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 ( )

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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. This work will show the enhancement gained by ∂un(ξ; t) −(6 + 4ξ2 − cos t)un(ξ; t) − dξ; using the VIM. ∂t λ ξ − x 4.1. Time-dependent Emden–Fowler type where we used = ( ). Considering the given initial values, we can select u x; t xesin t Example 1 0( ) = . Using this selection into (22) we obtain the following successive approximations We ﬁrst study the homogeneous time-dependent Emden– u x; t xesin t ; 0( ) = Fowler type u x; t xesin t x2 1 x4 ; 1( ) = (1 + + 5 ) u x; t xesin t x2 1 x4 13 x6 ; 2( ) = (1 + + 2! + 105 ) 2 t yxx yx − x2 − t y yt ; u x; t xesin x2 1 x4 1 x6 4 x8 ··· ; + x (6 + 4 cos ) = (15) 3( ) = (1 + + 2! + 3! + 105 + ) u x; t xesin t x2 1 x4 1 x6 1 x8 ··· ; 4( ) = (1 + + 2! + 3! + 4! + ) . with initial conditions . ; ∞ sin t P 1 2n un(x; t) = xe n n x : t =0 ! y(0; t) = esin ; yx (0; t) = 0: (16) (23)

101 A reliable iterative method for solving the time-dependent singular Emden-Fowler equations

Recall that where noise terms that appear between various terms van- u x; t un x; t ; ( ) =n lim→∞ ( ) (24) ish in the limit. Using the indirect approach that gives the exact solution of (21) by Using the transformation (9) in (27) and (28) gives t x2 u(x; t) = xesin + ; (25) uxx − (6 + 4x2)u = ut + x(6 − 5x2 − 4x4); where noise terms vanish in the limit. u(0; t) = 0; (33) Using (9) gives the exact solution of (15) by t ux (0; t) = e :

t x2 y(x; t) = esin + : (26) The correction functional for (33) reads Example 2 Z x ∂2un(ξ; t) un x; t un x; t ξ − x +1( ) = ( ) + ( ) ∂ξ2 We next study the inhomogeneous time-dependent Lane- 0 Emden type ∂un ξ; t − ξ2 u ξ; t − ( ) (6 + 4 ) n( ) ∂t (34) y 2 y − x2 y y − x2 − x4 ; xx + x x (5 + 4 ) = t + (6 5 4 ) (27) − ξ(6 − 5ξ2 − 4ξ4) dξ;

with initial conditions where we used λ = (ξ − x). y ; t et ; y ; t : Considering the given initial values, we can select (0 ) = x (0 ) = 0 (28) u x; t xet 0( ) = . Using this selection into (34) we obtain the Using the direct approach following successive approximations The correction functional for (33) reads u x; t xet ; 0( ) = 3 t 2 1 4 Z x 2 2 u x; t x xe x x ; ξ ∂ yn(ξ; t) 1( ) = + (1 + + 5 ) yn x; t yn x; t − ξ u x; t x3 xet x2 1 x4 13 x6 ; +1( ) = ( ) + ( x ) ∂ξ2 2( ) = + (1 + + + ) 0 2! 105 u x; t x3 xet x2 1 x4 1 x6 4 x8 ··· ; 2 ∂yn(ξ; t) 3( ) = + (1 + + 2! + 3! + 105 + ) − ξ2 yn ξ; t (29) u x; t x3 xet x2 1 x4 1 x6 1 x8 ··· ; + ξ ∂ξ (5 + 4 ) ( ) 4( ) = + (1 + + 2! + 3! + 4! + ) . ∂un ξ; t . ; − ( ) − − ξ2 − ξ4 dξ; . (6 5 4 ) t P∞ n ∂t un x; t x3 xe 1 x2 : ( ) = + n=0 n! (35) ξ2 Recall that where we used λ = ( x − ξ). u x; t un x; t ; Considering the given initial values, we can select ( ) =n lim→∞ ( ) (36) y x; t et 0( ) = . Using this selection into (29) we obtain the following successive approximations that gives the exact solution of (33) by

y x; t et ; 3 t+x2 0( ) = u(x; t) = x + xe : (37) y x; t x2 et x2 − 1 x4 ; 1( ) = + (1 + 4 + other terms) y x; t x2 et x2 1 x4 13 x6 ; 2( ) = + (1 + + 2! + 105 + other terms) y x; t x2 et x2 1 x4 1 x6 ; Using (4) gives the exact solution of (33) by 3( ) = + (1 + + 2! + 3! + other terms) . . ; 2 . y x; t x2 et+x : t P∞ n ( ) = + (38) yn x; t x2 e 1 x2 : ( ) = + n=0 n! (30) Recall that 4.2. Singular Emden-Fowler wave-type equa- y x; t yn x; t ; ( ) =n lim→∞ ( ) (31) tions

that gives the exact solution of (33) by In this part we examine two singular wave-type equations. Each example will be handled by the direct approach and t x2 y(x; t) = x2 + e + ; (32) the indirect approach.

