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

Authors: Ali Zakeri; Azim Aminataei; Qodsiyeh Jannati
Journal: Discrete Dynamics in Nature and Society
 (2010)

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## 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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## Body

## 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. Section2 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

Letϕ(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:(2.1)Au=ϕ(x,t)inΩ0≡(0,l)×(0,T),
subject to the initial conditions:(2.2)u(0,t)=f(t),0≤t≤T,ux(0,t)=g(t),0≤t≤T,
where A is defined as(2.3)A(u(x,t))=∂tu(x,t)-∂x{(a(t)u(x,t)+b(t))∂xu(x,t)},
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 thatA, 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:(3.1)L(u)+N(u)-ϕ(x,t)=0.He [35] constructed a homotopy H:Ω×[0,1]→ℝ𝕖 which satisfies(3.2)H(v,p)=(1-p)[L(v)-L(v0)]+p[A(v)-ϕ(x,t)]=0,
or(3.3)H(v,p)=L(v)-L(v0)+pL(v0)+p[N(v)-ϕ(x,t)]=0.
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(3.4)H(v,0)=L(v)-L(v0)=0,H(v,1)=A(v)-ϕ(x,t)=0.Now, the changing process ofp 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 that0≤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:(3.5)v=v0+pv1+p2v2+p3v3⋯,
when p→1; (3.2) or (3.3) corresponds to (3.1) and becomes the approximate solution of (3.1). That is,(3.6)u=limp→1v=v0+v1+v2+v3+⋯.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(4.1)∂2u∂x2-{1b(t)∂u∂t-a(t)b(t)(∂∂x(u∂u∂x))}=-1b(t)Φ(x,t).
Then we can write (4.1) as follows:(4.2)Lxu-Nu=Ψ(x,t),
where Ψ(x,t)=(-1/b(t))Φ(x,t),    Lxu=∂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:(4.3)u(x,t)-xg(t)-f(t)-∫0x∫0xNudxdx=∫0x∫0xΨ(x,t)dxdx=-1b(t)∫0x∫0xΦ(x,t)dxdx.
Consequently, we obtain(4.4)u(x,t)=xg(t)+f(t)+∫0x∫0xNudxdx-1b(t)∫0x∫0xΦ(x,t)dxdx.By HPM, letF(u)=u(x,t)-h(x,t)=0, where h(x,t)=xg(t)+f(t)+∬0xΨ(x,t)dxdx. That is, h(x,t)=xg(t)+f(t)-(1/b(t))∬0xΦ(x,t)dxdx.Hence, we may choose a convex homotopy such that [23](4.5)H(v,p)=v(x,t)-h(x,t)-p∫0x∫0xNvdxdx=0,
where(4.6)F(u)=u(x,t)-h(x,t)=0,h(x,t)=xg(t)+f(t)+∫0x∫0xΨ(x,t)dxdx.
By using (4.5), we find(4.7)v(x,t)=h(x,t)+p∫0x∫0xNvdxdx.By combining (4.1) and (4.7), we obtain(4.8)v(x,t)=xg(t)+f(t)-1b(t)∫0x∫0xΦ(x,t)dxdx+p∫0x∫0x(1b(t)∂∂tv(x,t)-a(t)b(t)∂∂x(v(x,t)∂∂xv(x,t)))dxdx,
or(4.9)v0(x,t)=h(x,t)=xg(t)+f(t)-1b(t)∫0x∫0xΦ(x,t)dxdx,v1(x,t)=∫0x∫0x{1b(t)∂∂tv0-a(t)b(t)∂∂x(v0∂∂xv0)}dxdx,v2(x,t)=∫0x∫0x{1b(t)∂∂tv1-a(t)b(t)∂∂x(v0∂∂xv1+v1∂∂xv0)}dxdx,v3(x,t)=∫0x∫0x{1b(t)∂∂tv2-a(t)b(t)∂∂x(v0∂∂xv2+v1∂∂xv1+v2∂∂xv0)}dxdx,
where the above relations are obtained by equating the terms with identical powers of p in (4.8).Therefore, the approximation solution is(4.10)u(x,t)≃v0+v1+v2+v3.In Section5, 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(5.1)ut-∂∂x{(16e-tu+(t+5)e-t)∂u∂x}=-73t-9,(x,t)∈[0,1]×[0,1],
with initial conditions:(5.2)u(0,t)=t,0≤t≤1,ux(0,t)=0,0≤t≤1.
If we want to use our last notation, we have(5.3)Φ(x,t)=-73t-9,a(t)=16e-t,b(t)=(t+5)e-t.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 pointsx = 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 Section3: (5.4)H(v,p)=u(x,t)-h(x,t)-p∫0x∫0x{et(t+5)∂u∂t-16(t+5)∂∂x(u∂u∂x)}dxdx=0.By substituting (3.5) into the above equation, and equating the terms with identical powers of p, we have (5.5)v0(x,t)=16(t+5)(6t2+30t+7etx2t+27etx2),v1(x,t)=-etx2432(t+5)3(-129etx2+7etx2t2+6etx2t+528t2-540t-5400+84t3),v2(x,t)=etx277760(t+5)5(-12177e2tx4+705e2tx4t2-2865e2tx4t+161e2tx4t3+6930etx2t2-255150etx2t-526500etx2+16830etx2t3+1890etx2t4+2520t5+28440t4+63000t3-243000t2-810000tx4),v3(x,t)=-etx278382080(t+5)7(187092e3tx6t3-131550e3tx6t2-3508884e3tx6t+2457e3tx6t4-6173667e3tx6-49829472e2tx4t2-219217320e2tx4t-217954800e2tx4+357268e2tx4t4+713160e2tx4t3+31164e2tx4t5+84672etx2t6+1155168etx2t5+35592480etx2t4-131997600etx2t3-805140000etx2t2-578340000etx2t+1360800000etx2+423360t7+6894720t6+3447360t5+1209600t4-34020000t3-680400000t2).
