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

Home , Jonathan Homer Lane

Appl. Math. Inf. Sci. 9, No. 5, 2429-2436 (2015) 2429 Applied Mathematics & Information Sciences An International Journal

http://dx.doi.org/10.12785/amis/090526

Abdul-Majid Wazwaz∗ Department of Mathematics, Saint Xavier University, Chicago, IL 60655, U.S.A

Received: 27 Jan. 2015, Revised: 28 Apr. 2015, Accepted: 29 Apr. 2015 Published online: 1 Sep. 2015

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, by using the systematic variational iteration method. We corroborate this study by investigating several Emden–Fowler type examples with initial value conditions.

Keywords: Emden–Fowler equation; variational iteration method; Lagrange multipliers.

1 Introduction isothermal gas sphere [9–22] Moreover, the Lane–Emden equation (2) describes the temperature variation of a Many scientific applications in the literature of spherical gas cloud under the mutual attraction of its mathematical physics and fluid mechanics can be molecules and subject to the laws of thermodynamics. In distinctively described by the Emden–Fowler equation addition, the Lane-Emden equation of the first kind appears also in other context such as in the case of k radiatively cooling, self-gravitating gas clouds, in the y + y + f (x)g(y)= 0,y(0)= y ,y (0)= 0, (1) ′′ x ′ 0 ′ mean-field treatment of a phase transition in critical adsorption and in the modeling of clusters of where f (x) and g(y) are some given functions of x and y, galaxies [17–19]. 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 (x) = 1 and g(y) = ym, Eq. (1) becomes the standard Lane–Emden equation of the first order and The Lane-Emden equation was first studied by index m, given by astrophysicists Jonathan Homer Lane and Robert Emden, where they considered the thermal behavior of a spherical k cloud of gas acting under the mutual attraction of its y + y + ym = 0,y(0)= y ,y (0)= 0, (2) ′′ x ′ 0 ′ molecules and subject to the classical laws of thermodynamics. The well-known Lane-Emden equation The Lane–Emden equation (2) models the thermal has been used to model several phenomena in behavior of a spherical cloud of gas acting under the mathematical physics and astrophysics such as the theory mutual attraction of its molecules [1–12] and subject to of stellar structure, the thermal behavior of a spherical the classical laws of thermodynamics. Moreover, the cloud of gas, isothermal gas spheres, the theory of Lane–Emden equation of first order is a useful equation in thermionic currents, and in the modeling of clusters of astrophysics for computing the structure of interiors of galaxies. The Emden–Fowler equation was studied by polytropic stars. On the other side, for f (x) = 1 and Fowler [2] to describe a variety of phenomena in fluid g(y) = ey, Eq. (1) becomes the standard Lane–Emden mechanics and relativistic mechanics among others. The equation of the second order that models the singular behavior that occurs at x = 0 is the main non-dimensional density distribution y(x) in an difficulty of Equations (1)and (2).

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We note that (1) was derived by using the equation or equivalently

2k k(k 1) y′′′ y′′ − y′ f x g y 0,y 0 y ,y 0 y 0 0. k d k d + x + x2 + ( ) ( )= ( )= 0 ′( )= ′′( )= x− x y + f (x)g(y)= 0,y(0)= y0,y′(0)= 0, dx dx (10) (3) Notice that the singular point x = 0 appears twice as x and x2 with shape factors 2k and k(k 1), respectively. where k is called the shape factor. − The Lane–Emden equation and the Emden–Fowler Moreover, the third term vanishes for k = 1 and the shape equation were subjected to a considerable size of factor in this case reduces to 2. investigation, both numerically and analytically. A variety For f (x)= 1, Eqs.(10) becomes the Lane–Emden type of useful methods were used to obtain exact and equation of the third-order given by approximate solutions as well. Examples of the methods that were applied are the Adomian decomposition k(k 1) ′′′ 2k ′′ ′ , , . y + x y + x−2 y + g(y)= 0 y(0)= y0 y′(0)= y′′(0)= 0 method [6, 7, 18], the variational iteration (11) method [8, 10, 14, 23], the homotopy perturbation In the other case, we substitute m = 1,n = 2in(4) to method [22], the rational Legendre pseudospectral obtain approach [17], and other methods as well [19, 20]. 2 k d k d We aim in this work to establish two kinds of x− x y + f (x)g(y)= 0. (12) dx dx2 Emden–Fowler type equations of third-order. Our approach depends mainly on using different orders of the This in turn gives the second kind of Emden–Fowler type differential operators involved in the Emden–Fowler equation of third order of the form sense given in (3). Our next goal of this work is to apply d3y k d2y the variational iteration method to handle the developed 3 + 2 + f (x)g(y)= 0,y(0)= y0,y′(0)= y′′(0)= 0, Emden–Fowler type equations of third-order. Several dx x dx (13) numerical examples, with specified initial conditions, of or equivalently each model will be examined to handle the singular point that exists in each model. k y′′′ + y′′ + f (x)g(y)= 0,y(0)= y ,y (0)= y (0)= 0. x 0 ′ ′′ (14) 2 Constructing Emden–Fowler type Unlike the first kind, the singular point in the second case x = 0 appears once with shape factor k. Moreover, the first- equations of third-order order term y′(x) term vanishes in the second kind. For f (x)= 1, Eqs.(14) becomes the Lane–Emden type To derive the Emden-Fowler type equations of third-order, equations of the third–order given by we use the sense of (3) and set