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Example 3 where we used λ = (ξ − x). Considering the given initial values, we can select We study here the homogeneous singular wave type equation u x; t xet2 0( ) = . Using this selection into (46) we obtain the following successive approximations y 2 y − x − x4 t2 y y ; xx + x x + (2 12 9 + 4 ) = tt (39) u x; t xet2 ; with initial conditions 0( ) = u x; t xet2 x3 3 x6 ; 1( ) = (1 + + 14 ) t2 t2 3 1 6 9 9 y ; t e ; yx ; t : u x; t xe x x x ··· ; (0 ) = (0 ) = 0 (40) 2( ) = (1 + + 2! + 70 + ) u x; t xet2 x3 1 x6 1 x9 141 x12 ··· ; 3( ) = (1 + + 2! + 3! + 3640 ) Using the direct approach . . ; The correction functional for (45) reads t2 P∞ n un x; t xe 1 x3 : ( ) = n=0 n! Z x ξ2 ∂2yn(ξ; t) (47) yn x; t yn x; t − ξ +1( ) = ( ) + ( x ) ∂ξ2 0 Recall that ∂yn ξ; t u x; t un x; t ; 2 ( ) − ξ − ξ4 ( ) =n lim→∞ ( ) (48) + ξ ∂ξ + (2 12 9 (41) ∂2yn(ξ; t) that gives the exact solution of (45) by t2 yn ξ; t − dξ; + 4 ) ( ) ∂t2

t2+x3 ξ2 u(x; t) = xe : (49) where we used λ = ( x − ξ). Considering the given initial values, we can select y x; t et2 0( ) = . Using this selection into (41) we obtain the Using (4) gives the exact solution of (39) by following successive approximations t2 y x; t e ; t2+x3 0( ) = y(x; t) = e : (50) y x; t et2 x3 3 x6 ; 1( ) = (1 + + 14 ) y x; t et2 x3 1 x6 9 x9 ··· ; 2( ) = (1 + + 2! + 70 + ) y x; t et2 x3 1 x6 1 x9 141 x12 ··· ; 3( ) = (1 + + 2! + 3! + 3640 ) Example 4 . . ; t2 P∞ n We close our analysis by studying the inhomogeneous yn x; t e 1 x3 : ( ) = n=0 n! singular wave type equation (42) Recall that y x; t yn x; t : y 2 y − x2 y y x − x3 − x5 ; ( ) =n lim→∞ ( ) (43) xx + x x (5 + 4 ) = tt + (12 5 4 ) (51) This gives the exact solution of (39) by

t2 x3 with initial conditions y(x; t) = e + : (44)

−t Using the indirect approach y(0; t) = e ; yx (0; t) = 0: (52) Using the transformation (4) in (39) and (40) gives Using the direct approach uxx + x(2 − 12x − 9x4 + 4t2)u = utt ; The correction functional for (52) reads u(0; t) = 0; (45) t2 ux (0; t) = e : Z x ξ2 ∂2yn(ξ; t) yn x; t yn x; t − ξ +1( ) = ( ) + ( x ) ∂ξ2 The correction functional for (45) reads 0 2 ∂yn(ξ; t) − ξ2 yn ξ; t Z x ∂2u ξ; t + ξ ∂ξ (5 + 4 ) ( ) (53) u x; t u x; t ξ − x n( ) n+1( ) = n( ) + ( ) ∂ξ2 ∂2yn ξ; t 0 − ( ) − ξ − ξ3 − ξ5 dξ; 4 2 ∂t2 (12 5 4 ) + ξ(2 − 12ξ − 9ξ + 4t )un(ξ; t) (46) ∂2u ξ; t − n( ) dξ; ∂t2 ξ2 where we used λ = ( x − ξ).

