# An Inverse Problem for Parabolic Partial Differential Equations with  Nonlinear Conductivity Term

Authors: Ali Zakeri; Q. Jannati
Journal: Scholarly Research Exchange
 (2009)

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## Abstract

We consider an inverse problem for partial differential equation with nonlinear conductivity
term in one-dimensional space within a finite interval. In the considered problem, a temperature
history is unknown in a boundary of domain. The homotopy perturbation technique is
used. Moreover, we have presented a numerical example.


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## Body

## 1. Introduction

Inverse heat
conduction problems (IHCPs) rely on temperature heat flux measurements for
estimating unknown quantities in the analysis of physical problems in thermal
engineering. As an example, inverse problems dealing with heat conduction have
been generally associated with estimating an unknown boundary heat flux by
using temperature measurements taken below the boundary surfaces. Therefore,
while in the classical direct heat conduction problem the cause (boundary heat
flux) is given and the effect (temperature field in the body) is determined, the inverse problem involves the estimation of the cause from the knowledge of
the effect. An advantage of IHCP is that it enables a much closer collaboration
between experimental and theoretical researchers in order to obtain the maximum
of information regarding the physical problem under study.Difficulties encountered in the solution of IHCPs
should be recognized. IHCPs are mathematically classified as ill-posed in a
general sense because their solutions may become unstable, as a result of the
errors inherent to the measurements used in the analysis. Inverse problems were
initially taken as not of physical interest due to their ill-posedness.In recent years, some new methods for estimating
surface heat flux in IHCPs developed in the theory and practice. Consequently,
some methods for approximate solution of these problems have developed. To
obtain stable results, special numerical techniques should be used. Examples
include iterative gradient methods, optimization algorithms, regularization
methods, function specification methods, space-marching method, conjugate
gradient method, Levenberg-Marquardt method, iterative techniques, variational
iteration, finite elements, finite volumes, boundary elements methods, and
so on. In some works, the numerical methods have been used for IHCPs [1–14].
For instance, in [14]
the variational iteration method is used to find the exact solution of a
control parameter in parabolic equations. Unfortunately, most of these methods
are useful in the linear IHCPs merely.The homotopy perturbation method (HPM) was first
proposed by the Chinese mathematician He [15–17]. The essential idea of this method is to introduce a
homotopy parameter, say p, which takes values from 0 to 1. When p=0, the system of equations usually reduces to a
sufficiently simplified form, which normally admits a rather simple solution.
As p is gradually increased to 1, the system goes through a sequence of
deformations, the solution for each of which is close to that at the previous
stage of deformation. Eventually, at p=1, the system takes the original form of the
equation and the final stage of deformation gives the desired solution. One of
the most remarkable features of the HPM is that usually just few perturbation
terms are sufficient for obtaining a reasonably accurate solution. Considerable
research works have been conducted recently in applying this method to a class
of linear and nonlinear equations [14–23]. The interested reader
can see [21–23] for last development of
HPM. This homotopy perturbation method will become a much more interesting
method to solving nonlinear differential equations in science and engineering.In this paper, we consider a one-dimensional nonlinear
inverse heat conduction problem with nonlinear diffusivity term that
temperature history is unknown in a boundary. Then, using finite difference method
and discrete time variable, the partial
differential equation converts to a system of nonlinear ordinary differential
equations. Consequently, by applying the homotopy perturbation technique and
estimate solution, the temperature distribution in domain at discrete times
will be found. Finally, a numerical experiment is given.


## 2. Homotopy Perturbation Technique

We begin with
the following definition which is presented in [24].Definition 1.
LetX and Y be topologic spaces. If f and g are continuous maps of the space X into space Y,
it is said that f is homotopic to g,
if there is a continuous map F:X×I=[0,1]→Y such that
(1)F(x,0)=f(x),F(x,1)=g(x)foreachx∈X.
The map F is called a homotopy between f and g.To illustrate homotopy perturbation method, we
consider the following nonlinear equation:(2)A(u)−f(r)=0,(r∈Ω),
with the boundary
conditions:
(3)B(u,∂u∂n)=0,(r∈Γ),
where A is a general differential operator, B is a boundary operator,
f(r) is a known analytic function, and Γ is the boundary of domain Ω. The operator A can be generally divided in to two parts F and N, where F and N are linear and nonlinear parts of A, respectively. However, (2) converts to the following form:(4)L(u)+N(u)−f(r)=0.In [15], the author constructs a homotopy V:Ω×[0,1]→ℝ which satisfies
(5)H(v,p)=(1−p)[L(v)−L(v0)]+p[A(v)−f(r)]=0,
or
(6)H(v,p)=L(v)−L(v0)+pL(v0)+p[N(v)−f(r)]=0,
where r∈ℝ and p∈[0,1]. In this situation, the parameter p is called homotopy parameter and v0 is an initial approximation of (2) which satisfies boundary conditions. When p=0 or p=1, we have
(7)H(v,0)=L(v)−L(v0)=0,H(v,1)=A(v)−f(r)=0.
On the other hand, if p∈(0,1),
then the homotopy H(v,p) changes from L(v)−L(v0) to A(v)−f(r).Noticing that0≤p≤1 can be considered as a small parameter,
applying the perturbation technique, we may assume that the solution of (5) or
(6) can be expressed as a series in p, as follows:
(8)v=v0+pv1+p2v2+⋯.
When p→1, (5) or (6) corresponds to (4) and becomes the approximate solution of (4), that
is, we have
(9)u=limp→1v=v0+v1+v2+⋯.
Series (9) is convergent for
most cases and the rate of convergence depends on A(v) (for more details see [16, 17]).