m n k k d k d ′′′ ′′ , , . x− x y + f (x)g(y)= 0. (4) y + y + g(y)= 0 y(0)= y0 y′(0)= y′′(0)= 0 (15) dxm dxn x

To determine third-order equations, it is clear that we should select 3 Analysis of the method m + n = 3,m,n 1 (5) ≥ In this section, we will present the analysis for the use of that leads to the following two choices the variational iteration method (VIM) for a reliable treatment of the two models of the Emden–Fowler type m = 2,n = 1, (6) equations of third-order. The main focus will be, as will be seen later, on the derivation of a variety of Lagrange and multipliers for each models and investigate several m = 1,n = 2. (7) examples as well. Substituting m = 2,n = 1in(4) gives

2 3.1 The ﬁrst model: the VIM and the Lagrange k d k d x− x y + f (x)g(y)= 0. (8) multipliers dx2 dx In this section we will present the essential steps for using This in turn gives the ﬁrst Emden–Fowler type equation of the variational iteration method and the determination of third order in the form the Lagrange multipliers for various values of k. Consider the differential equation d3y 2k d2y k(k 1) dy − f x g y 0,y 0 y ,y 0 y 0 0, dx3 + x dx2 + x2 dx + ( ) ( )= ( )= 0 ′( )= ′′( )= (9) Ly + Ny = h(t), (16)

c 2015 NSP Natural Sciences Publishing Cor. Appl. Math. Inf. Sci. 9, No. 5, 2429-2436 (2015) / www.naturalspublishing.com/Journals.asp 2431

