy n+1 =δy n +δ λ,ty n +[ λ,k t λ,t ] [ y n + λ,t 2k t λ,t+ 2k+kk 1 y n ] t= + λ,t+ 2k t λ,t 4k+kk 1 λ,t+ 4k+2kk 1 λ,t δy t n. 3 For arbitrary δy n, we obtain the stationary condition λ,t+ 2k t λ,t 4k+kk 1 and the boundary conditions 1+λ,t 2k t λ,t+ 2k+kk 1 27 λ,t+ 4k+2kk 1 λ,t= t 3 28 λ,t t= =, 2k t λ,t λ,t t= =, λ,t t= =. 29 To determine λ,t, three cases will be eamined: i For the general case where k 1,2: solving gives λ,t= k 1k 2 k 1 2 t k+1+ 2 t k. k 2 3 ii For k= 1: solving gives λ,t= t+ 1 ln t. 31 iii For k=2: solving gives λ,t= t3 + t2 1+ln t. 32 The successive approimations y n+1, for n, of the solution y will be readily obtained upon using any selective function y. Consequently, the solution y= lim n y n. 33 In other words, the correction functional 18 will give several approimations, and therefore the eact solution is obtained in the limit of the resulting successive approimations. 3.2 Numerical eamples for the first kind of the Emden Fowler type equations + 2k y + kk 1 2 y + fgy=,y=α,y =y = 34 We will study this equation for a variety of values for the shape factor k and for the given functions fgy. For comparison reasons we will solve the same eamples solved in [1] using the Adomian decomposition method. c 215 NSP

4 2432 A. M. Wazwaz: Solving Two Emden Fowler Type Equations of Third... Eample 1. We first consider the Emden Fowler type equation + 2 y y 5 =,y=1,y =y =, 35 obtained substituting k = 1 in 34 and setting fgy= y 5. From 31, the Lagrange multiplier for k = 1 is given λ,t= t+ 1 ln t. 36 The correction functional for 35 becomes y n+1 =y n + t+ 1 ln t n t+ 2 t y n t 9 8 t6 + 8y 5 n 37 where n. By selecting the zeroth approimation y = 1, approimations y = 1, y 1 = , y 2 = , y 3 = , y 4 = , To facilitate the computational work, we used few terms of the eponential y n t,n 1. This in turn gives the series solution y= , 38 that converges to the eact solution Eample 2. y= We net consider the linear Emden Fowler type equation + 4 y + 2 y y=,y=1,y =y =, 2 4 obtained substituting k = 2 in 34 and setting fgy= y. From 32, the Lagrange multiplier for k = 2 is given λ,t= t3 + t2 1+ln t. 41 The correction functional for 4 becomes y n+1 =y n + t3 + t2 1+ln t nt+ 4 t y nt+ 2 y t nt 94+1t 3 + 3t 6 y nt 2 42 for n. By selecting the zeroth approimation y = 1, we obtain the following calculated solution approimations y = 1, y 1 = , y 2 = , y 3 = , y 4 = ! ! 12 +, y= ! ! ! 12 +, 43 that converges to the eact solution Eample 3. y=e We now consider the nonlinear Emden Fowler type equation + 6 y + 6 y e 3y =,y=,y =y =, 2 45 obtained substituting k = 3 in 34 and setting fgy= e 3y. From 3, the Lagrange multiplier for k = 3 is given λ,t= t t 4+ 2 t The correction functional for 45 becomes y n+1 =y n t 4+ 2 t 3 nt+ 6 t y nt+ 6 y nt 61+2t 3 + t 6 e 3ynt 47 for n. By selecting the zeroth approimation y =, approimations y =, y 1 = , y 2 = , y 3 = , y 4 = , y= , 48 c 215 NSP

