Abstract
The similarity transformation is introduced for studying free boundary value problems for a class of generalized convection-diffusion 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 , convection functional coefficient , power law exponent , and parameter on the flux transport characteristics are presented graphically and analyzed in detail.
1. 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 [1–4]. 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 [5–13]). 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 [15–17], 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 [23–25] 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 [26–30]). 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 , variable thermal conduction , and power law index on the flux transport characteristics are discussed by graph in detail.
2. Mathematical Formulation
In the paper, we consider a class of generalized convection-diffusion equation with free boundary conditions as follows: Here , is an unknown function and must be determined as a part of the solution; is given positive power law exponent; and is given free parameter. is assumed to be a real-valued continuous differential function defined on , , for . and are supposed to be real-valued nonnegative continuous differential functions defined on is convection functional coefficient, and is variable thermal conduction. is the heat density per unit area.
3. Transformation of (1)–(4)
Introducing the similarity transformation [31–34],
Substituting (5) into (1)–(4) and choosing , , , the nonlinear ordinary differential equation boundary value problem with respect to is derived as
Conversely, if is a solution of (6)–(8), then must be a similarity solution to (1)–(4).
Let be a solution to (6)–(8) and it is strictly decreasing in ; then its inverse function exists. Setting , then (6)–(8) can be converted into the following singular nonlinear ordinary differential equation of two-point boundary value problem
Here is diffusion flux. The derivation process indicates that only the positive solutions of (9)-(10) are physically significant.
4. Adomian Approximate Results and Discussion
Integrate equation (9) twice from 0 to for and to obtainwhere and is a twofold integral operator. Letting is presented as follows according to ADM:
The iterative formula of is presented as follows:where . We can obtain Adomian polynomial for and 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.
It is obvious that we can promptly obtain the value of by applying in (17) for each fixed and each fixed . Based on (17) and the corresponding we can easily obtain the graph of the diffusion flux distribution for , , and different values of and . The results are given in Figures 1–3.
It can be seen from Figure 1 that the diffusion flux decreases with increase of for , , and specific . Figure 2 indicates that for , , and specific , the diffusion flux appears with an utterly different characteristic with the changing .There exists a subinterval such that diffusion flux increases with the increase of for but decreases with the increase of for . Figure 3 describes the behavior of with the changes of parameter ; from Figure 3 we note that the diffusion flux increases sharply with the increasing of .
Similarly, (9)-(10) can be solved for , , . The corresponding approximate solutions are, respectively, given in (18)–(20).
It is very easy to obtain the value of by applying in (18)–(20). Based on (18)–(20) and the corresponding we can easily describe the graph of the diffusion flux distribution for and . The results are illustrated in Figure 4.
Considering , , we have . Figure 4 indicates that the diffusion flux decreases with the decrease of .
Similarly, it is easy to solve (9)-(10) for , , , . The corresponding approximate solutions are, respectively, given in (18) and (21)-(22).
We can obtain easily the value of by employing in (21)-(22). Based on (18), (21)-(22), and the corresponding , it is easy to get the graph of the diffusion flux distribution for and , . The results are illustrated in Figure 5.
Considering for and , it is seen that from Figure 5 the diffusion flux increases with the decrease of in the domain near zero but increases with the increase of 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.
5. 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.
Data Availability
All the data in this paper are correct and reliable. They can be obtained by using MATLAB.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
Both authors read and approved the final paper.
Acknowledgments
The research was supported by a grant from the National Natural Science Foundation of China (No. 11501496), the Natural Science Basic Research Plan in Shaanxi Province of China (No. 2014JQ2-1003), and the Doctor Start-up Research Fund of Yulin University (No. 13GK04).