where L and N are linear and nonlinear operators In view of (21)–(26), Eq. (20) becomes respectively, and h(t) is the source term. δyn+1(x)= δyn(x) To use the VIM, a correction functional for equation 2k 2k 2k+k(k 1) +δ λ(x,t)y + λ(x,t) λ (x,t) y + λ (x,t) λ (x,t)+ − )yn t=x (16) should be used in the form n′′ t − ′ n′ ′′ − t ′ t2 | x 2k 4k+k(k 1) h 4k+2k(k 1) i + λ (x,t)+ λ (x,t) − λ (x,t)+ − λ(x,t) δyn. 0 − ′′′ t ′′ − t2 ′ t3 x R (27) λ ξ ξ ξ yn+1(x)= yn(x)+ (Lyn( )+ N ˜yn( )) d , (17) For arbitrary δy , we obtain the stationary condition Z0 n λ 2k λ 4k+k(k 1) λ 4k+2k(k 1) λ where λ is a general Lagrange’s multiplier, which can be ′′′(x,t)+ ′′(x,t) 2 − ′(x,t)+ 3 − (x,t)= 0 − t − t t identiﬁed optimally via the variational theory, and ˜yn is a (28) δ restricted variation, which means ˜yn = 0. and the boundary conditions For Eq. (10) the correction functional reads λ(x,t) t=x = 0, | x 2k k(k 1) 2k λ λ y (x)= y (x)+ λ(x,t) y′′′ (t)+ y′′ (t)+ − y′ (t)+ f (t)˜g(y (t)) dt, (x,t) ′(x,t) t=x = 0, n+1 n 0 n t n t2 n n t − | R λ 2k λ 2k+k(k 1) λ (18) 1 + ′′(x,t) ′(x,t)+ 2 − (x,t) t=x = 0. δ − t t | where (˜g(yn(t)))= 0. (29) λ To determine the optimal value of (x,t), we take the To determine λ(x,t), three cases will be examined: variation for both sides with respect to yn(x) to obtain (i) For the general case where k = 1,2: solving (28)–(29) gives 6 x k(k 1) δ δ δ λ , ′′′ 2k ′′ ′ , yn+1(x)= yn(x)+ 0 (x t) yn (t)+ t yn(t)+ t−2 yn(t)+ f (t)˜g(yn(t)) dt R (19) 2 2 k+1 2 k λ t x t x t or equivalently (x,t)= + . −(k 1)(k 2) − k 1 x k 2 x − − − − (30) x k(k 1) δ δ δ λ , ′′′ 2k ′′ ′ , yn+1(x)= yn(x)+ 0 (x t) yn (t)+ t yn(t)+ t−2 yn(t) dt (ii) For k = 1: solving (28)–(29) gives R (20) δ t where we used (˜g(yn(t)))= 0. λ(x,t)= xt +t2 1 ln . (31) For illustrative purposes, we evaluate the integral at the − − x right side in steps. We ﬁrst integrate (iii) For k = 2: solving (28)–(29) gives x λ 3 I1 = (x,t)yn′′′(t)dt. (21) t 2 t Z0 λ(x,t)= +t 1 + ln . (32) − x x Integrating I1 by parts three times gives The successive approximations yn+1, for n 0, of the solution y x will be readily obtained upon≥ using any λ λ λ x λ ( ) I1 = ( (x,t)yn′′ ′(x,t)yn′ + ′′(x,t)yn) t=x ′′′(x,t)yn(t)dt. − | − 0 selective function y0(x). Consequently, the solution R (22) We next integrate the second term of the integral in (20), y(x)= lim yn(x). (33) n ∞ therefore we set → x In other words, the correction functional (18) will give λ 2k I2 = (x,t) yn′′(t)dt. (23) several approximations, and therefore the exact solution is Z0 t obtained in the limit of the resulting successive Integrating I2 by parts twice gives approximations.

λ λ λ 2k 2kt ′(x,t) 2k (x,t) I2 = (x,t) yn′ −2 yn(t) t=x t − t | (24) 2λ λ λ 3.2 Numerical examples for the ﬁrst kind of the x 2kt ′′(x,t) 4kt ′(x,t)+4k (x,t) . + 0 − t3 yn(t)dt Emden–Fowler type equations R We ﬁnally integrate the third term of the integral in (20), k(k 1) therefore we set ′′′ 2k ′′ ′ , α, y + x y + x−2 y + f (x)g(y)= 0 y(0)= y′(0)= y′′(0)= 0 (34) x k(k 1) λ We will study this equation for a variety of values for the I3 = (x,t) −2 yn′ (t)dt. (25) Z0 t shape factor k and for the given functions f (x)g(y). For comparison reasons we will solve the same examples Integrating I by parts once gives 3 solved in [1] by using the Adomian decomposition λ λ method. k(k 1) x k(k 1)t ′(x,t) 2k(k 1) (x,t) I3 = λ(x,t) − yn t=x − − − yn(t)dt. t2 | − 0 t3 R (26)

c 2015 NSP Natural Sciences Publishing Cor. 2432 A. M. Wazwaz: Solving Two Emden–Fowler Type Equations of Third...

