Cauchy Problem for the Heat Equation in a Bounded Domain without Initial Value

1 Introduction

In this paper,we consider an inverse heat conduction problem to determine the heat flux in a bounded domain without initial value.To our know ledge,this kind of inverse problem is very important for applications in science,engineering and bioengineering which has attracted great attention of many researchers in recent years.In this case,our goal is to determine the interior and surface heat flux on an inaccessible from Cauchy data on the accessible boundary.As we know,this kind of Cauchy problem is severely ill-posed in Hadamard’s sense[Eldén(1987);Eldén,Berntsson,and Reginska(2000);Qian and Fu(2007);Hào,Reinhardt,and Schneider(2001);Weber(1981);Liu and Zhang(2013)],that is,small perturbations in Cauchy data can result in dramatically large errors in the solution.Hence,regularization techniques should be considered to stabilize the computations[Engl,Hanke,and Neubauer(1996);Groetsch(1984)].In the past years,many regular-ization methods have been developed for the heat equation in an unbounded domain[Carasso(1982);Eldén,Berntsson,and Regi′nska(2000);Seidman and Elden(1990);Fu and Qiu(2003);Tautenhahn(1997);Xiong and Fu(2008);Berntsson(1999);Eldén(1995)].These methods include Tikhonov method[Carasso(1982)],wavelet and spectral method[Eldén,Berntsson,and Regi′nska(2000);Fu and Qiu(2003);Xiong and Fu(2008)],conjugate gradient method[Lee,Yang,Chang,and Wu(2009)],optimal schemes[Tautenhahn(1997);Seidman and Elden(1990);Chang and Liu(2012)],boundary particle method and singular meshless method[Fu,Chen,and Zhang(2012);Chen and Fu(2009);Gu,Chen,and Fu(2013)],etc.In this paper,we propose a quasi-reversibility regularization method to solve the Cauchy problem of the heat equation.The method of quasi-reversibility was first proposed by Lattès and Lions to deal with some ill-posed problems[Lattès and Lions(1969)].The main idea of this method is by perturbing the equation in the illposed to obtain a well-posed problem.The similar regularization method was used in Eldén’s papers[Eldén(1987,1988)]where the author used the Fourier transform to get the exact solution for the sideways heat equation problem in a quarter plane.Qian et al.[Qian,Fu,and Xiong(2007)]rectified the defect of Eldén’s papers and got the convergence in the whole solution domain for the heat flux distribution by the Fourier transform.

In many situations we do not know the initial condition because the heat process has already started before we estimate this problem.As we know,there are very few works to deal with the Cauchy problem without initial value[Dorroh and Ru(1999);Wang,Cheng,Nakagawa,and Yamamoto(2010)].Based on the existing theory,Wang et al.[Wang,Cheng,Nakagawa,and Yamamoto(2010)]proved the uniqueness in determining both a boundary value and an initial value.Cannon and Douglas[Cannon and Douglas(1967)]established H?lder continuous dependence on the Cauchy data for solutions of the heat equation with an a priori bound.Dorroh and Ru[Dorroh and Ru(1999)]proved that the regularized solution for the exact Cauchy data converges the exact solution without initial value and did not provide a convergence estimate for the regularized solution corresponding to the noisy Cauchy data.In this paper,we apply a fourth-order modified method to obtain a regularized solution in a bounded domain without initial value.Convergence estimates are given based on the Fourier series.For numerical implementation,we apply a method of lines to obtain a stable approximate solution.

The outline of the paper is as follows.In Section 2,the formulation of the heat conduction problem and a quasi-reversibility regularization method are given.Section 3 gives the convergence estimates for the regularized solution.The method of lines is applied to obtain an approximate solution in Section 4.Several numerical examples are presented in Section 5 to illustrate the efficiency of the proposed method.Finally,in Section 6 we give some concluding remarks.

2 Formulation of the heat conduction prob lem and a quasi-reversibility regularization method

We consider the heat conduction problem as follows

Suppose thatsof(t)can be written in its Fourier series.For the detail of this inverse problem,we refer to[Cannon(1984)Chap.2].Uniqueness of the solution of problem(1)follows from the analyticity of the solution of the heat equation in the spatial variablex.Properties of uniqueness and continuous dependence are discussed in[Cannon(1984)Chap.11].