## 3. Solution of Nonlinear Parabolic Problem by Homotopy Perturbation Technique

LetΦ(x,t) be given function in Ω≡[0,l]×[0,T],f(t),g(t) and s(x) known functions on [0,T] and [0,l],
respectively.Now, consider the following nonlinear differential
equation:(10)ut−{(a(t)u+b(t))ux}x=Φ(x,t),(x,t)∈Ω0≡(0,l)×(0,T),
with initial condition:
(11)u(x,0)=s(x),x∈[0,l],
and boundary conditions:
(12)u(0,t)=f(t),t∈[0,T],ux(0,t)=g(t),t∈[0,T],
where a,b,f,g,
and s are known function such that b(t) is far from zero in [0,T].Suppose thatu0(x)=u(x,0)=s(x),  nΔt=T,  k=1/Δt,  tj=jΔt,  uj(x)=u(x,tj), and Φj(x)=Φ(x,tj) for j∈Jn={1,2,…,n}. By using the backward finite difference scheme
for term ut in (10) in the form(13)ut(x)≃k(uj(x)−uj−1(x)),j∈Jn,
and substituting to (10), we find
a system of second-order differential equations with respect to x. Then, we obtain(14)k(uj(x)−uj−1(x))−ddx{(a(tj)uj(x)+b(tj))ddxuj(x)}=Φj(x),1≤j≤n,
or
(15)d2dx2uj(x)−{kb(tj)(uj(x)−uj−1(x))−a(tj)b(tj)(ddx(uj(x)ddxuj(x)))}=−1b(tj)Φj(x).Now, we can write (2) as follows:
(16)Au=Lxu−Nu=Ψ(x,t),where Ψ(x,t)=(−1/b(tj))Φj(x),
and Lx=d2/dx2 and Nu=(k/b(tj))(uj(x)−uj−1(x))+(a(tj)/b(tj))((d/dx)(uj(x)(d/dx)uj(x))) are the linear and nonlinear parts of operator A, respectively.For simplicity, defineu(x)=(u1(x),u2(x),…,un(x))T and D(u(x),v(x))=(u1(x)(d/dx)v1(x),u2(x)(d/dx)v2×(x),…,un(x)(d/dx)vn(x))T. Now, by putting (17)m=(−kb(t1)s(x),0,…,0),M1=(kb(t1)0−kb(t2)kb(t2)⋱⋱0−kb(tn)kb(tn)),M2=diag(a(t1)b(t1),a(t2)b(t2),…,a(tn)b(tn)),  and D(u(x),u(x))=(u1(x)(d/dx)u1(x),…,un(x)(d/dx)un(x))T, where uj(x) are the values of u at t=tj for 1≤j≤n, and the notation diag(α1,α2,…,αn) refers to a diagonal matrix in the form
(18)(α10⋯00α2⋱⋮⋮⋱⋱00⋯0αn). We can express (16) to the matrix form(19)Nu=−M2ddx(D(u,u))+M1u+m.By twice integration of (10) with respect to x and applying the initial conditions, we
obtain(20)u(x)−xg(t)−f(t)−∫0x∫0xNu(x)dxdx=∫0x∫0xΨ(x)dxdx=−∫0x∫0xΦb(x)dxdx,where g(t)=(g(t1),…,g(tn))T, f(t)=(f(t1),…,f(tn))T, Ψ(x)=diag(Ψ1(x),…,Ψn(x)), and Φb(x)=diag(Φ1(x)/b(t1),…,Φn(x)/b(tn)).By homotopy perturbation method, we may choose a
convex homotopy such that [18, 20](21)H(v(x),p)=v(x)−h(x)−p∫0x∫0xNv(x)dxdx=0,(22)F(u(x))=u(x)−h(x)=0,where(23)h(x)=xg(t)+f(t)−∫0x∫0xΨ(x)dxdx,v(x)=(v(t1),…,v(tn))T, and h(x)=(h(t1),…,h(tn))T. By using (16), we can write(24)v(x)=h(x)+p∫0x∫0xNv(x)dxdx.By combining (10) and (22), we
obtain following results:   (25)v(x)=xg(t)+f(t)−∫0x∫0xΦb(x)dxdx+p∫0x∫0x{M1(v(x)−us(x))−M2ddxD(u(x),u(x))}dxdx,where us(x)=(s(x),u1(x),…,un−1x))T,
or
(26)v0=h(x)=xg(t)+f(t)−∫0x∫0xΦb(x)dxdx,v1=∫0x∫0x{M1(v0−us(x))−M2ddxD(v0,v0)}dxdx,v2=∫0x∫0x{diag(kb(t1),…,kb(tn))v1−M2ddx(D(v0,v1)+D(v1,v0))}dxdx,
where the above
relations are obtained of equating the terms with
identical powers of p in (25). Obviously, if p→1, then the approximate solution
is(27)u(x)≃v0+v1+v2.
In Section 5, we give a numerical sample. By using of homotopy perturbation technique, an approximate
solution for nonlinear diffusion problem is obtained [15, 21].


