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.

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 𝜙(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:𝐴𝑢=𝜙(𝑥,𝑡)inΩ₀≡(0,𝑙)×(0,𝑇), (2.1) subject to the initial conditions:𝑢𝑢(0,𝑡)=𝑓(𝑡),0≤𝑡≤𝑇,_𝑥(0,𝑡)=𝑔(𝑡),0≤𝑡≤𝑇, (2.2) where 𝐴 is defined as𝐴(𝑢(𝑥,𝑡))=𝜕_𝑡𝑢(𝑥,𝑡)−𝜕_𝑥{(𝑎(𝑡)𝑢(𝑥,𝑡)+𝑏(𝑡))𝜕_𝑥𝑢(𝑥,𝑡)}, (2.3) 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:𝐿(𝑢)+𝑁(𝑢)−𝜙(𝑥,𝑡)=0. (3.1)

He [35] constructed a homotopy 𝐻∶Ω×[0,1]→ℝ𝕖 which satisfies𝐻(𝑣,𝑝)=(1−𝑝)[𝐿(𝑣)−𝐿(𝑣₀)]+𝑝[𝐴(𝑣)−𝜙(𝑥,𝑡)]=0, (3.2) or𝐻(𝑣,𝑝)=𝐿(𝑣)−𝐿(𝑣₀)+𝑝𝐿(𝑣₀)+𝑝[𝑁(𝑣)−𝜙(𝑥,𝑡)]=0. (3.3) 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 that𝐻(𝑣,0)=𝐿(𝑣)−𝐿(𝑣₀)=0,𝐻(𝑣,1)=𝐴(𝑣)−𝜙(𝑥,𝑡)=0. (3.4)

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)−𝜙(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:𝑣=𝑣₀+𝑝𝑣₁+𝑝²𝑣₂+𝑝³𝑣₃⋯, (3.5) when 𝑝→1; (3.2) or (3.3) corresponds to (3.1) and becomes the approximate solution of (3.1). That is,𝑢=lim_𝑝→1𝑣=𝑣₀+𝑣₁+𝑣₂+𝑣₃+⋯. (3.6)

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𝜕²𝑢𝜕𝑥²1−{𝑏(𝑡)𝜕𝑢−𝜕𝑡𝑎(𝑡)(𝜕𝑏(𝑡)𝜕𝑥(𝑢𝜕𝑢𝜕𝑥))}=−1𝑏(𝑡)Φ(𝑥,𝑡). (4.1) Then we can write (4.1) as follows:𝐿_𝑥𝑢−𝑁𝑢=Ψ(𝑥,𝑡), (4.2) where Ψ(𝑥,𝑡)=(−1/𝑏(𝑡))Φ(𝑥,𝑡),   𝐿_𝑥𝑢=𝜕²𝑢/𝜕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:𝑢(𝑥,𝑡)−𝑥𝑔(𝑡)−𝑓(𝑡)−^𝑥₀^𝑥₀𝑁_𝑢=𝑑𝑥𝑑𝑥_𝑥0𝑥0=Ψ(𝑥,𝑡)𝑑𝑥𝑑𝑥−1𝑏(𝑡)^𝑥₀^𝑥₀Φ(𝑥,𝑡)𝑑𝑥𝑑𝑥. (4.3) Consequently, we obtain𝑢(𝑥,𝑡)=𝑥𝑔(𝑡)+𝑓(𝑡)+^𝑥₀^𝑥₀𝑁_𝑢1𝑑𝑥𝑑𝑥−𝑏(𝑡)^𝑥₀^𝑥₀Φ(𝑥,𝑡)𝑑𝑥𝑑𝑥. (4.4)

By HPM, let 𝐹(𝑢)=𝑢(𝑥,𝑡)−ℎ(𝑥,𝑡)=0, where ℎ(𝑥,𝑡)=𝑥𝑔(𝑡)+𝑓(𝑡)+∬^𝑥₀Ψ(𝑥,𝑡)𝑑𝑥𝑑𝑥. That is, ℎ(𝑥,𝑡)=𝑥𝑔(𝑡)+𝑓(𝑡)−(1/𝑏(𝑡))∬^𝑥₀Φ(𝑥,𝑡)𝑑𝑥𝑑𝑥.