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 ofu(x,t) at the time t = 0.25
xExact solutionApproximate solutionAbsolute errorRelative error0.10.26284025420.26284025420.000.000.20.30136101670.30136100838.20×10-92.75×10-80.30.36556228750.36556223125.62×10-81.54×10-70.40.45544406670.45544384712.20×10-74.81×10-70.50.57100635420.57100570316.51×10-71.14×10-60.60.71224915010.71224752581.62×10-62.28×10-60.70.87917245430.87916887163.58×10-64.07×10-60.81.0717762671.0717690747.19×10-66.71×10-60.91.2900605881.2900471891.34×10-61.03×10-511.5340254171.5340019592.34×10-51.52×10-5(b)
Exact solution, approximate solution, absolute error, and relative error ofu(x,t) at the time t = 0.50
xExact solutionApproximate solutionAbsolute errorRelative error0.10.51648721270.51648721613.40×10-96.19×10-90.20.56594885080.56594884812.70×10-94.77×10-90.30.64838491440.64838484117.33×10-81.13×10-70.40.76379540340.76379507233.31×10-74.33×10-70.50.91218031780.91217931151.01×10-61.10×10-60.61.0935396581.0935371682.49×10-62.27×10-60.71.3078734231.3078680435.38×10-64.11×10-60.81.5551816131.5551710741.05×10-56.77×10-60.91.8354642301.8354451111.91×10-51.04×10-512.1487212712.1486887173.26×10-51.51×10-5(c)
Exact solution, approximate solution, absolute error, and relative error ofu(x,t) at the time t = 0.75
xExact solutionApproximate solutionAbsolute errorRelative error0.10.77117000020.77117001271.19×10-81.54×10-80.20.83468000070.83468003022.88×10-83.45×10-80.30.94053000150.94052997702.45×10-82.60×10-80.41.0887200031.0887196933.11×10-72.85×10-70.51.2792500041.2792488911.11×10-68.70×10-70.61.5121200061.5121171042.90×10-61.91×10-60.71.7873300081.7873236556.35×10-63.55×10-60.82.1048800112.1048676511.24×10-55.87×10-60.92.4647700142.4647480392.20×10-58.91×10-612.8670000172.8669637503.62×10-51.26×10-5(d)
Exact solution, approximate solution, absolute error, and relative error ofu(x,t) at the time t = 1
xExact solutionApproximate solutionAbsolute errorRelative error0.11.0271828181.0271828493.07×10-82.98×10-80.21.1087312731.1087313741.00×10-79.04×10-80.31.2446453641.2446454951.31×10-71.04×10-70.41.4349250921.4349250504.23×10-82.94×10-80.51.6795704571.6795697557.01×10-74.17×10-70.61.9785814581.9785791972.26×10-61.14×10-60.72.3319580962.3319528715.22×10-62.24×10-60.82.7397003702.7396903211.00×10-53.66×10-60.93.2018082813.2017914261.69×10-55.26×10-613.7182818283.7182569142.49×10-56.70×10-6Table 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)∥2Maximum absolute errorMaximum relative error0.250.0019345040620.000023459000000.000015291793560.50.0023208868660.000032554000000.000015150406170.750.0024860490620.000036267000000.0000126498080910.0021724093950.000024914649240.00000670058118In 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.Figure1 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 Figure3, 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 Section5. 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.



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*Source: 780207-2010-06-02.xml*