AMP Advances in Mathematical Physics 1687-9139 1687-9120 Hindawi Publishing Corporation 10.1155/2016/9504829 9504829 Research Article A Lipschitz Stability Estimate for the Inverse Source Problem and the Numerical Scheme http://orcid.org/0000-0003-0132-5184 Jia Xianzheng 1 Salahshour Soheil Shandong University of Technology Shandong 255000 China sdut.edu.cn 2016 2072016 2016 04 05 2016 27 06 2016 2016 Copyright © 2016 Xianzheng Jia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We consider the inverse source problem for heat equation, where the source term has the form f(t)ϕ(x). We give a numerical algorithm to compute unknown source term f(t). Also, we give a stability estimate in the case that f(t) is a piecewise constant function.

Natural Science Foundation of China 11371231 11071148 Natural Science Foundation of Shandong Province ZR2011AQ014
1. Introduction to the Problem

The inverse source problems in heat equation are solving the heat equation with unknown heat sources. They are well known to be ill-posed. There were many studies to identify different types of heat sources. For example, Cannon and DuChateau  estimated the nonlinear temperature dependent heat source. In [2, 3], the method of fundamental solutions has been presented for an inverse time-dependent heat source problem. In , several numerical schemes have been proposed to determine a space-dependent heat source. In , these numerical methods were considered to solve some two- and three-dimensional inverse heat source problems. In Savateev , Trong et al. , and Shidfar et al. , the authors considered the source as a function of both space and time, but it was additive or separable.

Throughout this paper, we will assume that Ω is an open set in Rn. We consider the following initial-boundary value problem:(1)ut=Δu+ftϕxx,tΩ×0,T,ux,0=0xΩ,ux,t=0x,tΩ×0,T.

Problems of the above equations include inverse problems, heat conduction processes, hydrology, material sciences, and heat transfer problems. In the context of heat conduction and diffusion, u represents either temperature or concentration. In this system, ϕ(x) is spatial density and unknown f(t) is interpreted as either a heat or a material source, respectively, in a chemical or a biochemical application.

When f(t)>0, it means that term f(t)ϕ(x) is a source. We assume that ϕ is a given function and satisfies the following condition:(2)ϕ0,0in  Ωϕ  has  compact  support  in  Ω.

Our problem is to derive f(t) and a conditional stability in the determination of f(t),0tT, from the observation (3)ux0,t,0<t<T,where x0Ω/suppϕ¯.

2. Some Existing Results

To derive the stability estimate, we need the following inequalities, which can be found in . Here, we just list the result.

Theorem 1.

Let p1, δ>0, 0α<T, and f,gL(0,T) satisfy(4)0f,gM<,0<t<T.

Then, (5)fLpα,TgLp0,δM2p-2/pαT+δαtfsgt-sdsdt1/p.

In particular, for(6)fgt=0tft-sgsds,0<t<Tand for α=0, one has (7)fLpα,TgLp0,δM2p-2/pfgL10,T+δ1/p.

In , by using the above theorem, a Hölder estimate was obtained for the case that Ω=Rn and fU, where (8)U=fC0,T;fC0,TM,f  changes  the  signs  at  most  N-times.

Theorem 2.

Let ϕ satisfy (2), and set U to be defined as above definition.

Then, for arbitrarily given δ>0, there exists a constant C=Cx0,ϕ,T,p,δ,U>0 such that (9)fLp0,TCux0,·L10,T+δ1/pN,for any fU.

Corollary 3.

In Theorem 2, if one replaces U by (10)PN=f;fisapolynomialwhoseorderisatmost  N,fC0,TM,

also the same result can be obtained.

But we are more interested in the case that f(t) is a piecewise constant function. In that case, f(t) can have infinity number of zeros. For that case, we obtained the similar result, and we will claim the result and the proof in the next section.

3. The Estimate

For problem (1), we can see that solution u(x,t) can be expressed in the following form: (11)ux,t=0tΩKx,y,t-sfsϕydyds,xΩ,  t>0,where K(x,y,t) is the fundamental function for problem (1). See .

Also from , we can see that fundamental solution K(x,y,t) satisfies the following conditions: (12)Kx,y,t>0,t>0.

Therefore, setting(13)μx0t=ΩKx0,y,tϕydy,t>0,we have(14)ux0,t=0tμx0t-sfsds,0<t<T,which is a Volterra integral equation of the first kind with respect to f.

Here, we consider the case that fF={f  is  a  piecewise  constant  function  in  [0,T]}.

Theorem 4 (uniqueness).

For problem (1), given u(x0,t), then fF is uniquely determined except for finite point in (0,T).

Proof.

Suppose that there are two functions f1(t),f2(t)F satisfying (1). We assume that the discontinuous points of f1 and f2 are 0=t01<t11<<tI11=T and 0=t02<t12<<tI22=T, respectively, and f1(t)=fj1,t(tj-11,tj1] and f2(t)=fj2,t(tj-12,tj2].