5 Appl. Math. Inf. Sci. 9, No. 5, / that converges to the eact solution Eample 4. y=ln We conclude this section considering the Lane Emden type equation + 8 y y + y m =,y=1,y =y =, 5 obtained substituting k= 4 in 34 and setting gy= y m, f=1. From 3, the Lagrange multiplier for k = 4 is given λ,t= 1 6 t t t The correction functional for 5 becomes y n+1 =y n t t t 4 n t+ 8 t y 12 n t+ y m n t 52 for n. By selecting the zeroth approimation y = 1, approimations y = 1, y 1 = , y 2 = m , y 3 = m m17m , y 4 = 1 1 m m17m m679m2 1182m , y= m m17m m679m2 1182m Substituting m = gives the eact solution as y= The second model: the VIM and the Lagrange multipliers In this section we will follow the analysis presented for the first kind. For Eq. 14 the correction functional reads y n+1 =y n + λ,t nt+ k t y nt+ ft gy n t 55 where δ gy n t=. To determine the optimal value of λ,t, we take the variation for both sides with respect to y n to obtain δy n+1 =δy n +δ λ,t n t+ k t y n t+ ft gy nt 56 or equivalently δy n+1 =δy n +δ λ,t n t+ k t y n t 57 where we used δ gy n t=. Proceeding as before, we evaluate the integral at the right side in steps. We first integrate I 1 = λ,t n tdt. 58 Integrating I 1 parts three times gives I 1 =λ,ty n λ,ty n + λ,ty n t= λ,ty n tdt. 59 We net integrate the second term of the integral in 57, therefore we set I 2 = λ,t k t y nt dt. 6 Integrating I 2 parts twice gives I 2 = λ,t k t y n ktλ,t kλ,t y n t t= + k λ,t 2ktλ,t+2kλ,t y t 3 n tdt. In view of 58 61, Eq. 57 becomes 61 δy n+1 =δy n [ [ +δ λ,ty n + λ,t k = λ,t ]y n + λ,t k t λ,t+ k y n ] t= + λ,t+ k t λ,t 2k λ,t+ 2k λ,t δy t 3 n. For arbitrary δy n, we obtain the stationary condition 62 λ,t+ k t λ,t 2k λ,t+ 2k λ,t= 63 t3 and the boundary conditions λ,t t= =, k λ,t+λ,t t= =, 1+λ,t k t λ,t+ k λ,t t= =. 64 To determine λ,t, three cases will be eamined: i For the general case where k 1,2: solving gives λ,t= t2 k 2 t k 1 k 1k 2 2 t k. 65 ii For k= 1: solving gives λ,t= + t 1+ln t iii For k=2: solving gives λ,t= t+ 1 ln t c 215 NSP

6 2434 A. M. Wazwaz: Solving Two Emden Fowler Type Equations of Third... The successive approimations y n+1, for n, of the solution y will be readily obtained upon using any selective function y. Consequently, the solution y= lim n y n. 68 In other words, the correction functional 56 will give several approimations, and therefore the eact solution is obtained in the limit of the resulting successive approimations. 3.4 Numerical eamples for the second kind of the Emden Fowler type equations In this section, we study several numerical eamples for the second case of the Emden Fowler type equations of the third-order in the form + k y + fgy=,y=α,y =y = 69 We will study this equation for a variety of values for the shape factor k and for the given functions fgy. Eample 1. We first consider the nonlinear Emden Fowler type equation + 1 y e 3y =,y=,y =y =, 7 obtained substituting k = 1 in 69 and setting fgy= e 3y. From 66, the Lagrange multiplier for k = 1 is given λ,t= + t 1+ln t. 71 The correction functional for 7 becomes y n+1 =y n + + t 1+ln t n t+ 1 t y n t+4t9+22t4 + t 8 e 3ynt 72 for n. By selecting the zeroth approimation y =, approimations y =, y 1 = , y 2 = , y 3 = , y 4 = , To facilitate the computational work, we used few terms of the eponential e 3y nt,n 1. This in turn gives the series solution y= , 73 that converges to the eact solution Eample 2. y=ln We net consider the nonlinear Emden Fowler type equation + 2 y + 2 y y 7 =,y=1,y =y =, 75 obtained substituting k = 2 in 69 and setting fgy= y 7. From 67, the Lagrange multiplier for k = 2 is given λ,t= t+ 1 ln t. 76 The correction functional for 75 becomes y n+1 =y n + t+ 1 ln t n t+ 2 t y n t 25 8 t t 5 + 7t 1 y 7 n t 77 for n. By selecting the zeroth approimation y = 1, approimations y = 1, y 1 = , y 2 = , y 3 = , y= , 78 that converges to the eact solution Eample 3. y= We now consider the nonlinear Emden Fowler type equation + 3 y y y 8 =,y=1,y =y =, 8 c 215 NSP