Example 1. for n 0. ≥ By selecting the zeroth approximation y0 = 1, we We ﬁrst consider the Emden–Fowler type equation obtain the following calculated solution approximations 2 9 6 5 y (x)= 1, y′′′ + y′′ (x +8)y− = 0,y(0)= 1,y′(0)= y′′(0)= 0, 0 x − 8 3 5 6 1 9 (35) y1(x)= 1 + x + x + x , obtained by substituting k = 1 in (34) and by setting 14 30 f (x)g(y)= 9 (x6 + 8)y 5. 3 1 6 101 9 44 12 8 − y2(x)= 1 + x + x + x + x + , From (31−), the Lagrange multiplier for k = 1 is given 2 630 1365 ··· by 3 1 6 1 9 1361 12 t y3(x)= 1 + x + x + x + x + , λ(x,t)= xt +t2 1 ln . (36) 2 6 32760 ··· − − x 3 1 6 1 9 1 12 y4(x)= 1 + x + x + x + x + , The correction functional for (35) becomes 2 3! 4! ··· .... x 2 t 2 9 6 5 yn 1(x)= yn(x)+ xt +t 1 ln y′′′ (t)+ y′′ (t) (t + 8)y dt, + 0 − − x n t n − 8 n− R (37) This in turn gives the series solution where n 0. By selecting the zeroth approximation 3 1 6 1 9 1 12 y = 1, we≥ obtain the following calculated solution y(x)= 1 + x + x + x + x + , (43) 0 2! 3! 4! ··· approximations that converges to the exact solution y x 1, 3 0( )= y(x)= ex . (44) 1 3 1 9 y1(x)= 1 + x + x , 2 576 Example 3. 1 3 1 6 31 9 2705 12 y2(x)= 1 + x x + x x + , 2 − 8 576 − 101376 ··· We now consider the nonlinear Emden–Fowler type 1 3 1 6 1 9 3935 12 equation y3(x)= 1 + x x + x x + , 2 − 8 16 − 101376 ··· ′′′ 6 ′′ 6 ′ 3 6 3y , , , y + x y + x2 y 6(10 + 2x + x )e− = 0 y(0)= 0 y′(0)= y′′(0)= 0 1 3 1 6 1 9 5 12 − y4(x)= 1 + x x + x x + , (45) 2 − 8 16 − 128 ··· obtained by substituting k = 3 in (34) and by setting .... 3 6 3y f (x)g(y)= 6(10 + 2x + x )e− . To facilitate the computational work, we used few terms From (30−), the Lagrange multiplier for k = 3 is given of the exponential yn(t),n 1. This in turn gives the series by ≥ solution t2 1 t 4 t 3 λ(x,t)= x2 + x2 . (46) 1 1 1 5 2 − 2 x x y(x)= 1 + x3 x6 + x9 x12 + , (38) 2 − 8 16 − 128 ··· The correction functional for (45) becomes that converges to the exact solution yn+1(x)= yn(x) x 2 4 3 + t 1 x2 t + x2 t y′′′ (t)+ 6 y′′ (t)+ 6 y′ (t) 6(10 + 2t3 +t6)e 3yn(t) dt, 0 2 − 2 x x n t n t2 n − − y(x)= 1 + x3. (39) R (47) p for n 0. By selecting the zeroth approximation y0 = 0, Example 2. we≥ obtain the following calculated solution approximations We next consider the linear Emden–Fowler type equation y0(x)= 0, ′′′ 4 ′′ 2 ′ 3 6 y + y + 2 y 9(4 + 10x + 3x )y = 0,y(0)= 1,y′(0)= y′′(0)= 0, x x − 3 1 6 1 9 (40) y1(x)= x + x + x , 28 165 obtained by substituting k = 2 in (34) and by setting 3 6 3 1 6 1 9 199 12 f (x)g(y)= 9(4 + 10x + 3x )y. y2(x)= x + x + x x + , From (32−), the Lagrange multiplier for k = 2 is given 28 165 − 2002 ··· by 3 1 6 1 9 3391 12 y3(x)= x + x + x x + , t3 t 28 165 − 14014 ··· λ(x,t)= +t2 1 + ln . (41) 3 1 6 1 9 1 12 − x x y4(x)= x + x + x x + , 28 165 − 4 ··· The correction functional for (40) becomes .... This in turn gives the series solution yn+1(x)= yn(x) 3 x t 2 t ′′′ 4 ′′ 2 3 6 + +t 1 + ln yn (t)+ yn(t)+ 2 yn′ (t) 9(4 + 10t + 3t )yn(t) dt, 0 − x x t t − 3 1 6 1 9 1 12 R (42) y(x)= x x + x x + , (48) − 2 3 − 4 ···