We can get the following formal solution of the problem(1),refer to[Dorroh and Ru(1999)],

and the heat flux is given by

where

and

Supposeis measured data and satisfies

wheredenotes theL2-norm and the constantδ>0 represents a noise level.Refer to[Dorroh and Ru(1999)],we know that the following formal solution of problem(4)

and the heat flux is given by

where

It is well known that for an ill-posed problem an a priori assumption on the exact solution is necessary.To get a more sharp convergence rates for the regularized solution,the following a priori bound on the exact solution is needed

whereEis a finite positive constant.

To obtain convergence estimates,we should choose a suitable regularization parameterα.It is difficult to choose parameterαby an a-priori method.In this paper,we choose parameterαsimilar toμin[Eldén(1987)]by

whereEandδare given in(8)and(5),respectively.

3 Convergence estimates

In this section,we give some error estimates for the heat flux in the interior of domain 0

Theorem 3.1Let u(x,t)be the solution of problem(1)given by(2).Let vδ(x,t)be the solution of problem(4)given by(6).The regularization parameter α is given by(9).Let the measurement temperature history at x=0,fδ(t),satisfies(5),and let the a priori assumption(8)hold.Then for fixed x∈(0,1),we have

PROOF.Letv(x,t)be the solution of problem(4)with noise-free data,i.e.,δ=0.By using the triangle inequality,we know

We start by estimating the second term on the right-hand side of(11).From(2)and(6),we have

where

In terms of the condition(5),we have

Consequently,

i.e.,

where

and combing with(9),we have

Now we estimate the first term on the right-hand side of(11).From(3)and(7),we have

From(8),the a priori assumption is equivalent to

Consequently,

where

where

We should estimateA(n)andC(n),respectively.To estimateA(n),we rewrite

where

For estimatingA(n),we should estimateA1andA2,respectively.We have

where

and

Inserting the inequalities(19)and(20)into equation(18),we have

We apply the method with the same asA1to estimateA2.Since 0<τ≤1,we get

Com ing with(21),(22)and(17),we then have

similarly,we get

forand,respectively.Sinceαis given by(9),we get

where

Combing(26)and(15),we get

The theorem 3.1 now follows by combing(14)and(27). ?

From Theorem 3.1,we know that(6)is a stable approximation of the exact solutionu(x,t).However,the accuracy of the regularized solution becomes progressively lower asTo obtain the continuous dependence of the solution atx=1,we need to introduce a stronger a priori assumption

wherep>1 is an integer.

Theorem 3.2Let u(x,t)be the solution of problem(1)which is given by(2)with exact data f.Let vδ(x,t)be the solution of problem(4)which is given by(6)with measurement data fδ.The measurement data fδsatisfies(5)and let the a priori assumption(28)hold.The regularization parameter α is chosen as

Then for p>1,we get the error bound

where

PROOF.From(2)and(28),we have

Since the procedure of the proof is completely similar wheneverpis even or odd,thus we only discuss the case thatpis even.

Taking a similar procedure of the proof of Theorem 3.1.From(5)and(28),we get

where

We also start by estimating the second term on the right-hand side of(31).Letx=1 in(13)and note thatαis given by(29),we have

To estimate the first term on the right-hand side of(31),we rewriteas

To estimate(33),we distinguish two cases.

Case 1:when

Taking a similar procedure of the estimation of.Letx=1 in(25),we get

If 1

If 5≤p<6,from(36),we have

Ifp≥6,note that,from(36),we have

Summarizing(34)-(39),we complete the estimate of the first term on the right-hand side of(31),i.e.,

The theorem 3.2 now follows from(31),(32)and(40). ?

Remark 1Since the regularization parameteras the measured errorwe can easily find that,forThus

Remark 2Note that the regularization parameter in Theorem 3.1 differs from in Theorem 3.2.However,we can use only one regularization parameter in Theorem 3.1 and Theorem 3.2 by making no more efforts.In Theorem 3.1,if we let the regularization parameter α be given by(29)and a stronger priori bound E given by(28).Using the procedure of the proof of Theorem 3.1,we can easily get the similar error estimate as

where p≥0andec(x)is the similar constant c(x)in Theorem 3.1.If we take p=0in(42),we haveec(x)=c(x)and can easily get the same error estimate as Theorem 3.1.Thus we conclude that(42)generalizes Theorem 3.1.

4 The method of lines

In order to obtain the approximate computed solution for the heat flux in a bounded domain without initial value,we use a method of lines[Eldén(1997)]to solve problem(4).

