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A Reliable Iterative Method for Solving the Time-Dependent Singular Emden-Fowler Equations

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A Reliable Iterative Method for Solving the Time-Dependent Singular Emden-Fowler Equations Cent. Eur. J. Eng. • 3(1) • 2013 • 99-105 DOI: 10.2478/s13531-012-0028-y Central European Journal of Engineering A reliable iterative method for solving the time-dependent singular Emden-Fowler equations Research Article Abdul-Majid Wazwaz Department of Mathematics Saint Xavier University, Chicago, IL 60655 Received 12 April 2012; accepted 18 June 2012 Abstract: Many problems of physical sciences and engineering are modelled by singular boundary value problems. In this paper, the variational iteration method (VIM) is used to study the time-dependent singular Emden-Fowler equations. This method overcomes the difficulties of singularity behavior. The VIM reveals quite a number of features over numerical methods that make it helpful for singular and nonsingular equations. The work is supported by analyzing few examples where the convergence of the results is observed. Keywords: Emden-Fowler equation • Variational iteration method • Lagrange multiplier © Versita sp. z o.o. 1. Introduction n Singular boundary value problems constitute an important For f( x) =1 and g(y) = y , Equation (1) is the standard class of boundary value problems and are frequently en- Lane-Emden equation that has been used to model several countered in the modeling of many problems of physical phenomena in mathematical physics and astrophysics and engineering sciences [1–8], such as reaction engineer- such as the theory of stellar structure and the thermal ing, heat transfer, fluid mechanics, quantum mechanics behavior of a spherical cloud of gas [11–14]. The Lane- and astrophysics [8]. In chemical engineering too, several Emden equation was first studied by the astrophysicists processes, such as isothermal and non-isothermal reac- Jonathan Homer Lane and Robert Emden, where they tion diffusion process and heat transfer are represented by considered the thermal behavior of a spherical cloud of singular boundary value problems [8]. gas acting under the mutual attraction of its molecules Many problems in the literature of the diffusion of heat and subject to the classical laws of thermodynamics [11]. perpendicular to the surfaces of parallel planes are modeled For other special forms of g(y), the well-known Lane- by the time-dependent Emden–Fowler equation [1–10]: Emden equation was used to model several phenomena in mathematical physics and astrophysics such as the 2 yxx + yx + af(x; t)g(y) + h(x; t) = yt ; theory of stellar structure, the thermal behavior of a x (1) spherical cloud of gas, isothermal gas spheres, and theory 0 < x ≤ L; 0 < t < L: of thermionic currents [11–14]. A substantial amount of where f(x; t)g(y) + h(x; t) is the nonlinear heat source, work has been done on this type of problems for various g y y(x; t) is the temperature, and t is the dimensionless time structures of ( ) in[1–10]. variable. 99 A reliable iterative method for solving the time-dependent singular Emden-Fowler equations On the other hand, the wave type of equations with singular where λ is a general Lagrange’s multiplier, which can be behavior of the form identified optimally via the variational theory, and ˜yn as a restricted variation which means δ˜yn = 0. 2 yxx + yx + af(x; t)g(y) + h(x; t) = ytt ; It is obvious now that the main steps of the He’s variational x (2) < x ≤ L; < t < L; iteration method require first the determination of the 0 0 Lagrange multiplier λ that will be identified optimally. For Equation (1), the correction functional reads will be studied by using the VIM, where af(x; t)g(y) + h x; t t ( ) is a nonlinear source, is the dimensionless time Z x y x; t ∂2yn(ξ; t) variable, and ( ) is the displacement of the wave at the yn x; t yn x; t λ +1( ) = ( ) + ∂ξ2 position x and at time t. 0 ∂y ξ; t (5) The singularity behavior that occurs at x = 0 is the main 2 n( ) y ξ; t dξ; + ξ ∂ξ + ˜g( n( )) difficulty of the Equations (1) and (2). The motivation for the analysis presented in this paper comes actually from the aim of extending our previous works in [1]. A reliable where δ(˜g(yn(ξ; t))) = 0. framework depends mainly on the variational iteration Integrating the integral at the right side by parts yields method [5–7, 9, 10] will be employed to handle this type the stationary conditions of singular models. A lot of attention has been devoted