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

Authors: Yunbin Xu; Meihua Wei
Journal: Advances in Mathematical Physics
 (2019)

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## 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 conductionk(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.


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## Body

## 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 h(z), variable thermal conduction k(z), and power law index n 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:(1)∂u∂t=p′tpnt∂∂xku∂u∂xn-1∂u∂x+p′thu∂u∂x,x,t∈G,(2)ux=φt=1,t>0,(3)ku∂u∂xn-1∂u∂xx=φt=αφ′tp′tpnt,(4)ut=0=0,x>φ0=0.Here G=x,t;x>φt,t>0, φ(t) is an unknown function and must be determined as a part of the solution; n is given positive power law exponent; and α≥0 is given free parameter. p(t) is assumed to be a real-valued continuous differential function defined on [0,+∞), p(0)=0, p(t)>0 for t>0. k(z) and h(z) are supposed to be real-valued nonnegative continuous differential functions defined on [0,1],h(z)=(1-z)q is convection functional coefficient, and k(z) is variable thermal conduction. -k(u)uxn-1ux is the heat density per unit area.


## 3. Transformation of (1)–(4)

Introducing the similarity transformation [31–34],(5)ux,t=uη,η=Axβpγt,φt=η0pt,η0>0.Substituting (5) into (1)–(4) and choosing A=1, β=1, γ=-1, the nonlinear ordinary differential equation boundary value problem with respect to η is derived as (6)kuηu′ηn-1u′η′+η+huηu′η=0,(7)uη0=1,u+∞=0,(8)kuηu′ηn-1u′ηη=η0=αη0.Conversely, ifu(η) is a solution of (6)–(8), then (u(x,t),φ(t)) must be a similarity solution to (1)–(4).Letz=u(η) be a solution to (6)–(8) and it is strictly decreasing in [η0,+∞); then its inverse function η=y(z) exists. Setting g(z)=k(z)(-y′(z))-n, then (6)–(8) can be converted into the following singular nonlinear ordinary differential equation of two-point boundary value problem(9)g′′z=-k1/nzg-1/nz+h′z,(10)g0=0,g1+αg′1=0.Hereg(z) 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 z for k(z)=z and h(z)=1-z to obtain(11)gz=σz-z22-L-1z1/ng-1/n,where σ=g′(0) and L-1(·)=∫0z∫0z(·)dzdz is a twofold integral operator. Letting(12)gz=∑n=0∞gnz,gn is presented as follows according to ADM:(13)g0=σz,g1=-z22-L-1z1/nA0,g2=-L-1z1/nA1,⋮gn+1=-L-1z1/nAnn=1,2,⋯.The iterative formula ofAn is presented as follows:(14)A0=fg0,A1=g1ddg0fg0,A2=g2ddg0fg0+g122!d2dg02fg0,A3=g3ddg0fg0+g1g2d2dg02fg0+g133!d3dg03fg0,⋮where f(g)=g-1/n. We can obtain Adomian polynomial for k(z)=z and h(z)=1-z according to the iterative formula of An as follows:(15)A0=1σ1/nz-1/n,A1=12nσ1+1/n+12nσ1+2/nz1-1/n,A2=n+18n2σ2+1/n+3n+412n2σ2+2/n+3n+524n2σ2+3/nz2-1/n,A3=2n2+3n+148n3σ3+1/n+12n2+23n+1196n3σ3+2/n+18n2+42n+25144n3σ3+3/n+12n2+33n+23288n3σ3+4/nz3-1/n.Substituting the derived Adomian polynomial (15) into (13), we obtain(16)g0=σz,g1=-12+12σ1/nz2,g2=-112nσ1+1/n+112nσ1+2/nz3,g3=-n+196n2σ2+1/n+3n+4144n2σ2+2/n+3n+5288n2σ2+3/nz4,g4=-2n2+3n+1960n3σ3+1/n+12n2+23n+111920n3σ3+2/n+18n2+42n+252880n3σ3+3/n+12n2+33n+235760n3σ3+4/nz5.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.(17)g=g0+g1+g2+g3+g4=σz-12+12σ1/nz2-112nσ1+1/n+112nσ1+2/nz3-n+196n2σ2+1/n+3n+4144n2σ2+2/n+3n+5288n2σ2+3/nz4-2n2+3n+1960n3σ3+1/n+12n2+23n+111920n3σ3+2/n+18n2+42n+252880n3σ3+3/n+12n2+33n+235760n3σ3+4/nz5It is obvious that we can promptly obtain the value ofσ by applying g(1)+αg′(1)=0 in (17) for each fixed α and each fixed n. Based on (17) and the corresponding σ we can easily obtain the graph of the diffusion flux distribution for k(z)=z, h(z)=1-z, and different values of α and n. The results are given in Figures 1–3.Figure 1