c 2015 NSP Natural Sciences Publishing Cor. Appl. Math. Inf. Sci. 9, No. 5, 2429-2436 (2015) / www.naturalspublishing.com/Journals.asp 2433

that converges to the exact solution or equivalently

y(x)= ln(1 + x3). (49) x k δ δ δ λ ′′′ ′′ yn+1(x)= yn(x)+ (x,t) yn (t)+ yn(t) dt, Z0 t Example 4. (57) where we used δ(˜g(yn(t)))= 0. We conclude this section by considering the Lane–Emden Proceeding as before, we evaluate the integral at the type equation right side in steps. We ﬁrst integrate

8 12 m x y′′′ + y′′ + y′ + y = 0,y(0)= 1,y′(0)= y′′(0)= 0, λ x x2 I1 = (x,t)yn′′′(t)dt. (58) (50) Z0 obtained by substituting k = 4in(34) and by setting g(y)= Integrating I1 by parts three times gives ym, f (x)= 1. x From (30), the Lagrange multiplier for k = 4 is given I1 = (λ(x,t)y λ (x,t)y + λ (x,t)yn) t=x λ (x,t)yn(t)dt. n′′ − ′ n′ ′′ | − 0 ′′′ by R (59) We next integrate the second term of the integral in (57), 1 1 t 5 1 t 4 λ(x,t)= t2 x2 + x2 . (51) therefore we set −6 − 3 x 2 x x λ k The correction functional for (50) becomes I2 = (x,t) yn′′(t)dt. (60) Z0 t yn+1(x)= yn(x) x 5 4 1t2 1 x2 t 1 x2 t y′′′ t 8 y′′ t 12 ym t dt, Integrating I2 by parts twice gives + 0 6 3 x + 2 x n ( )+ t n( )+ t2 n ( ) R − − (52) λ λ λ k kt ′(x,t) k (x,t) for n 0. By selecting the zeroth approximation y = 1, I2 = (x,t) yn′ 2− yn(t) t=x 0 t − t | (61) we≥ obtain the following calculated solution 2λ λ λ x kt ′′(x,t) 2kt ′(x,t)+2k (x,t) . approximations + 0 − t3 yn(t)dt y (x) = 1, R 0 In view of (58)–(61), Eq. (57) becomes 1 3 y1(x) = 1 x , − 90 δyn+1(x)= δyn(x) 1 3 m 6 y2(x) = 1 x x , k k k +δ λ(x,t)y + λ(x,t) = λ (x,t) y + λ (x,t) λ (x,t)+ )yn t=x − 90 2592 n′′ t2 ′ n′ ′′ − t ′ t2 | x h k 2k i h2k i 1 3 m 6 m(17m 12) 9 + λ (x,t)+ λ (x,t) λ (x,t)+ λ(x,t) δy . y3(x) = 1 x x − x , 0 ′′′ t ′′ t2 ′ t3 n − 90 3880 − 230947200 R − − 2 (62) 1 3 m 6 m(17m 12) 9 m(679m 1182m + 528) 12 y4(x) = 1 x x − x + − x , δ − 90 3880 − 230947200 2909934720000 For arbitrary yn, we obtain the stationary condition .... k 2k 2k This in turn gives the series solution λ ′′′(x,t)+ λ ′′(x,t) λ ′(x,t)+ λ(x,t)= 0 (63) − t − t2 t3 1 3 m 6 m(17m 12) 9 m(679m2 1182m+528) 12 y(x)= 1 x + x − x + − x + . − 90 38880 − 230947200 2909934720000 ··· and the boundary conditions (53) Substituting m = 0 gives the exact solution as λ(x,t) t=x = 0, k λ λ | 1 2 (x,t)+ ′(x,t) t=x = 0, y(x)= 1 x3. (54) t | (64) − 90 λ , k λ , k λ , . 1 + ′′(x t) t ′(x t)+ t2 (x t) t=x = 0 − | λ 3.3 The second model: the VIM and the To determine (x,t), three cases will be examined: (i) For the general case where k = 1,2: solving (63)–(64) Lagrange multipliers gives 6