to the study of VIM, λ(ξ; ξ) = 0; developed by He [15–18] to investigate various models, ∂λ(ξ;x) | ; singular and nonsingular, linear and nonlinear, and ordi- ∂ξ ξ=x = 1 (6) ∂λ(ξ;x) nary differential equations (ODEs) and partial differential 2 ξ −λ ξ;x ∂ λ(ξ;x) − ∂ξ ( ) : equations (PDEs) as well. The variational iteration method ∂ξ2 2 ξ2 = 0 has been employed by many investigators in the solution of theoretical shell and plate problems, differential and in- Consequently, the Lagrange multiplier λ is given by tegral equations, linear and nonlinear. The VIM accurately computes the solution in a series solution, that converges ξ2 λ − ξ: to the exact solution if such a solution exists. = x (7) We aim in this work to apply the VIM for these two types of Emden-Fowler equations with heat type and wave type. This result is consistent with the result in Yildirim et. al Two distinct approaches will be used to achieve the goal [6, 7]. set for this work. The first approach will applied in a direct y ; n ≥ The successive approximations n+1 0 of the solution manner to the standard form of the Emden–Fowler equation. y(x; t) will be readily obtained upon using any selective However, the second approach will be applied after using y x; t function 0( ). Consequently, the solution transformation formulae to overcome the difficulty of the singular point at x = 0. y x; t yn x; t : ( ) =n lim→∞ ( ) (8) 2. The direct approach In other words, the correction functional (5) will give several approximations, and therefore the exact solution is obtained In this section we will summarize the direct approach at the limit of the resulting successive approximations. where the variational iteration method will be employed in a direct manner to the Emden-Fowler type of Equations (1) and (2). Consider the differential equation 3. The indirect approach Ly Ny g t ; + = ( ) (3) The variational iteration method can be used in an indirect way, where we use transformation formulae to overcome L N where and are linear and nonlinear operators respec- the singularity behavior first. The transformation formulae g t tively, and ( ) is the source inhomogeneous term. are defined by To use the VIM, a correction functional for Equation (3) should be used in the form u(x; t) = xy(x; t); Z t ux (x; t) = xyx (x; t) + y(x; t); (9) yn t yn t λ Lyn ξ N n ξ − g ξ dξ; +1( ) = ( )+ ( ( ) + ˜y ( ) ( )) (4) uxx x; t xyxx x; t yx x; t ; 0 ( ) = ( ) + 2 ( ) 100 A.M.Wazwaz that will carry out (1) and (2) to Using the direct approach The correction functional for (21) reads uxx + axf(x; t)g(u/x) + xh(x; t) = ut ; (10) Z x 2 2 ξ ∂ yn(ξ; t) yn x; t yn x; t − ξ +1( ) = ( ) + ( x ) ∂ξ2 (17) 0 and ∂yn(ξ; t) − ξ2 − t yn ξ; t − dξ; uxx + axf(x; t)g(u/x) + xh(x; t) = utt ; (11) (6 + 4 cos ) ( ) ∂t respectively. It is obvious that (10) and (11) do not include ξ2 where we used λ = ( x − ξ). the singularity at x = 0 as in (1) and (2). Considering the given initial values, we can select In this case, we apply the variational iteration method to y x; t esin t 0( ) = . Using this selection into (17) we obtain the transformed Equations (10) and (11). The correction the following successive approximations functional for these equations is of the form y x; t esin t ; 0( ) = u x; t u x; t y x; t esin t x2 1 x4 ; n+1( ) = n( ) 1( ) = (1 + + 5 ) t Z x y x; t esin x2 1 x4 13 x6 ; ∂2un(ξ; t) (12) 2( ) = (1 + + 2! + 105 ) + λ + ˜g(un(ξ; t)) dξ: y x; t esin t x2 1 x4 1 x6 4 x8 ··· ; ∂ξ2 3( ) = (1 + + 2! + 3! + 105 + ) 0 y x; t esin t x2 1 x4 1 x6 1 x8 ··· ; 4( ) = (1 + + 2! + 3! + 4! + ) . Integrating the integral at the right side by parts yields . ; t P∞ n the stationary conditions yn x; t esin 1 x2 : ( ) = n=0 n! (18) ∂λ 1 − ∂ξ = 0; Recall that λ| ; y x; t yn x; t ; ξ=x = 0 (13) ( ) =n lim→∞ ( ) (19) ∂2λ : ∂ξ2 = 0 that gives the exact solution of (21) by y x; t esin t+x2 ; This in turn gives the Lagrange multiplier by ( ) = (20) where noise terms vanish in the limit. λ = ξ − x (14) Using the indirect approach Using the transformation (4) in (15) and (16) gives 4. Applications uxx − (6 + 4x2 − cos t)u = ut ; t (21) u ; t ; ux ; t esin : In this section we examine some distinct models with (0 ) = 0 (0 ) = x singular behavior at = 0, two linear time–dependent The correction functional for (21) reads Emden–Fowler type of equations, and two linear models Z x ∂2un(ξ; t) of wave-type equation. These models were examined in un x; t un x; t ξ − x +1( ) = ( ) + ( ) ∂ξ2 (22) our work in [1] where Adomian decomposition method was 0 used.
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