## 4. Numerical Results

Let(28)ut−∂∂x{(16e−tu+(t+5)e−t)∂u∂x}=−73t−9,(x,t)∈[0,1]×[0,1],u(x,0)=x2,0≤x≤1,u(0,t)=t,0≤t≤1,ux(0,t)=0,0≤t≤1.
If we want to use our last notation, we have
(29)Φ(x,t)=−73t−9,a(t)=16e−t,b(t)=(t+5)e−t.Obviously, the above assumptions satisfy consideration
of conditions. The exact solution isu(x,t)=x2et+t.
In this sample, we obtain the solution in x=0.2,0.4,0.8,1 at t=0.25,0.5,1. Assume that Δt=0.25, then we construct a homotopy as the same form
as we describe in previous sections. Consequently, the solution will be
constructed by
(30)v0j(x)=h(x,tj)=tj−1(tj+5)e−tj(−73t−9)x22,v1j(x)=∫0x∫0x{4etj(tj+5)(v0j(x)−u(x,tj−1))−1/6(tj+5)ddx(v0j(x)ddxv0j(x))}dxdx,v2j(x)=∫0x∫0x{4etj(tj+5)v1j(x)−1/6(tj+5)ddx×(v0j(x)ddxv1j(x)+v1j(x)ddxv0j(x))}dxdx,
for j=1,2,3,4.Exact solution, approximate solution, and relative
error for the above problem are given in Table1 at t=tj=jΔt, j=1,2,3,4.Table 1(a)
Exact and approximate solution ofuj(x) at tj=0.25.
xExact solutionApproximate solutionRelative error0.10.26284025420.26283991221.30×10−60.20.30136101670.30135639381.53×10−50.30.36556228750.36553969256.18×10−50.40.45544406670.45537359951.54×10−40.50.57100635420.57083550862.29×10−40.60.71224915010.71189651324.95×10−40.70.87917245430.87852151637.40×10−40.81.0717762671.0706693761.03×10−30.91.2900605881.2882930711.37×10−311.5340254171.5313398911.75×10−3(b)
Exact and approximate solution ofuj(x) at tj=0.5.
xExact solutionApproximate solutionRelative error0.10.51648721270.51648650281.37×10−60.20.56594885080.56594093741.39×10−50.30.64838491440.64834815085.67×10−50.40.76379540340.76368309541.47×10−40.50.91218031780.91191114602.95×10−40.61.0935396581.0929885315.03×10−40.71.3078734231.3068628857.72×10−40.81.5551816131.5534739051.09×10−30.91.8354642301.8327541121.47×10−312.1487212712.1446297111.90×10−3(c)
Exact and approximate solution ofuj(x) at tj=0.75.
xExact solutionApproximate solutionRelative error0.10.77117000020.77116868281.70×10−60.20.83468000070.83466676651.58×10−50.30.94053000150.94047047696.32×10−50.41.0887200031.0885405981.64×10−40.51.2792500041.2788230533.33×10−40.61.5121200061.5112497355.75×10−40.71.7873300081.7857395208.89×10−40.82.1048800112.1021994881.27×10−30.92.4647700142.4605262941.72×10−312.8670000172.8606077032.22×10−3(d)
Exact and approximate solution ofuj(x) at tj=1.
xExact solutionApproximate solutionRelative error0.11.0271828181.0271806022.15×10−60.21.1087312731.1087098411.93×10−50.31.2446453641.2445502327.64×10−50.41.4349250921.4346400741.98×10−40.51.6795704571.6788945794.02×10−40.61.9785814581.9772074176.94×10−40.72.3319580962.3294526611.07×10−30.82.7397003702.7354870361.53×10−30.93.2018082813.1951525022.07×10−313.7182818283.7082790352.69×10−3


## 5. Conclusion

In the obtained results of problem, we see that the
approximate solutions for small incrementΔt have relative error at least of order O(E−3). This technique applied for some inverse problems
and results of this approach are acceptable for small mesh size Δt. In [25–27], the stability and convergency of HPM for heat transfer, KdV equation, and couple of systems of
reaction-diffusion equations are discussed. The authors' aim is to find the
stability conditions of boundaries and initial data such that the solution is
stable.



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