Application of He's Homotopy Perturbation Method for Cauchy Problem of Ill-Posed Nonlinear Diffusion Equation

Abstract

We consider a Cauchy problem of unidimensional nonlinear diffusion equation on finite interval. This problem is ill-posed and its approximate solution is unstable. We apply the He's homotopy perturbation method (HPM) and obtain the third-order asymptotic expansion. We show that if the conductivity term in diffusion equation has a specified condition, the above solution can be estimated. Finally, a numerical experiment is provided to illustrate the method.

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Recommended by Marko Robnik

Department of Mathematics, K. N. Toosi University of Technology, P.O. Box 16315-1618, 1541849611 Tehran, Iran

Received 18 July 2009; Revised 14 March 2010; Accepted 31 March 2010

1. Introduction

The diffusion equation, one of the classical partial differential equations (PDEs), describes the process of diffusivity propagation. It has a great deal of application in different branches of sciences which have found a considerable amount of interest in recent years. This kind of equation arises naturally in a variety of models from theoretical physics, chemistry, and biology [1-8]. For instance, diffusion equations are used to investigate heat conduction, steady states and hysteresis, spatial patterns, blood oxygenation, moving fronts, pulses, and oscillations phenomena. Without any excessive simplification, these problems are all nonlinear. Therefore one needs to use a variety of different methods from different areas of mathematics such as numerical analysis, bifurcation and stability theory, similarity solutions, perturbations, topological methods, and many others, in order to study them [9-16].

Recently HPM is widely applied to linear and nonlinear problems. The method was proposed first by He in 1997 and systematical description in 2000 which is, in fact a coupling of traditional perturbation method and homotopy in topology. The application of the HPM to nonlinear problems has been developed, because this method continuously deforms the difficult problem under study into a simple one which is easy to solve. The method yields a very rapid convergence of the solution series in the most cases. Because of this rapid convergency, HPM has become a powerful mathematical tool, when it is successfully coupled with the perturbation theory. Also, HPM was used to solve variational problems by different investigators before [17-29]. One can find the recent developments of the HPM in [30-33].

This work is concerned to the nonlinear Cauchy diffusion problem and the HPM is applied to solve it. The organization of this paper is as follows. Section 2 is devoted to introduce the statement of Cauchy problem. In Section 3, we give the concepts of HPM. In Section 4, we derive the solution of Cauchy equation of nonlinear diffusion problem by HPM. In Section 5, we present an experiment wherein its numerical results illustrate the accuracy and efficiency of the proposed method, and finally in Section 6, some conclusions are considered.

2. Statement of the Cauchy Problem

Let [varphi](x,t) be a smooth function in Ω≡[0,l]×[0,T] where l and T are constant values, and f(t) , g(t) , a(t) , and b(t) are known functions in [0,T] . Now, we assume that u(x,t) satisfies the nonlinear Cauchy diffusion equation: [figure omitted; refer to PDF] subject to the initial conditions: [figure omitted; refer to PDF] where A is defined as [figure omitted; refer to PDF] such that a(t)u(x,t)+b(t) is positive [3-6], and u(x,t) is an unknown.

According to [34], we express HPM for the nonlinear problems in general case. Then, we apply this method to approximate the solution of the problem (2.1)-(2.3).

3. Description of the HPM

Suppose that A , a , b , Φ , f , and g satisfy to the above conditions. The operator A can be generally divided into two parts L and N , where L is a linear operator, and N is a nonlinear one. Therefore (2.1) can be rewritten as follows: [figure omitted; refer to PDF]

He [35] constructed a homotopy H:Ω×[0,1][arrow right]...... which satisfies [figure omitted; refer to PDF] or [figure omitted; refer to PDF] where p∈[0,1] , that is called a homotopy parameter, and _v0 is an initial approximation of (2.1) which satisfies initial conditions.

Hence, it is obvious that [figure omitted; refer to PDF]

Now, the changing process of p from 0 to 1 is just that of H(v,p) from L(v)-L(_v0 ) to A(v)-[varphi](x,t) .

Applying the perturbation technique due to the fact that 0≤p≤1 can be considered as a small parameter, we can assume that the solution of (3.2) or (3.3) can be expressed as a series in p , as follows: [figure omitted; refer to PDF] when p[arrow right]1; (3.2) or (3.3) corresponds to (3.1) and becomes the approximate solution of (3.1). That is, [figure omitted; refer to PDF]

The series solution (3.6) is convergent for different terms of v , and the rate of convergence depends on A(v) [36-38].

