Semi-discretization Difference Approximation for a Cauchy Problem of Heat Equation in Two-dimensional Space

LI Jinmei and XIONG Xiangtuan

Department of Mathematics,Northwest Normal University,Lanzhou730070,China.

Semi-discretization Difference Approximation for a Cauchy Problem of Heat Equation in Two-dimensional Space

LI Jinmei and XIONG Xiangtuan?

Department of Mathematics,Northwest Normal University,Lanzhou730070,China.

.In this paper we consider a semi-descretization difference scheme for solving a Cauchy problem of heat equation in two-dimensional setting.Some error estimates are proved for the semi-descretization difference regularization method which cannot be fltted into the framework of regularization theory presented by Engl,Hanke and Neubauer.Numerical results show that the proposed method works well.

2D inverse heat conduction problem;Ill-posedness;regularization;error estimate; finite difference.

1 Introduction

In practical many dynamic heat transfer situations,it is sometimes necessary to estimate the surface temperature or heat flux on a body from a measured temperature history at a fixed location inside the body.This so-called inverse heat conduction problem(IHCP)[1] has been investigated by many authors.It is well known that IHCP is an exponentially ill-posed problem[2].All kinds of the regularization strategies were proposed to obtain a stable numerical solution for the problem.These include the Tikhonov regularization method[3],difference approximation method[4],wavelet method[5],Fourier cut-off method[6],hyperbolic approximation method[7-8],optimal filtering method[9],mollification methods[10-12],and optimal stable approximation methods[13].The reader can refer to http://www.mai.liu.se/frber/ip/index.html for more details.However,as saidabove,for the IHCP most of these authors restricted themselves to the case of the onedimensional space.Recently IHCP in two-dimensional space has been studied by Qian and his co-works,please refer to[14-16].In this paper we present many different difference schemes and in detail discuss a central difference scheme as a model to stabilize the ill-posed IHCP by giving the error estimates.This class of methods is important because they can easily be adapted to treat nonlinear problems that often occur in applications.

The emphasis of this work lies in the numerical approximation of the problem,the proof of the stability of the proposed difference schemes and the presentation of numerical results.We stress that the current work can be considered as an extension of[15],in which Qian and Zhang analyzed the temperature.However in this work we discussed the heat flux and implemented the numerical experiment.

2 The problem and methods

2.1 The model problem

Inthispaper,weconsiderthefollowing Cauchyproblemforheatequationintwo-dimensional space[14]:

For the purpose of simpliflcation,we only consider the case of h(y,t)=0,u0(x,y)=0. In practice,the measured data function gδ(y,t)is available at hand.

2.2 Ill-posedness in L2space

Inthis subsection,weanalyze theill-posednessofproblem(2.1)inthefrequencydomain. In order to use the Fourier transform technique,we extend the functions u(x,y,t),g(y,t), gδ(y,t)to the whole t axis by deflning them to be the zero everywhere in t＜0.Thus we wish to determine the temperature u(x,y,t)∈L2(R2)and heat flux ux(x,y,t)∈L2(R2)for 0＜x≤L from the temperature measurements gδ(y,t)∈L2(R2).

We also assume that these functions are in L2(R2)and use the corresponding L2-norm.as deflned below,

We now could assume that the measured data function gδ(·,·)satisfles

where the constant δ＞0 represents a bound on the measurement error.

Let

be the Fourier transform of a function g(y,t)∈L2(R2).

Applying the Fourier transform to Eq.(2.1a)with respect to y and t,we obtain

which is a second-order ordinary differential equation for flxed ξ and τ.Now using the boundary conditions in the frequency domain.We can easily get the solution of problem:

and then taking the inverse Fourier transform.The solution to problem(2.1)is

wheredenotes the principal value of the square root

where σ=sign(τ).

2.3 Different difference schemes and their analysis

By using a simple central flnite difference scheme to approximate the time derivative and the space derivative,for 0≤x＜L,y∈R,t≥0,we get the following semi-discretization scheme:

with initial boundary conditions

where k:=δt and h:=δy are the time step length and the spatial length of the variable y,respectively.In the following,we shall assume 0＜k＜1 and 0＜h＜1 without loss of generality.