Rewrite(4)in a block operator equation in which the subscript x denotes the spatial derivative

and Cauchy conditions become

Partition the time interval[0,2π]as0=t0

According to(44),we have

The time derivative can be approximated by the forward difference scheme,we

have

Since the initial data is unknown,we should apply some neighborhood points to approximate the initial data.Thus we have the following approximation

By the similar method,we get

From(48),(49)and(47),the unbounded operator can be expressed by

where the coefficient matrixΨis given by

The second order derivative can be approximated by the central differencescheme as

whereΦis a three-diagonal matrix,with nonzero elements as follows

and the first and the last row of the matrixΦcan be obtained by the following deductions.The1-st component of the vector Vtt(x)is given by

thus the first row of the matrixΦis

The(n+1)-th component of the vector Vtt(x)is given by the same method

thus we can obtain the last row of the matrixΦas follows

Therefore,the coefficient matrixΦis given by

In terms of(50)and(52),the problem(43)can be discreted to be a system of ordinary differential equations

where I is the identity matrix.Further from(59),we can prove thatΦhas nonpositive eigenvalues,i.e.,Φis a semi-negative definite matrix,therefore the matrix I?α2Φis invertible.We finally get

The Cauchy conditions(44)become

There are many feasible methods to solve the ODES(60).In our numerical implementation,we use the fourth order Kutta method for solving the system of equations(60).Therefore,we get

whereis a step size for spatial variable and

Combining(61)with(62),it is easy to obtain the heat flux in the solution domain.

5 Numerical experiments

In this section,we test numerical examples to demonstrate the feasible of our approach.In order to check the effect of numerical computations,we compute the root mean square error at fixed x by the following formula

where?ux is the regularized solution,ux is the exact solution,and{tj}is a set of discrete times in internal[0,2π].

The noise Cauchy data are generated by

where f(tj)is the exact data,rand(j)is a random number uniformly distributed in[?1,1]and the magnitude ε indicates a relative noise level.Therefore,we take δ=ε k f(t)k in the proof of Theorem.

5.1 Examples

In this section,we will present three examples to illustrate the effectiveness of the proposed method.All numerical results show that the proposed numerical approach is feasible and stable.

Example 1:Let the exact solution for the problem(1)be

We apply two methods to recover the surface heat flux in a bounded domain.One method is the method of lines(ML)given in Section 5 and the other method is Fourier series method(MS)given by(3),refer to[Dorroh and Ru(1999)].Figure 1 shows the numerical comparison of the exact solution and its approximations with ML and MS where we take the regularization α≈0.0158from(29).For the MS solution we choose n=10and for the computation of the ML solution the step size for x is1/100,for t is2π/380.The root mean square errors are eML=0.0235and eMS=0.013for ε=0.001,respectively.Since the exact solution u(x,t)is periodic function with t,the MS solution converges the exact solution everywhere.Both methods work very well for such a periodic example.

Figure 1:Approximate solutions with the method of lines(ML)and the method of Fourier Series(MS)at x=1 for ε=0.001.

Figure 2:The ML solutions compared with the exact solution for different x.(a)x=0.2(b)x=0.6(c)x=1(d)x=1

Table 1:The root mean square errors for the ML solutions for Example 1 with ε=0.001 and ε=0.005.

Numerical results at different locations x for two noise levels ε=0.001,0.005are computed by ML,see Figure 2.We choose the regularization parameters α≈0.0037,0.0058from(9)for noise levels ε=0.001,0.005in Figure 2(a)-(c),respectively and in 2(d)we take p=2and the regularization parameters α≈0.0158,0.0316chosen by(29)for ε=0.001,0.005,respectively.In Table 1,we display the root means square errors in line with Figure 2.

We can see that the accuracy of the regularized solution becomes lower for the same noise level from Figure 2(a)to Figure 2(c).The far the distance between x and Cauchy data is,the large the root mean square error of heat flux between approximation solution and exact solution is from the second column to the seventh column in Table 1.

From Figure 2(c)and the seventh column in Table 1,we know that the accuracy is worst and the root mean square error is largest on the boundary x=1.These results are consistent with the conclusion of Theorem 3.1,that is the accuracy of the regularized solution becomes progressively lower as x→1.As we know,it is difficult to recover the heat flux far away from Cauchy data without initial value.In order to obtain fairly accurate approximate solution,we use a stronger a priori bound(28)and the regularization parameter(29)to solve this Cauchy problem.Compared with Figure 2(c)and 2(d)or the last column in Table 1,it can seen that the numerical solution is more accurate for recovering the heat flux on the boundary x=1with(28)and(29).These results are consistent with the conclusion of Theorem 3.2.