Hence, we may choose a convex homotopy such that [23]𝐻(𝑣,𝑝)=𝑣(𝑥,𝑡)−ℎ(𝑥,𝑡)−𝑝^𝑥₀^𝑥₀𝑁_𝑣𝑑𝑥𝑑𝑥=0, (4.5) where𝐹(𝑢)=𝑢(𝑥,𝑡)−ℎ(𝑥,𝑡)=0,ℎ(𝑥,𝑡)=𝑥𝑔(𝑡)+𝑓(𝑡)+^𝑥₀^𝑥₀Ψ(𝑥,𝑡)𝑑𝑥𝑑𝑥. (4.6) By using (4.5), we find𝑣(𝑥,𝑡)=ℎ(𝑥,𝑡)+𝑝^𝑥₀^𝑥₀𝑁_𝑣𝑑𝑥𝑑𝑥. (4.7)

By combining (4.1) and (4.7), we obtain1𝑣(𝑥,𝑡)=𝑥𝑔(𝑡)+𝑓(𝑡)−𝑏(𝑡)^𝑥₀^𝑥₀Φ(𝑥,𝑡)𝑑𝑥𝑑𝑥+𝑝^𝑥₀^𝑥₀(1𝜕𝑏(𝑡)𝜕𝑡𝑣(𝑥,𝑡)−𝑎(𝑡)𝜕𝑏(𝑡)𝜕𝜕𝑥(𝑣(𝑥,𝑡)𝜕𝑥𝑣(𝑥,𝑡)))𝑑𝑥𝑑𝑥, (4.8) or𝑣₀1(𝑥,𝑡)=ℎ(𝑥,𝑡)=𝑥𝑔(𝑡)+𝑓(𝑡)−𝑏(𝑡)^𝑥₀^𝑥₀𝑣Φ(𝑥,𝑡)𝑑𝑥𝑑𝑥,₁(𝑥,𝑡)=^𝑥₀^𝑥₀{1𝜕𝑏(𝑡)𝑣𝜕𝑡⁰−𝑎(𝑡)𝜕𝑏(𝑡)𝜕𝑥(𝑣₀𝜕𝑣𝜕𝑥⁰𝑣)}𝑑𝑥𝑑𝑥,₂(𝑥,𝑡)=^𝑥₀^𝑥₀{1𝜕𝑏(𝑡)𝑣𝜕𝑡¹−𝑎(𝑡)𝜕𝑏(𝑡)𝜕𝑥(𝑣₀𝜕𝑣𝜕𝑥¹+𝑣₁𝜕𝑣𝜕𝑥⁰𝑣)}𝑑𝑥𝑑𝑥,₃(𝑥,𝑡)=^𝑥₀^𝑥₀{1𝜕𝑏(𝑡)𝑣𝜕𝑡²−𝑎(𝑡)𝜕𝑏(𝑡)𝜕𝑥(𝑣₀𝜕𝑣𝜕𝑥²+𝑣₁𝜕𝑣𝜕𝑥¹+𝑣₂𝜕𝑣𝜕𝑥⁰)}𝑑𝑥𝑑𝑥, (4.9) where the above relations are obtained by equating the terms with identical powers of 𝑝 in (4.8).

Therefore, the approximation solution is𝑢(𝑥,𝑡)≃𝑣₀+𝑣₁+𝑣₂+𝑣₃. (4.10)

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 equation𝑢_𝑡−𝜕1𝜕𝑥{(6𝑒^−𝑡𝑢+(𝑡+5)𝑒^−𝑡)𝜕𝑢7𝜕𝑥}=−3𝑡−9,(𝑥,𝑡)∈[0,1]×[0,1], (5.1) with initial conditions:𝑢(0,𝑡)=𝑡,0≤𝑡≤1,𝑢_𝑥(0,𝑡)=0,0≤𝑡≤1. (5.2) If we want to use our last notation, we have7Φ(𝑥,𝑡)=−31𝑡−9,𝑎(𝑡)=6𝑒^−𝑡,𝑏(𝑡)=(𝑡+5)𝑒^−𝑡. (5.3)