103 A reliable iterative method for solving the time-dependent singular Emden-Fowler equations

Considering the given initial values, we can select 5. Conclusion y x; t e−t 0( ) = . Using this selection into (54) we obtain the following successive approximations This present analysis exhibits the reliable applicability y x; t e−t ; of the variational iteration method to solve the singular 0( ) = y x; t x3 e−t x2 1 x4 ··· ; Emden-Fowler type of equations. In this work we demon- 1( ) = + (1 + + 5 + ) y x; t x3 e−t x2 1 x4 13 x6 ··· ; strate that this method can be well suited to attain an 2( ) = + (1 + + 2! + 105 + ) y x; t x3 e−t x2 1 x4 1 x6 4 x8 ··· ; analytic solution to the type of examined equations. The 3( ) = + (1 + + 2! + 3! + 105 + ) y x; t x3 e−t x2 1 x4 1 x6 1 x8 ··· ; difficulty in this type of equations, due to the existence of 4( ) = + (1 + + 2! + 3! + 4! + ) x . singular point at = 0 can be easily overcome here. . ; −t P∞ n To support our work, we examined four distinct examples, yn x; t x3 e 1 x2 : ( ) = + n=0 n! two are of heat type and the other two are of wave type (54) equations. Two reliable approaches were employed. The Recall that first approach relies of the direct approach, whereas as the y x; t yn x; t ; ( ) =n lim→∞ ( ) (55) other one depends on using a transformation formulae to that gives the exact solution of (57) by overcome the singular behavior. The results demonstrate reliability and effectiveness of the schemes used. −t x2 y(x; t) = x3 + e + : (56) Using the indirect approach References Using the transformation (4) in (51) and (52) gives

uxx − x2 u utt x x − x3 − x5 ; (5 + 4 ) = + (12 5 4 ) [1] Wazwaz A.M., Analytical solution for the timedepen- u(0; t) = 0; (57) dent Emden Fowler type of Equations by Adomian de- −t ux (0; t) = e : composition method, Appl. Math. Comput., 166, 2005, 638–651 The correction functional for (57) reads [2] Wazwaz A.M., A new method for solving differential Z x 2 Equations of the Lane-Emden type, Appl. Math. Com- ∂ un(ξ; t) un x; t un x; t ξ − x +1( ) = ( ) + ( ) ∂ξ2 put., 118(2/3), 2001, 287–310 0 (58) [3] Wazwaz A.M., A new method for solving singular initial ∂2un(ξ; t) − ξ2 un ξ; t − dξ; (5 + 4 ) ( ) ∂t2 value problems in the second order ordinary differential equations, Appl. Math. Comput., 128, 2002, 47–57 where we used λ = (ξ − x). [4] Wazwaz A.M., Adomian decomposition method for Considering the given initial values, we can select a reliable treatment of the Emden-Fowler equation, u x; t xe−t 0( ) = . Using this selection into (58) we obtain Appl.Math. Comput., 161, 2005, 543–560 the following successive approximations [5] Wazwaz A.M., Partial differential Equations and Soli- tary waves theory, HEP and Springer, Beijing and u x; t xe−t ; 0( ) = Berlin, 2009 u x; t x4 xe−t x2 1 x4 ··· ; 1( ) = + (1 + + 5 + ) [6] Yildirim A., Ozis T., Solutions of singular IVPs of u x; t x4 xe−t x2 1 x4 13 x6 ··· ; 2( ) = + (1 + + 2! + 105 + ) Lane-Emden type by homotopy perturbation method, u x; t x4 xe−t x2 1 x4 1 x6 4 x8 ··· ; 3( ) = + (1 + + 2! + 3! + 105 + ) Phys.Lett. A, 369, 2007, 70–76 u x; t x4 xe−t x2 1 x4 1 x6 1 x8 ··· ; 4( ) = + (1 + + 2! + 3! + 4! + ) [7] Yildirim A., Ozis T., Solutions of singular IVPs of . . ; Lane-Emden type by the variational iteration method, −t P∞ n un x; t x4 xe 1 x2 : Nonlinear Analysis 70, 2009, 2480–2484 ( ) = + n=0 n! (59) [8] Danish M., Kumar S., Kumar S., A note on the solu- Recall that tion of singular boundary value problems arising in u x; t un x; t ; engineering and sciences: Use of OHAM, Computers ( ) =n lim→∞ ( ) (60) and Chemical Engineering, 36, 2012, 57–67 that gives the exact solution of (57) by [9] Dehghan M., Shakeri F., Approximate solution of a 4 −t+x2 differential equation arising in astrophysics using the u(x; t) = x + xe : (61) variational iteration method, New Astronomy, 13, 2008, Using (4) gives the exact solution of (51) by 53–59 [10] Parand K., Dehghan M., Rezaeia A.R., Ghaderi S.M., y x; t x3 e−t+x2 : ( ) = + (62) An approximation algorithm for the solution of the

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