In Table 2,we display the root means square errors for different noise levels at the location x=0.4.For the second row in Table 2,a priori bound and the regularization parameter are given by(8)and(9),respectively.For the third row in Table 2,a stronger a priori bound and the regularization parameter are given by(28)and(29),respectively.From Table 2,we can see that the larger the noise levels are,the larger the root means square errors are between the approximate solution and exact solution.The root means square errors of the third row are less than errorsof the second row for the same noise level.Thus a stronger a priori bound(28)and the regularization parameter(29)can obtain better convergence and stability which is consistent with Remark 2 in Section 3.

Table 2:The root mean square errors for the different noise levels in Example 1 with location at x=0.4.

Example 2:Take the exact solution for the problem(1)as

The Cauchy data can be calculated as and q(t)=0.We consider toimpose the stronger a priori bound onwhere p=2.We can calculateby Matlab that so we might as well choose E=2.6.

Figure 3:Approximate solutions with the method of lines(ML)and the method of Fourier Series(MS)at x=1 for ε=0.001.

Figure 3 shows the comparison of the exact solution and the ML solution and MS solution at x=1for noise level ε=0.001.In the computation of the MS solution,we take the regularization parameter α≈0.0179from(29)and n=30.For theML solution,the step size for x is1/100,for t is2π/300.The root mean square errors are eML=0.0171and eMS=2.2278for ε=0.001,respectively.Since the exact solution u(x,t)is not periodic to t,the computed surface heat flux for MS is drastically oscillatory on the boundary,especially at the neighbourhood two endpoints.Therefore,MS fails to recover the surface heat flux in a bounded domain.From Figure 3,it can be seen that ML is much more effective.

Figure 4:The ML solutions compared with the exact solution for different x.(a)x=0.2(b)x=0.4(c)x=0.9(d)x=1

From the analysis of Remark 2 in Section 3,we let the regularization parameter α be given by(29)and a stronger priori bound E given by(28),then we can easily get the same error estimate for both of interior and boundary heat flux.For reconstructing the interior and surface heat flux,we take p=2and E=2.6and the regularization parameters α≈0.0179,0.0373given by(3.19)for ε=0.001,0.005,respectively.Numerical results by ML for noise levels ε=0.001,0.005are presented in Figure 4 for different fixed x.We can see that numerical approximationsare satisfactory for both of interior and surface heat flux.Meanwhile,numerical results are stable to the increase of noise levels.

Example 3:In this example,we consider a more complicated problem.The exact solution for problem(1)is unknown and the surface heat flux is a piecewise smooth function as follows

The Neumann boundary data q(t)=0and the Dirichlet data at x=0is obtained by solving a direct problem

We apply the finite difference method of Crank-Nicolson scheme to solve this direct problem to get f(t),then use ML to solve the inverse problem.

Figure 5 shows the comparison of the exact solution and the ML solution and MS solution at x=1for noise level ε=0.001.In the computation of the MS solution,we take the regularization parameter α≈0.0199from(29)and n=10.The root mean square errors are eML=0.3520and eMS=17.9424for ε=0.001,respectively.From Figure 5,it can be seen that MS fails and ML is much more effective to recover the heat flux on the boundary for problem(1)without exact solution.

Numerical results for various levels δ of relative noises are computed by ML in Figure 6.From(29),we choose the regularization parameters α≈0.0199,0.0329,0.0 430for ε=0.001,0.003,0.005,respectively.The root mean square errors are e0.001=0.3520,e0.003=0.6354and e0.005=0.8868for ε=0.001,0.003,0.005,respectively.We can see that the numerical results at x=1are convergent to the exact boundary value if choosing the regularization parameter α from(29)which is consistent with Theorem 3.2.

From the numerical results,we can see that the proposed ML is much more effective than MS.

Figure 5:Approximate solutions with the method of lines(ML)and the method of Fourier Series(MS)at x=1 for ε=0.001.

Figure 6:Exact and computed solution at x=1.

6 Conclusion

In this paper,we study an inverse heat conduction problem in a bounded domain without initial value.This problem is severely ill-posed,we apply a quasireversibility regularization method to reconstruct heat flux.Under a certain choice of the regularization parameter,we can obtain some logarithmic convergence estimates with respect to the noise level in the Cauchy data.With a stronger assumption on the regularity of the solution,the convergence estimate is obtained for the whole domain,including boundary.The numerical results are consistent with our theoretic results and also show that the proposed method is reasonable,feasible and stable.

The research of J.C.Liu was supported by the Fundamental Research Funds for the Central Universities(2014QNA57).

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