4. Solution of Cauchy Equation of Nonlinear Diffusion Problem by HPM

Consider the nonlinear differential equation (2.1), with the indicated initial conditions (2.2). From (2.1) we have [figure omitted; refer to PDF] Then we can write (4.1) as follows: [figure omitted; refer to PDF] where Ψ(x,t)=(-1/b(t))Φ(x,t), _Lx u=^∂2 u/∂^x2 and Nu=(1/b(t))(∂u/∂t)-(a(t)/b(t))((∂/∂x)(u∂u/∂x)) are the linear and nonlinear parts of Au , respectively.

By twice integration of (4.1) with respect to x , and applying the initial conditions (2.3), we obtain: [figure omitted; refer to PDF] Consequently, we obtain [figure omitted; refer to PDF]

By HPM, let F(u)=u(x,t)-h(x,t)=0, where h(x,t)=xg(t)+f(t)+^...0x Ψ(x,t)dx dx. That is, h(x,t)=xg(t)+f(t)-(1/b(t))^...0x Φ(x,t)dx dx.

Hence, we may choose a convex homotopy such that [23] [figure omitted; refer to PDF] where [figure omitted; refer to PDF] By using (4.5), we find [figure omitted; refer to PDF]

By combining (4.1) and (4.7), we obtain [figure omitted; refer to PDF] or [figure omitted; refer to PDF] where the above relations are obtained by equating the terms with identical powers of p in (4.8).

Therefore, the approximation solution is [figure omitted; refer to PDF]

In Section 5, we explain a numerical experiment. By using the HPM, an approximate solution for nonlinear diffusion equation is obtained.

5. Numerical Experiment

Let us consider the following nonlinear differential equation [figure omitted; refer to PDF] with initial conditions: [figure omitted; refer to PDF] If we want to use our last notation, we have [figure omitted; refer to PDF]

Obviously, the above assumptions satisfy to consideration of aforesaid conditions. In addition, the exact solution of the problem is: u(x,t)=^x2et +t .

In this experiment, we have obtained the solution of Cauchy problem at the points x = 0.1, 0.2, 0.3,...,1 , where t = 0.25, 0.50, 0.75 and 1.

We construct a homotopy in the same form as we have described in Section 3: [figure omitted; refer to PDF]

By substituting (3.5) into the above equation, and equating the terms with identical powers of p, we have [figure omitted; refer to PDF] The exact solution, approximate solution, absolute error, relative error, _L2 -norm error, maximum absolute error, and maximum relative error at some time levels are presented in Tables 1 and 2.

Table 1

(a) Exact solution, approximate solution, absolute error, and relative error of u(x,t) at the time t = 0.25

x	Exact solution	Approximate solution	Absolute error	Relative error
0.1	0.2628402542	0.2628402542	0.00	0.00
0.2	0.3013610167	0.3013610083	8.20×10-9	2.75×10-8
0.3	0.3655622875	0.3655622312	5.62×10-8	1.54×10-7
0.4	0.4554440667	0.4554438471	2.20×10-7	4.81×10-7
0.5	0.5710063542	0.5710057031	6.51×10-7	1.14×10-6
0.6	0.7122491501	0.7122475258	1.62×10-6	2.28×10-6
0.7	0.8791724543	0.8791688716	3.58×10-6	4.07×10-6
0.8	1.071776267	1.071769074	7.19×10-6	6.71×10-6
0.9	1.290060588	1.290047189	1.34×10-6	1.03×10-5
1	1.534025417	1.534001959	2.34×10-5	1.52×10-5

(b) Exact solution, approximate solution, absolute error, and relative error of u(x,t) at the time t = 0.50

x	Exact solution	Approximate solution	Absolute error	Relative error
0.1	0.5164872127	0.5164872161	3.40×10-9	6.19×10-9
0.2	0.5659488508	0.5659488481	2.70×10-9	4.77×10-9
0.3	0.6483849144	0.6483848411	7.33×10-8	1.13×10-7
0.4	0.7637954034	0.7637950723	3.31×10-7	4.33×10-7
0.5	0.9121803178	0.9121793115	1.01×10-6	1.10×10-6
0.6	1.093539658	1.093537168	2.49×10-6	2.27×10-6
0.7	1.307873423	1.307868043	5.38×10-6	4.11×10-6
0.8	1.555181613	1.555171074	1.05×10-5	6.77×10-6
0.9	1.835464230	1.835445111	1.91×10-5	1.04×10-5
1	2.148721271	2.148688717	3.26×10-5	1.51×10-5