Similarly we can approximate the time derivative by a forward difference

or a backward difference

Like the previous analysis,taking the Fourier transform for(2.5)with respect to(y,t), one can easily get

Nowusingboundaryconditionsinthefrequencydomain,onecan easilygetthesolution and its x-derivative of the problem:

and then taking the inverse Fourier transform,we get the solution and its x-derivative of problem:

wheredenotes the principal value of the square root.Let

For the forward difference approximation,we can see that the solution is given by

And for the backward difference approximation,we can also see that the solution is given by

Now we analyze the possibility ofv(x,y,t)approximatingu(x,y,t).Note that if the step lengths(k,h)are small,then the variablesτc(orτb,τf)≈τandξc≈ξin the lowfrequencycomponent(i.e.,small|τ|andsmall|ξ|).Consequently,cosh(ρcx)(orcosh(ρbx), cosh(ρfx))is close to cosh(ρx)andv(x,y,t)is close tou(x,y,t)with thesame exact datag. Thisfact correspondstotheconsistency.Moreover,forflxed(k,h),ρc(orρb,ρf)isbounded even ifτandξtend to inflnity,i.e.,cosh(ρcx)(or cosh(ρbx),cosh(ρfx))is bounded.This point guarantees the stability.Actually,the larger the parameterskandh,the better the stability.However,the smaller the parameterskandh,the better the consistency.Thus, we need a strategy to choose the parameterskandhto balance the stability and consistency.These discussions hint the regularization role ofkandh.This is consistent with the general regularization theory[17-18].

3 Some useful lemmas

In this section,we give some auxiliary results that be used in the proof of the next section. These results are obvious and the proofs can be found in[15].

Lemma 3.1.If a≥b≥0,x≥0,σ=sign(τ),τ∈R,we have

Lemma 3.2.If(ξ,τ)∈R2,we have

Lemma 3.3.If a and b are given according to(2.4),acand bcaccording to(2.9),and τcand ξcaccording to(2.9),there holds

Lemma 3.4.Let τcand ξcbe given by(2.7),we get

Lemma 3.5.For|γ|≤1,we get

Lemma 3.6.For a flxed constant x＞0,a flxed integer n＞0,and a variable γ≥0,

4 Stability and convergence estimate

In this section we prove some stability and convergence results,which also hints how to choose the step lengthshandk.

Theorem 4.1.Let v(x,y,t)and vδ(x,y,t)be the solution of(2.5)with the exact and noisy data g(y,t)and gδ(y,t),respectively,and‖g(·,·)?gδ(·,·)‖≤δ,then we have

for flxed0＜x＜L.

Proof.Using Parseval’s identity and(2.8)with‖g(·,·)?gδ(·,·)‖≤δ.We have

where

Note thatρcis given by(2.9),and due to Lemmas 3.1,3.2 and 3.4,we have

Thus,we complete the proof.

Theorem 4.2.Let u(x,y,t)be the solution of(2.1)and v(x,y,t)be the solution(2.5)with the same exact data g(y,t).Assume the a-priori bound‖u(L,·,·)‖≤E,where E is a positive constant, then we have

for flxed0＜x＜L.

Proof.From(2.2)we get

consequently the condition‖u(L,·,·)‖≤E,leads to

Now using(2.2)and(2.8)with the same exact datag(y,t)and(4.7),we have

where

We deflne

We now distinguish two case(ξ,τ)∈?0and(ξ,τ)∈R2?0to estimate(4.9).

Case 1.(ξ,τ)∈R2?0.Due to Lemma 3.2 and note thata≥ac,we can estimate(4.9) as

where we used|1?e?x(ρ?ρc)|≤2,|1?e?x(ρ+ρc)|≤2.

Case 2.(ξ,τ)∈?0.We estimate(4.9)as

Through simple calculus,we have

Now by Lemma 3.3 the estimate(4.12)become

Thus,we can further estimate(4.14)as

Now combining(4.8),(4.11)and(4.17),we arrive at the conclusion of the theorem.

Theorem 4.3.Let u(x,y,t)be the solution of(2.1)with the exact data g(y,t)and vδ(x,y,t)be the solution of(2.5)with the noise data gδ(y,t).Assume that‖g?gδ‖≤δ,‖u(L,·,·)‖≤E.Then we have

under the choice of

where0＜x＜L and C4=4L2(C2+4C3).

Proof.Using the triangle inequality and theorems 4.1 and 4.2,we have

Now insert(4.19)into(4.20),we get the statement of the theorem.

5 Error estimation for heat flux

In this section,we will prove error estimate between the semi-discretization difference regularization and the exact heat flux.

