Analytical solution for the time-dependent Emden–Fowler type of equations by Adomian decomposition method

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Abstract

In this paper, Adomian decomposition method is used to obtain analytical solution for the time-dependent Emden–Fowler type of equations and wave-type equation with singular behavior. The advantage of this single global method is employed to present a reliable framework is used to overcome the singularity behavior at the point x = 0 for both models. The proposed scheme reveals quite a number of remarkable features that will be helpful for identical problems. The work is supported by analyzing few examples where the convergence of the results is observed.

Introduction

Many problems in the literature of the diffusion of heat perpendicular to the surfaces of parallel planes are modeled by the heat equationx-r(xryx)x+af(x,t)g(y)+h(x,t)=yt,0<xL,0<t<T,r>0,or equivalentlyyxx+rxyx+af(x,t)g(y)+h(x,t)=yt,0<xL,0<t<T,r>0,where f(x, t)g(y) + h(x, t) is the nonlinear heat source, y(x, t) is the temperature, and t is the dimensionless time variable. For the steady-state case, and for k = 2, h(x, t) = 0, Eq. (2) is the Emden–Fowler equation [1], [2], [3] given byy+2xy+af(x)g(y)=0,y(0)=y0,y(0)=0,where f(x) and g(y) are some given functions of x and y respectively. For f(x) = 1 and g(y) = yn, Eq. (3) is the standard Lane–Emden equation that was used to model the thermal behavior of a spherical cloud of gas acting under the mutual attraction of its molecules [1] and subject to the classical laws of thermodynamics. For other special forms of g(y), the well-known Lane–Emden equation was 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, and theory of thermionic currents [1], [2], [3]. A substantial amount of work has been done on this type of problems for various structures of g(y) in [1], [2], [3], [4], [5], [6], [7], [8], [9].

On the other hand, the wave type of equations with singular behavior of the formx-r(xryx)x+af(x,t)g(y)+h(x,t)=ytt,0<xL,0<t<T,r>0,or equivalentlyyxx+rxyx+af(x,t)g(y)+h(x,t)=ytt,0<xL,0<t<T,r>0,will be examined as well, where f(x, t)g(y) + h(x, t) is a nonlinear source, t is the dimensionless time variable, and y(x, t) is the displacement of the wave at the position x and at time t.

The singularity behavior that occurs at the point x = 0 is the main difficulty in the analysis of Eqs. (2), (5). The motivation for the analysis presented in this paper comes actually from the aim of extending our previous works in [7], [8], [9]. A reliable framework depends mainly on Adomian decomposition method [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19] was introduced in [6], [7], [8], [9] to handle this type of models. The introduced algorithm in [6], [7], [8], [9] was mainly used for Lane–Emden equation and for the Emden–Fowler equation where there is no time dependence for y. It is the purpose of this paper to show that the proposed scheme in [6], [7], [8], [9] will effectively handle models of the type given in (2), (5) where y depends on the position x and on the time t as well.

In recent years a lot of attention has been devoted to the study of Adomian decomposition method to investigate various scientific models. Adomian decomposition method, which accurately computes the series solution, is of great interest to applied sciences. The method provides the solution in a rapidly convergent series solution, if the equation has a unique solution. The method provides realistic series solutions that converge very rapidly in applied sciences. The components of the series solution are elegantly computed.

The main advantage of the method is that it can be applied directly for all types of differential and integral equations, linear or nonlinear, homogeneous or inhomogeneous, with constant coefficients or with variable coefficients. Another important advantage is that the method is capable of greatly reducing the size of computational work, without the need of linearization, perturbation or discretization of the problem, while still maintaining the physical behavior of the solution. Convergence of Adomian decomposition scheme was established by many authors by using fixed point theorems in [18], [19]. The effectiveness and the usefulness of Adomian method are demonstrated by finding exact solutions, if these solutions exist, to the models that will be investigated.

In what follows we will present an outline for the alternative framework that can be employed to overcome the difficulty of the singular point at x = 0.

Section snippets

Alternative approach

The proposed framework depends mainly on Adomian decomposition method. Adomian decomposition method will be used in a straightforward manner, but with a new choice for the differential operator L compared to the standard manner that Adomian method is usually used.

The alternative approach rests mainly on defining the differential operator L in terms of the two derivatives, yxx+rxyx, contained in the problem. Following [6], [7], [8], [9], we first rewrite (2) in the formLy=-af(x,t)g(y)-h(x,t)+yt,

Applications

In this section we examine six distinct models with singular behavior at x = 0, two linear time-dependent Lane–Emden type of equations, two linear models of wave-type equation, and two nonlinear singular models.

Conclusion

This present analysis exhibits the reliable applicability of Adomian method to solve linear and nonlinear problems with singular feature. In this work we demonstrate that this method can be well suited to attain an analytic solution to the type of examined equations, linear and nonlinear as well. The difficulty in this type of equations, due to the existence of singular point at x = 0, is overcome here.

Our goal has been achieved by formally obtaining analytical verifiable solutions with a high

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