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

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Solving Two Emden–Fowler Type Equations of Third Order by the Variational Iteration Method 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 Solving Two Emden–Fowler Type Equations of Third Order by the Variational Iteration Method 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). ∗ Corresponding author e-mail: [email protected] c 2015 NSP Natural Sciences Publishing Cor. 2430 A. M. Wazwaz: Solving Two Emden–Fowler Type Equations of Third... 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.
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