Proof.By using the Parseval formula and(2.2),(2.8)and(4.7),we get

Therefore

To estimate(5.5),we have

Thus,

where we used

and Lemma 3.3.Because

where a and b are be given by(2.4),B1,B2are be given by(4.12),combing(5.3),(5.7) and(5.8),we get the error estimate(5.1).

6 Numerical experiments

Inthissection,wepresentsomenumerical examplestoillustratethepropertiesoftheproposed method with the flxed parameter L=1.Although the present problem is described in an unbounded domain in(x,y),we are interesting in the domain(x,y)∈[0,1]×[0,1]. Thisis reasonable becausetheproblemcanbe solvedby periodicextensionin(x,y)plane.

The numerical tests is performed in the following way:First we select a solution u(1,y,t)=f(y,t),0≤t≤1,0≤y≤1 and compute the exact solution u(x,y,t)and the data function u(0,y,t)=g(y,t)by solving a well-posed problem for the equation using a f inite difference scheme.Then we add a normally distributed perturbation of variance ??max{g(y,t)}to the data function,giving gδ.We compute the noise level δ by discrete L2-norm‖g?gδ‖and the a-priori bound‖f‖≈E.From the noisy data function we get the regularized solution v(x,y,t)and the regularized flux vx(x,y,t)with 0＜x＜1 and compare the results with the exact solution u(x,y,t)and the flux ux(x,y,t).In the process of reconstruction,we use 2D FFT(two-dimensional Fast Fourier Transform)algorithm to compute the closed form(2.8),then we use inverse 2D FFT to obtain the reconstructed solutions.

Generally the a-priori information E is hard to obtain,then h,k in(4.19)cannot be obtained easily.In numerical experiment we choose them artiflcially.We conduct two examples to show that the proposed method works well.

Figure 1:the noisy input data gδ(y,t).

Figure 2:(2a):the exact solution at x=0.2;(2b):the reconstructed solution at x=0.2.

Example 6.1.First we select the exact solution deflned on[0,1]×[0,1]:

First by solving a well-posed forward problem and adding the noise,we can get the input noisy data gδ(y,t)which is displayed in Fig.1,and then we computed the solutions at x=0.2 which is displayed in Fig.2 and the flux solutions at x=0.2 displayed in Fig.3. The parameters involved are listed as follows:?=1?10?2,δ=0.02,h=0.6,k=0.1.

In Fig.4 and Fig.5,we give the numerical results with the location at x=0.7.The parameters are not changed.We will see that the reconstruct effect are worse because the degree of ill-posedness at x=0.7 is more severe than the degree of ill-posedness at x=0.2.

Figure 3:(3a):the exact flux at x=0.2;(3b):the reconstructed flux at x=0.2.

Figure 4:(4a)the exact solution at x=0.7;(4b):the reconstructed solution at x=0.7.

Figure 5:(5a)the exact flux at x=0.7;(5b):the reconstructed flux at x=0.7.

Consider a non-smooth function

Figure 6:the noisy input data gδ(y,t).

Figure 7:(7a):the exact solution at x=0.3;(7b):the reconstructed solution at x=0.3.

In this example,similarly we can get the noisy input data gδ(y,t)which is displayed in Fig.6,and then we computed the solutions at x=0.3 which are displayed in Fig.7 and the fluxes at x=0.3 which are displayed in Fig.8.The parameters involved are listed as follows:?=1?10?2,δ=0.01,h=0.6,k=0.1.

We have testedourmethodto various otherexamples and got similar goodnumerical results.

7 Conclusions

We have proposed a new method for solving a Cauchy problem for heat equation in twodimensional space which is severely ill-posed.In this paper,we have proved that the method is stable and given an error bound.It is very simple and fast since the regularized solution has a closed form in the frequency domain.The numerical results for testexamples are convincing.

Figure 8:(8a):the exact flux at x=0.3;(8b):the reconstructed flux at x=0.3.

Acknowledgement

The authors would like to thank the reviewers for their very careful reading and for pointing out several mistakes as well as for their useful comments and suggestions.

The research was partially supported by a grant from the Key(Keygrant)Project of Chinese Ministry of Education(No 212179)and Natural Science Foundation of Gansu Province(No 145RJZA037).

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?Corresponding author.Email addresses:xiongxt@gmail.com(X.T.Xiong),992461300@qq.com(J.M.Li)

Received 27 July 2015;Accepted 24 August 2015

AMS Subject Classiflcations:65R35

Chinese Library Classiflcations:O175.26