After some resorting procedure, we get points 0=τ0<τ1<<τI3=T; then, for 0<t<τ1, (15)0tμx0t-sf1s-f2sds=ux0,t-ux0,t=0f11-f120tμx0t-sds=0and 0tμx0(t-s)ds>0,0<t<τ1, so we get f1(t)=f2(t),0<t<τ1.

For τ1<t<τ2, (16)0tμx0t-sf1s-f2sds=τ1tμx0t-sf1s-f2sds=0,therefore, f1(t)=f2(t),t(τ1,τ2).

Repeating the above procedure, we can get f1(t)=f2(t),t(τj-1,τj),j=1,2,,I3.

Thus, end the proof.

Suppose that f changes its values at 0=t0<t1<t2<<tI=T,IN, and f(t)=fj,t[tj-1,tj),j=1,2,,I. Here, we assume that there exists a constant c>0, such that t(i)-t(i-1)c,i=1,2,,I.

Theorem 5.

If ϕ satisfies (2), x0Ω/suppϕ¯.

Then, for p>1, there exists a constant C=C(x0,ϕ,T,p,F,N)>0 such that (17)fLp0,TCux0,·Lp0,T,for any fF. Here, N is the number of intervals in [0,T] that f0.

Proof.

Since μx0(t) is positive and bounded and (13), we can see that(18)ux0,t=f10tμx0t-sds,0tt1.Choosing Lp norm from 0 to t1, we obtain(19)uLp0,t1=f10tμx0t-sdsLp0,t1.Thus, we can see that(20)f1C1uLp0,t1,where C1>(0tμx0(t-s)dsLp(0,t1))-1 is a constant.

For t1tt2, we can see that (21)ux0,t=0tμx0t-sfsds=f10t1μx0t-sds+f2t1tμx0t-sds,ux0,t-f10t1μx0t-sds=f2t1tμx0t-sds.Then, we can obtain(22)f2·t1tμx0t-sdsLpt1,t2=ux0,t-f10t1μx0t-sdsLpt1,t2ux0,tLpt1,t2+f10t1μx0t-sdsLpt1,t2ux0,tLpt1,t2+C2ux0,tLp0,t1C3ux0,tLp0,t2.Then (23)f2Cux0,tLp0,t2.Therefore, in the case of ti-ti-1c>0,i=1,2,,I and (20), we get (24)fLp0,t2Cux0,tLp0,t2.

Continuing this argument until ti=T, we get the proof of this theorem.

4. Numerical Scheme

For problem (1), if ux0,t is given, we need to reconstruct function ft.

Consider an auxiliary problem:(25)ut=Δu+ϕxx,tΩ×0,T,ux,0=0xΩ,ux,t=0x,tΩ×0,T.

Actually, we can get the solution of problem (25) and suppose that u~ is the solution. Then,(26)u~x,t=0tΩKx,y,t-sϕydyds.

Given data u(x0,ti),i=0,1,2,,N, where 0=t0<t1<<tN=T, then, for every ti, we have(27)ux0,ti=0tiΩKx0,y,ti-sfsϕydyds,i=0,1,2,,N.

We suppose that f(t)=fi,t[ti-1,ti],i=1,2,,N; then(28)ux0,ti=0tiΩKx0,y,ti-sfsϕydyds=j=1ifjtj-1tjΩKx0,y,ti-sϕydyds=f10tiΩKx0,y,ti-sϕydyds+j=2ifj-f1tj-1tjΩKx0,y,ti-sϕydyds=f1u~x0,ti+f2-f1t1tiΩKx0,y,ti-sϕydyds+j=3ifj-f1-f2-f1tj-1tjΩKx0,y,ti-sϕydyds=f1u~x0,ti+f2-f10ti-t1ΩKx0,y,ti-t1-sϕydyds+j=3ifj-f2tj-1tjΩKx0,y,ti-sϕydyds==f1u~x0,ti+j=2ifj-fj-1u~x0,ti-tj-1.Thus, we can use the above formula to get the discrete data for function f(t).

Example 6.

We choose f(t) as the following function:(29)ft=1,t0,1-1,t1,2.3-0.5,t2.3,3.50.5,t3.5,4.

Choose Ω=(0,2), x0=1.5, ϕ(x)=1, x(0,1), and ϕ(x)0 outside interval (0,1). Given data u(x0,ti),i=1,2,,N. Then, we can get the numerical approximation of f(t) using the following formula: (30)ux0,ti=f1u~x0,ti+j=2ifj-f1u~x0,ti-tj-1,i=1,2,,N.

And the numerical result is as shown in Figure 1.

Numerical result.

Figure 1 is the numerical result and Figure 2 is the exact function.

Exact solution.