Obviously, the above assumptions satisfy to consideration of aforesaid conditions. In addition, the exact solution of the problem is: 𝑢(𝑥,𝑡)=𝑥²𝑒^𝑡+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: 𝐻(𝑣,𝑝)=𝑢(𝑥,𝑡)−ℎ(𝑥,𝑡)−𝑝^𝑥₀^𝑥₀{𝑒^𝑡(𝑡+5)𝜕𝑢−1𝜕𝑡𝜕6(𝑡+5)𝜕𝑥(𝑢𝜕𝑢𝜕𝑥)}𝑑𝑥𝑑𝑥=0. (5.4)

By substituting (3.5) into the above equation, and equating the terms with identical powers of 𝑝, we have 𝑣₀1(𝑥,𝑡)=6(𝑡+5)(6𝑡²+30𝑡+7𝑒^𝑡𝑥²𝑡+27𝑒^𝑡𝑥²𝑣),₁(𝑥,𝑡)=−𝑒^𝑡𝑥²432(𝑡+5)³(−129𝑒^𝑡𝑥²+7𝑒^𝑡𝑥²𝑡²+6𝑒^𝑡𝑥²𝑡+528𝑡²−540𝑡−5400+84𝑡³𝑣),₂𝑒(𝑥,𝑡)=^𝑡𝑥²77760(𝑡+5)⁵(−12177𝑒^2𝑡𝑥⁴+705𝑒^2𝑡𝑥⁴𝑡²−2865𝑒^2𝑡𝑥⁴𝑡+161𝑒^2𝑡𝑥⁴𝑡³+6930𝑒^𝑡𝑥²𝑡²−255150𝑒^𝑡𝑥²𝑡−526500𝑒^𝑡𝑥²+16830𝑒^𝑡𝑥²𝑡³+1890𝑒^𝑡𝑥²𝑡⁴+2520𝑡⁵+28440𝑡⁴+63000𝑡³−243000𝑡²𝑣−810000𝑡),₃𝑒(𝑥,𝑡)=−^𝑡𝑥²78382080(𝑡+5)⁷(187092𝑒^3𝑡𝑥⁶𝑡³−131550𝑒^3𝑡𝑥⁶𝑡²−3508884𝑒^3𝑡𝑥⁶𝑡+2457𝑒^3𝑡𝑥⁶𝑡⁴−6173667𝑒^3𝑡𝑥⁶−49829472𝑒^2𝑡𝑥⁴𝑡²−219217320𝑒^2𝑡𝑥⁴𝑡−217954800𝑒^2𝑡𝑥⁴+357268𝑒^2𝑡𝑥⁴𝑡⁴+713160𝑒^2𝑡𝑥⁴𝑡³+31164𝑒^2𝑡𝑥⁴𝑡⁵+84672𝑒^𝑡𝑥²𝑡⁶+1155168𝑒^𝑡𝑥²𝑡⁵[phone]𝑒^𝑡𝑥²𝑡⁴−131997600𝑒^𝑡𝑥²𝑡³−805140000𝑒^𝑡𝑥²𝑡²−578340000𝑒^𝑡𝑥²𝑡+1360800000𝑒^𝑡𝑥²+423360𝑡⁷+6894720𝑡⁶+3447360𝑡⁵+1209600𝑡⁴−34020000𝑡³−680400000𝑡²). (5.5)(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12) 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 

Table 2 

𝐿₂-norm error ‖𝑢_𝑒(𝑥,𝑡)−𝑢_𝑎(𝑥,𝑡)‖₂, maximum absolute error, and maximum relative error at the times 𝑡= 0.25, 0.50, 0.75, and 1.

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 2 

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

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].

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 10⁻⁵. That makes our technique remarkable and convenient. We have used Maple 11 Packages on common home PC for all of our computations.