Flux Transport Characteristics of Free Boundary Value Problems for a Class of Generalized Convection-Diffusion Equation

The similarity transformation is introduced for studying free boundary value problems for a class of generalized convectiondiffusion equation. A class of singular nonlinear boundary value problems are obtained and solved by using Adomian decomposition method (ADM). The approximate solution can be expressed in terms of a rapid convergent power series with easily computable terms. The efficiency and reliability of the approximate solution are verified by numerical ones. The effects of the variable thermal conduction k(z), convection functional coefficient h(z), power law exponent n, and parameter α on the flux transport characteristics are presented graphically and analyzed in detail.


Introduction
Convection-diffusion equation is a class of very important equations; it often appears in many research fields such as hydrodynamics, transport, electronics, energy, and environmental science [1][2][3][4].In the process of dealing with practical problems, for many mathematical models, especially partial differential equations, it is difficult to obtain their analytical solutions in general.On the one hand, some scholars consider the existence, uniqueness, or nonuniqueness of solutions for the convection-diffusion equations (for example, see [5][6][7][8][9][10][11][12][13]).On the other hand, others focus on the numerical solution of the convection-diffusion equation by all kinds of methods, for instance, the spectral element method [14], the finite element method [15][16][17], the finite difference method [18,19], and the Runge-Kutta method [20].
However, all of the above-mentioned problems paid attention only to the qualitative properties of the solutions, such as the existence and uniqueness, or of the numerical ones.The very important approximate solution [21,22] of convection-diffusion equation has not been well solved.In this paper we present similarity solutions for generalized convection-diffusion equation with free boundary conditions, which are then solved using ADM.ADM [23][24][25] has been shown as a useful way of obtaining accurate and computable solutions to operator equations involving nonlinear terms.The characteristic of ADM is to decompose the nonlinear terms in the equations into a peculiar series of polynomials which are the so-called Adomian polynomials.The solution of the equations is then considered as the sum of a series rapidly converging to an accurate solution.The convergence analysis and proof of ADM for nonlinear problems have been studied by some scholars (for example, see [26][27][28][29][30]).In this paper, we can find that the approximate solution agrees very well with the numerical solution, which shows the reliability and validity of the present work.The effects of the convection functional coefficient ℎ(), variable thermal conduction (), and power law index  on the flux transport characteristics are discussed by graph in detail.

Adomian Approximate Results and Discussion
Integrate equation ( 9) twice from 0 to  for () =  and ℎ() = 1 −  to obtain where  =   (0) and is presented as follows according to ADM: The iterative formula of   is presented as follows: Advances in Mathematical Physics 3 where () =  −1/ .We can obtain Adomian polynomial for () =  and ℎ() = 1 −  according to the iterative formula of   as follows: Substituting the derived Adomian polynomial ( 15) into (13), we obtain We express the solution of ( 10) and ( 13) by a four-ordered approximate solution and the higher order can be determined in a parallel manner.
Considering (1 − )  1 > (1 − )  2 for 0 <  < 1 and 0 <  1 <  2 , it is seen that from Figure 5 the diffusion flux increases with the decrease of ℎ() in the domain near zero but increases with the increase of ℎ() in the domain near one.In order to verify the efficiency and reliability of approximate solutions, approximate solutions obtained by the ADM are compared with the numerical ones obtained by the finite difference method in Figures 6-11.It is seen that the approximate solution is highly in agreement with the numerical solution.

Conclusions
This paper presented a similarity analysis for a class of generalized convection-diffusion with free boundary conditions.The partial differential equation together with the free boundary conditions was changed into a singular nonlinear ordinary differential equation of two-point boundary value problem by using similarity transformation.An efficient approximate technique of the problem was presented.The approximate solution of the problem was obtained and the corresponding transfer behavior is discussed.The numerical results show that the approximate solution proposed in this paper can be successfully used to reveal the physical nature of the studied problem.