7 Appl. Math. Inf. Sci. 9, No. 5, / obtained substituting k = 3 in 69 and setting fgy=24+ 3 y 8. From 65, the Lagrange multiplier for k = 3 is given λ,t= 1 2 t t The correction functional for 8 becomes y n+1 =y n t t 3 n t+ 3 t y n t+24+ t3 y 8 n 82 for n. By selecting the zeroth approimation y = 1, approimations y = 1, y 1 = , y 2 = , y 3 = , y= , 83 that converges to the eact solution Eample 4. y= We conclude this section considering the linear Emden Fowler type equation + 4 y y=,y=1,y =y =, 85 obtained substituting k = 4 in 69 and setting fgy= y. From 65, the Lagrange multiplier for k = 4 is given λ,t= 1 2 t2 1 3 t t The correction functional for 85 becomes y n+1 =y n t2 1 3 t t 4 n t+ 4 t y n t 1+1t3 + t 6 y n t 87 for n. By selecting the zeroth approimation y = 1, approimations y = 1, y 1 = , y 2 = , y 3 = , y= , 88 that gives the eact solution 4 Conclusion y=e In this work, we have presented a framework to establish two kinds of Emden Fowler type equations of third-order. Unlike the standard Emden Fowler equations where the shape factor is unique, we showed that there are more than one shape factor for equations of order greater than or equal to 3 for one case and only one shape factor for another case. Similarly, the singular point appears once in the standard form, whereas in the established cases, the singular point = may appear twice. We used the variational iteration method for treating linear and nonlinear problems to illustrate our analysis. A variety of Lagrange multipliers was derived where the shape factor s play a major role in its determination. The calculated results from the recursion scheme are effective for all shape factor values k greater than or equal to 1. The obtained results validate the reliability and rapid convergence of the VIM. References [1] I. J. H. Lane, On the theoretical temperature of the sun under the hypothesis of a gaseous mass maintaining its volume its internal heat and depending on the laws of gases known to terrestial eperiment,american Journal of Science and Arts, [2] R. H. Fowler, Further studies of Emdens and similar differential equations, Quarterly Journal of Mathematics, [3] A. M. Wazwaz, R. Rach, L. Bougoffa, J.-S. Duan, Solving the Lane Emden Fowler type equations of higher orders the Adomian decomposition method,, In Press. [4] A.M. Wazwaz, Analytical solution for the time dependent Emden Fowler type of equations Adomian decomposition method, Appl. Math. Comput c 215 NSP

8 2436 A. M. Wazwaz: Solving Two Emden Fowler Type Equations of Third... [5] A.M. Wazwaz, A new algorithm for solving differential equations of Lane Emden type, Appl. Math. Comput [6] A.M. Wazwaz, A new method for solving singular initial value problems in the second-order ordinary differential equations, Appl. Math. Comput [7] A.M. Wazwaz, Adomian decomposition method for a reliable treatment of the Emden Fowler equation, Appl. Math. Comput [8] A.M. Wazwaz, The variational iteration method for solving nonlinear singular boundary value problems arising in various physical models, Commun. Nonlinear Sci. Numer. Simulat [9] A.M.Wazwaz, Partial Differential Equations and Solitary Waves Theory, HEP and Springer, Beijing and Berlin, 29. [1] J.H.He, Variational iteration method for autonomous ordinary differential systems, Appl. math. Comput., 1142/ [11] J.H. He, Variational iteration method - Some recent results and new interpretations, J. Comput. Appl. Math., [12] C.M. Khalique, F. M. Mahomed, B. Muatjetjeja, Lagrangian formulation of a generalized Lane Emden equation and double reduction, J. Nonlinear Math. Phys [13] B. Muatjetjeja, C.M. Khalique, Eact solutions of the generalized Lane Emden equations of the first and second kind, Pramana [14] M. Dehghan, F. Shakeri, Approimate solution of a differential equation arising in astrophysics using the variational iteration method, New Astronomy [15] O.W. Richardson, The Emission of Electricity from Hot Bodies, Longmans Green and Co. London, [16] K. Parand, M. Dehghan, A.R. Rezaeia, S.M. Ghaderi, An approimation algorithm for the solution of the nonlinear Lane Emden type equations arising in astrophysics using Hermite functions collocation method, Comput. Phys. Commun [17] K. Parand, M. Shahini, M. Dehghan, Rational Legendre pseudospectral approach for solving nonlinear differential equations of Lane Emden type, J. Comput. Phys [18] G. Adomian, R. Rach, N.T. Shawagfeh, On the analytic solution of the Lane Emden equation, Found. Phys. Lett [19] H.T. Davis, Introduction to Nonlinear Differential and Integral Equations, Dover Publications, New York, [2] S. Chandrasekhar, An Introduction to the Study of Stellar Structure, Dover Publications, New York, [21] E. Momoniat, C. Harley, Approimate implicit solution of a Lane Emden equation, New Astronomy [22] A. Yildirim, T. Özis, Solutions of singular IVPs of Lane Emden type homotopy perturbation method, Phys. Lett. A [23] A. Yildirim, T. Özis, Solutions of singular IVPs of Lane Emden type the variational iteration method, Nonlinear Analysis Abdul-Majid Wazwaz is a Professor of Mathematics at Saint Xavier University in Chicago, Illinois, USA. He has both authored and co-authored more than 4 papers in applied mathematics and mathematical physics. He is the author of five books on the subjects of discrete mathematics, integral equations and partial differential equations. Furthermore, he has contributed etensively to theoretical advances in solitary waves theory, the Adomian decomposition method and the variational iteration method.. c 215 NSP