(c) Exact solution, approximate solution, absolute error, and relative error of u(x,t) at the time t = 0.75

x	Exact solution	Approximate solution	Absolute error	Relative error
0.1	0.7711700002	0.7711700127	1.19×10-8	1.54×10-8
0.2	0.8346800007	0.8346800302	2.88×10-8	3.45×10-8
0.3	0.9405300015	0.9405299770	2.45×10-8	2.60×10-8
0.4	1.088720003	1.088719693	3.11×10-7	2.85×10-7
0.5	1.279250004	1.279248891	1.11×10-6	8.70×10-7
0.6	1.512120006	1.512117104	2.90×10-6	1.91×10-6
0.7	1.787330008	1.787323655	6.35×10-6	3.55×10-6
0.8	2.104880011	2.104867651	1.24×10-5	5.87×10-6
0.9	2.464770014	2.464748039	2.20×10-5	8.91×10-6
1	2.867000017	2.866963750	3.62×10-5	1.26×10-5

(d) Exact solution, approximate solution, absolute error, and relative error of u(x,t) at the time t = 1

x	Exact solution	Approximate solution	Absolute error	Relative error
0.1	1.027182818	1.027182849	3.07×10-8	2.98×10-8
0.2	1.108731273	1.108731374	1.00×10-7	9.04×10-8
0.3	1.244645364	1.244645495	1.31×10-7	1.04×10-7
0.4	1.434925092	1.434925050	4.23×10-8	2.94×10-8
0.5	1.679570457	1.679569755	7.01×10-7	4.17×10-7
0.6	1.978581458	1.978579197	2.26×10-6	1.14×10-6
0.7	2.331958096	2.331952871	5.22×10-6	2.24×10-6
0.8	2.739700370	2.739690321	1.00×10-5	3.66×10-6
0.9	3.201808281	3.201791426	1.69×10-5	5.26×10-6
1	3.718281828	3.718256914	2.49×10-5	6.70×10-6

Table 2: _L2 -norm error _{||ue (x,t)-ua (x,t)||2} , maximum absolute error, and maximum relative error at the times t= 0.25, 0.50, 0.75, and 1.

t	||ue (x,t)-ua (x,t)||2	Maximum absolute error	Maximum relative error
0.25	0.001934504062	0.00002345900000	0.00001529179356
0.5	0.002320886866	0.00003255400000	0.00001515040617
0.75	0.002486049062	0.00003626700000	0.00001264980809
1	0.002172409395	0.00002491464924	0.00000670058118

In the tables fortunately, we do not have any diametrical sharp changes in our error bounds and it has no common difficulties that may appear in numerical approaches like Runge's phenomenon [39]. This means that our method works steady at all time levels. Also, the computed relative errors magnitude is acceptable and it makes our approach admissible.

Figure 1 represents the exact solution of the nonlinear diffusion problem on the interval [0,1] . As it is illustrated in Figure 2, the approximate solution gives the solution in function form. We would like to emphasize that we have presented the results in tables at some points, in order to compare our computed values with exact solution easily.

Figure 1: Exact solution of the nonlinear diffusion problem on the interval [0,1].

[figure omitted; refer to PDF]

Figure 2: Approximate solution of the nonlinear diffusion problem on the interval [0,1].

[figure omitted; refer to PDF]

In addition, it is possible to draw the absolute error graph because it is yield in function form too. We drew absolute error function in Figure 3, to show how little its magnitude is.

Figure 3: Absolute error of the nonlinear diffusion problem on the interval [0,1].

[figure omitted; refer to PDF]

6. Conclusions

In this study, we consider the Cauchy problem of unidimensional nonlinear diffusion equation. This problem is inherently ill-posed and unsteady. If the analytical solution exists, it needs some rigid and sophisticated computation in practice. We investigate this problem with a very modern acclaimed powerful method called HPM. Our simple rapid exact approach yields good results as we have reported in Section 5 . We have computed an approximate solution with acceptable error bounds which are at least of order 1^0-5 . That makes our technique remarkable and convenient. We have used Maple 11 Packages on common home PC for all of our computations.

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