In this section we will follow the analysis presented for the t2 xt x2 t k ﬁrst kind. For Eq. (14) the correction functional reads λ(x,t)= . (65) k 2 − k 1 − (k 1)(k 2) x x − − − − y (x)= y (x)+ λ(x,t) y′′′ (t)+ k y′′ (t)+ f (t)˜g(y (t)) dt, n+1 n 0 n t n n (ii) For k = 1: solving (63)–(64) gives R (55) δ where (˜g(yn(t)))= 0. λ 2 t λ (x,t)= t + xt 1 + ln (66) To determine the optimal value of (x,t), we take the − x variation for both sides with respect to yn(x) to obtain (iii) For k = 2: solving (63)–(64) gives δ δ δ x λ ′′′ k ′′ yn+1(x)= yn(x)+ 0 (x,t) yn (t)+ t yn(t)+ f (t)˜g(yn(t)) dt, 2 t R (56) λ(x,t)= xt +t 1 ln . (67) − − x

c 2015 NSP Natural Sciences Publishing Cor. 2434 A. M. Wazwaz: Solving Two Emden–Fowler Type Equations of Third...

The successive approximations yn 1, for n 0, of the To facilitate the computational work, we used few terms of + ≥ solution y(x) will be readily obtained upon using any the exponential e 3yn(t),n 1. This in turn gives the series − ≥ selective function y0(x). Consequently, the solution solution

y(x)= lim yn(x). (68) 4 1 8 1 12 1 16 1 20 n ∞ y(x)= x x x x x + , (73) → − − 2 − 3 − 4 − 5 ··· In other words, the correction functional (56) will give that converges to the exact solution several approximations, and therefore the exact solution is obtained in the limit of the resulting successive y(x)= ln(1 x4). (74) approximations. −

Example 2. 3.4 Numerical examples for the second kind of We next consider the nonlinear Emden–Fowler type the Emden–Fowler type equations equation

′′′ 2 ′′ 2 ′ 25 2 5 10 7 In this section, we study several numerical examples for y + y + 2 y x (16 + 52x 5 + 7x )y = 0,y(0)= 1,y′(0)= y′′(0)= 0, x x − 8 the second case of the Emden–Fowler type equations of (75) the third-order in the form obtained by substituting k = 2 in (69) and by setting f (x)g(y)= 25 x2(16 + 52x5 + 7x10)y7. k − 8 ′′′ ′′ From (67), the Lagrange multiplier for k 2 is given y + y + f (x)g(y)= 0,y(0)= α,y′(0)= y′′(0)= 0 = x by (69) 2 t We will study this equation for a variety of values for the λ(x,t)= xt +t 1 ln . (76) − − x shape factor k and for the given functions f (x)g(y). The correction functional for (75) becomes Example 1. yn+1(x)= yn(x) x We ﬁrst consider the nonlinear Emden–Fowler type + xt +t2 1 ln t y′′′ (t)+ 2 y′′ (t) 25 t2(16 + 52t5 + 7t10)y7(t) dt, 0 − − x n t n − 8 n equation R (77) for n 0. By selecting the zeroth approximation y0 = 1, ′′′ 1 ′′ 4 8 3y ≥ y + x y + 4x(9 + 22x + x )e− = 0,y(0)= 0,y′(0)= y′′(0)= 0, we obtain the following calculated solution (70) approximations obtained by substituting k = 1 in (69) and by setting 4 8 3y f (x)g(y)= 4x(9 + 22x + x )e− . y0(x)= 1, From (66), the Lagrange multiplier for k = 1 is given 1 13 1 y (x)= 1 + x5 + x10 + x15, by 1 2 72 144 2 t λ(x,t)= t + xt 1 + ln . (71) 1 5 3 10 377 15 y2(x)= 1 + x + x + x + , − x 2 8 1296 ··· The correction functional for (70) becomes 1 5 3 10 5 15 y3(x)= 1 + x + x + x + , 2 8 16 ··· yn+1(x)= yn(x) .... x + t2 + xt 1 + ln t y′′′ (t)+ 1 y′′ (t)+ 4t(9 + 22t4 +t8)e 3yn(t) dt, 0 − x n t n − R (72) This in turn gives the series solution for n 0. By selecting the zeroth approximation y 0, 0 = 1 3 5 35 we≥ obtain the following calculated solution y(x)= 1 + x5 + x10 + x15 + x20 + , (78) approximations 2 8 16 128 ··· that converges to the exact solution y0(x)= 0, 1 y(x)= . (79) 4 11 8 1 12 5 y1(x)= x x x , √1 x − − 49 − 363 − 4 1 8 11129 12 4721 16 y2(x)= x x x x + , Example 3. − − 2 − 35574 − 24200 ··· 4 1 8 1 12 295721 16 We now consider the nonlinear Emden–Fowler type y3(x)= x x x x + , − − 2 − 3 − 1185800 ··· equation 4 1 8 1 12 1 16 y4(x)= x x x x + , y′′′ + 3 y′′ + 6 y′ + 2(4 + x3)y 8 = 0,y(0)= 1,y (0)= y (0)= 0, − − 2 − 3 − 4 ··· x x2 − ′ ′′ .... (80)