Flux distribution fork(z)=z,h(z)=1-z,α=0.1,n=1.0,5.0,10.0.Figure 2
Flux distribution fork(z)=z,h(z)=1-z,α=1.0,n=0.2,0.5,0.8.Figure 3
Flux distribution fork(z)=z,h(z)=1-z,n=0.5,α=0.1,0.5,1.0,2.0.It can be seen from Figure1 that the diffusion flux g(z) decreases with increase of n(n≥1) for k(z)=z, h(z)=1-z, and specific α. Figure 2 indicates that for k(z)=z, h(z)=1-z, and specific α, the diffusion flux g(z) appears with an utterly different characteristic with the changing n(0<n<1).There exists a subinterval [0,z0]⊂[0,1] such that diffusion flux g(z) increases with the increase of n(0<n<1) for z∈[0,z0] but decreases with the increase of n(0<n<1) for z∈[z0,1]. Figure 3 describes the behavior of g(z) with the changes of parameter α; from Figure 3 we note that the diffusion flux g(z) increases sharply with the increasing of α.Similarly, (9)-(10) can be solved for n=1.0, h(z)=1-z, k(z)=z,z2,z3. The corresponding approximate solutions are, respectively, given in (18)–(20).(18)g=σz-12+12σz2-112σ2+112σ3z3-148σ3+7144σ4+136σ5z4-1160σ4+23960σ5+17576σ6+171440σ7z5,(19)g=σz-12z2-16σz3-124σ2z4-148σ3z5-190σ4z6-316048σ5z7-1713440σ6z8-25217728σ7z9,(20)g=σz-12z2-112σz4-140σ2z5-1120σ3z6-1504σ3+1336σ4z7-136720σ4z8-2317280σ5z9-110080σ5z10-3732217600σ6z11-619434880σ7z13.It is very easy to obtain the value ofσ by applying g(1)+αg′(1)=0 in (18)–(20). Based on (18)–(20) and the corresponding σ we can easily describe the graph of the diffusion flux distribution for h(z)=1-z and k(z)=z,z2,z3. The results are illustrated in Figure 4.Figure 4
Flux distribution forα=1.0, n=1.0, h(z)=1-z, k(z)=z,z2,z3.Considering0<p1<p2, 0<z<1, we have zp1>zp2. Figure 4 indicates that the diffusion flux decreases with the decrease of k(z).Similarly, it is easy to solve (9)-(10) for n=1.0, k(z)=z, h(z)=(1-z)q, q=1,2,3. The corresponding approximate solutions are, respectively, given in (18) and (21)-(22).(21)g=σz-1+12σz2+13-16σ2-112σ3z3+136σ2-112σ3-772σ4-136σ5z4+130σ3-23720σ4-23240σ5-17288σ6-171440σ7z5(22)g=σz-32+12σz2+1-14σ2-112σ3z3-14-112σ2+316σ3+748σ4+136σ5z4-180σ2-320σ3+1196σ4+69320σ5+17192σ6+171440σ7z5We can obtain easily the value ofσ by employing g(1)+αg′(1)=0 in (21)-(22). Based on (18), (21)-(22), and the corresponding σ, it is easy to get the graph of the diffusion flux distribution for k(z)=z and h(z)=(1-z)q, q=1,2,3. The results are illustrated in Figure 5.Figure 5
Flux distribution forα=1.0, n=1.0, k(z)=z, h(z)=1-z,(1-z)2,(1-z)3.Considering(1-z)q1>(1-z)q2 for 0<z<1 and 0<q1<q2, it is seen that from Figure 5 the diffusion flux increases with the decrease of h(z) in the domain near zero but increases with the increase of h(z) 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 Figures6–11. It is seen that the approximate solution is highly in agreement with the numerical solution.Figure 6
The comparison of the results of approximate solution and numerical solution forα=0.1, n=2.5, k(z)=z, h(z)=1-z.Figure 7
The comparison of the results of approximate solution and numerical solution forα=0.5, n=5.0, k(z)=z, h(z)=1-z.Figure 8
The comparison of the results of approximate solution and numerical solution forα=1.0, n=8.0, k(z)=z, h(z)=1-z.Figure 9
The comparison of the results of approximate solution and numerical solution forα=2.0, n=1.0, k(z)=z3, h(z)=1-z.Figure 10
The comparison of the results of approximate solution and numerical solution forα=3.0, n=1.0, k(z)=z, h(z)=(1-z)3.Figure 11
The comparison of the results of approximate solution and numerical solution forα=10.0, n=1.0, kz=z, hz=1-z2.


## 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.



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