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obtained by substituting k = 3 in (69) and by setting approximations f (x)g(y)= 2(4 + x3)y 8. − y (x)= 1, From (65), the Lagrange multiplier for k = 3 is given 0 1 1 1 by y (x)= 1 + x3 + x6 + x9, 1 3 24 792 2 1 1 2 t 3 λ(x,t)= t xt x ( ) . (81) 1 3 1 6 19 9 − 2 − 2 x y2(x)= 1 + x + x + x + , 3 18 3168 ··· 1 1 1 The correction functional for (80) becomes y (x)= 1 + x3 + x6 + x9, 3 3 18 162 .... yn+1(x)= yn(x) x + t2 1 xt 1 x2( t )3 y′′′ (t)+ 3 y′′ (t)+ 2(4 +t3)y 8 dt, This in turn gives the series solution 0 − 2 − 2 x n t n n− R (82) 1 3 1 6 1 9 1 12 for n 0. By selecting the zeroth approximation y0 = 1, y(x)= 1 + x + x + x + x + , (88) we≥ obtain the following calculated solution 3 18 162 1944 ··· approximations that gives the exact solution y0(x)= 1, x3 y(x)= e 3 . (89) 1 3 1 6 y1(x)= 1 x x , − 3 − 105 1 3 1 6 83 9 y2(x)= 1 x x x + , 4 Conclusion − 3 − 9 − 1575 ··· 1 3 1 6 5 9 In this work, we have presented a framework to establish y3(x)= 1 x x x + , − 3 − 9 − 81 ··· 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 This in turn gives the series solution than one shape factor for equations of order greater than or equal to 3 for one case and only one shape factor for 1 1 5 10 y(x)= 1 x3 x6 x9 x12 + , (83) another case. Similarly, the singular point appears once in − 3 − 9 − 81 − 243 ··· the standard form, whereas in the established cases, the singular point x = 0 may appear twice. We used the that converges to the exact solution variational iteration method for treating linear and nonlinear problems to illustrate our analysis. A variety of 3 1 y(x) = (1 x ) 3 . (84) 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 Example 4. shape factor values k greater than or equal to 1. The obtained results validate the reliability and rapid We conclude this section by consideringthe linear Emden– convergence of the VIM. Fowler type equation

′′′ 4 ′′ 3 6 y + x y (10 + 10x + x )y = 0,y(0)= 1,y′(0)= y′′(0)= 0, References − (85) obtained by substituting k = 4 in (69) and by setting [1] I. J. H. Lane, On the theoretical temperature of the sun under f (x)g(y)= (10 + 10x3 + x6)y. the hypothesis of a gaseous mass maintaining its volume by − From (65), the Lagrange multiplier for k = 4 is given its internal heat and depending on the laws of gases known to by terrestial experiment,American Journal of Science and Arts, 4 (1870) 57-74. 1 1 1 t λ(x,t)= t2 xt x2( )4. (86) [2] R. H. Fowler, Further studies of Emdens and similar 2 − 3 − 6 x differential equations, Quarterly Journal of Mathematics, 2(1) (1931) 259-288. The correction functional for (85) becomes [3] A. M. Wazwaz, R. Rach, L. Bougoffa, J.-S. Duan, Solving the Lane–Emden–Fowler type equations of higher orders by yn+1(x)= yn(x) the Adomian decomposition method, , In Press. x 1t2 1 xt 1 x2 t 4 y′′′ t 4 y′′ t 10 10t3 t6 y t dt, + 0 2 3 6 ( x ) n ( )+ t n( ) ( + + ) n( ) [4] A.M. Wazwaz, Analytical solution for the time– − − − R (87) dependent Emden–Fowler type of equations by Adomian for n 0. By selecting the zeroth approximation y0 = 1, decomposition method, Appl. Math. Comput. 166 (2005) we≥ obtain the following calculated solution 638–651.

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[5] A.M. Wazwaz, A new algorithm for solving differential Abdul-Majid Wazwaz equations of Lane–Emden type, Appl. Math. Comput. 118 is a Professor of (2001) 287–310. Mathematics at Saint [6] A.M. Wazwaz, A new method for solving singular initial Xavier University in Chicago, value problems in the second-order ordinary differential Illinois, USA. He has both equations, Appl. Math. Comput. 128 (2002) 47–57. authored and co-authored [7] A.M. Wazwaz, Adomian decomposition method for a more than 400 papers reliable treatment of the Emden–Fowler equation, Appl. in applied mathematics Math. Comput. 161 (2005) 543–560. and mathematical physics. He [8] A.M. Wazwaz, The variational iteration method for solving is the author of ﬁve books on nonlinear singular boundary value problems arising in various physical models, Commun. Nonlinear Sci. Numer. the subjects of discrete mathematics, integral equations Simulat. 16 (2011) 3881–3886. and partial differential equations. Furthermore, he has [9] A.M.Wazwaz, Partial Differential Equations and Solitary contributed extensively to theoretical advances in solitary Waves Theory, HEP and Springer, Beijing and Berlin, 2009. waves theory, the Adomian decomposition method and [10] J.H.He, Variational iteration method for autonomous the variational iteration method.. ordinary differential systems, Appl. math. Comput., 114(2/3) (2000) 115–123. [11] J.H. He, Variational iteration method - Some recent results and new interpretations, J. Comput. Appl. Math., 207(1)( 2007) 3–17. [12] C.M. Khalique, F. M. Mahomed, B. Muatjetjeja, Lagrangian formulation of a generalized Lane–Emden equation and double reduction, J. Nonlinear Math. Phys. 15 (2008) 152– 161. [13] B. Muatjetjeja, C.M. Khalique, Exact solutions of the generalized Lane–Emden equations of the ﬁrst and second kind, Pramana 77 (2011) 545–554. [14] M. Dehghan, F. Shakeri, Approximate solution of a differential equation arising in astrophysics using the variational iteration method, New Astronomy 13 (2008) 53– 59. [15] O.W. Richardson, The Emission of Electricity from Hot Bodies, Longmans Green and Co. London, 1921. [16] K. Parand, M. Dehghan, A.R. Rezaeia, S.M. Ghaderi, An approximation algorithm for the solution of the nonlinear Lane–Emden type equations arising in astrophysics using Hermite functions collocation method, Comput. Phys. Commun. 181 (2010) 1096–1108. [17] K. Parand, M. Shahini, M. Dehghan, Rational Legendre pseudospectral approach for solving nonlinear differential equations of Lane–Emden type, J. Comput. Phys. 228 (2009) 8830–8840. [18] G. Adomian, R. Rach, N.T. Shawagfeh, On the analytic solution of the Lane–Emden equation, Found. Phys. Lett. 8 (1995) 161–181. [19] H.T. Davis, Introduction to Nonlinear Differential and Integral Equations, Dover Publications, New York, 1962. [20] S. Chandrasekhar, An Introduction to the Study of Stellar Structure, Dover Publications, New York, 1967. [21] E. Momoniat, C. Harley, Approximate implicit solution of a Lane–Emden equation, New Astronomy 11 (2006) 520– 526. [22] A. Yildirim, T. Ozis,¨ Solutions of singular IVPs of Lane– Emden type by homotopy perturbation method, Phys. Lett. A 369 (2007) 70–76. [23] A. Yildirim, T. Ozis,¨ Solutions of singular IVPs of Lane– Emden type by the variational iteration method, Nonlinear Analysis 70 (2009) 2480–2484.

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