Figure 3 shows the error between the numerical result and exact solution.

The error between numerical result and exact solution.

Example 7.

Now, we will give a numerical result for the 2D case. Suppose the problem is also (1), and the domain Ω=(-1,1)×(-1,1), and(31)ϕx=1,x0.50,xΩ,  x>0.5ft=1,t0,0.80,t0.8,2.2-2,t2.2,3.

Let x0=(0.7,0.8) and T=3, and the numerical result is as shown in Figure 4.

Numerical result.

Figure 4 is the numerical result and Figure 5 is the exact function.

Exact solution.

Figure 6 is the error between exact f and the numerical result.

The error between numerical result and exact solution.

Competing Interests

The author declares that there are no competing interests regarding the publication of this paper.

Acknowledgments

This work was supported by the Natural Science Foundation of China Grant no. 11371231 and Grant no. 11071148 and the Natural Science Foundation of Shandong Province, China, Grant no. ZR2011AQ014.

Cannon J. R. DuChateau P. Structural identification of an unknown source term in a heat equation Inverse Problems 1998 14 3 535 551 10.1088/0266-5611/14/3/010 MR1629991 2-s2.0-0001634697 Yan L. Fu C.-L. Yang F.-L. The method of fundamental solutions for the inverse heat source problem Engineering Analysis with Boundary Elements 2008 32 3 216 222 Chantasiriwan S. Methods of fundamental solutions for time-dependent heat conduction problems International Journal for Numerical Methods in Engineering 2006 66 1 147 165 10.1002/nme.1549 2-s2.0-33645461386 Yan L. Yang F.-L. Fu C.-L. A meshless method for solving an inverse spacewise-dependent heat source problem Journal of Computational Physics 2009 228 1 123 136 10.1016/j.jcp.2008.09.001 MR2464071 2-s2.0-55549138009 Ahmadabadi M. N. Arab M. Ghaini F. M. M. The method of fundamental solutions for the inverse space-dependent heat source problem Engineering Analysis with Boundary Elements 2009 33 10 1231 1235 10.1016/j.enganabound.2009.05.001 MR2520489 2-s2.0-67649338397 Johansson T. Lesnic D. Determination of a spacewise dependent heat source Journal of Computational and Applied Mathematics 2007 209 1 66 80 10.1016/j.cam.2006.10.026 MR2384372 ZBL1135.35097 2-s2.0-34548022129 Farcas A. Lesnic D. The boundary-element method for the determination of a heat source dependent on one variable Journal of Engineering Mathematics 2006 54 4 375 388 10.1007/s10665-005-9023-0 MR2243847 2-s2.0-33745082425 Johansson B. T. Lesnic D. A variational method for identifying a spacewise-dependent heat source IMA Journal of Applied Mathematics 2007 72 6 748 760 10.1093/imamat/hxm024 MR2372086 ZBL1135.65034 2-s2.0-36549083034 Johansson B. T. Lesnic D. A procedure for determining a spacewise dependent heat source and the initial temperature Applicable Analysis 2008 87 3 265 276 10.1080/00036810701858193 MR2401821 Jin B. Marin L. The method of fundamental solutions for inverse source problems associated with the steady-state heat conduction International Journal for Numerical Methods in Engineering 2007 69 8 1570 1589 10.1002/nme.1826 MR2291586 ZBL1194.80101 2-s2.0-33846990077 Wang F. Chen W. Ling L. Combinations of the method of fundamental solutions for general inverse source identification problems Applied Mathematics and Computation 2012 219 3 1173 1182 10.1016/j.amc.2012.07.027 MR2981312 ZBL1287.35103 2-s2.0-84867333631 Mierzwiczak M. Kolodziej J. A. Application of the method of fundamental solutions and radial basis functions for inverse transient heat source problem Computer Physics Communications 2010 181 12 2035 2043 10.1016/j.cpc.2010.08.020 MR2727689 2-s2.0-77957909451 Mierzwiczak M. Kołodziej J. A. Application of the method of fundamental solutions with the Laplace transformation for the inverse transient heat source problem Journal of Theoretical and Applied Mechanics 2012 50 4 1011 1023 2-s2.0-84868332004 Savateev E. G. On problems of determining the source function in a parabolic equation Journal of Inverse and Ill-Posed Problems 1995 3 1 83 102 10.1515/jiip.1995.3.1.83 MR1332879 ZBL0828.35142 Trong D. D. Long N. T. Alain P. N. Nonhomogeneous heat equation: identification and regularization for the inhomogeneous term Journal of Mathematical Analysis and Applications 2005 312 1 93 104 10.1016/j.jmaa.2005.03.037 MR2175207 ZBL1087.35095 2-s2.0-27644472267 Shidfar A. Jazbi B. Alinejadmofrad M. Inverse estimation of the pulse parameters of a time-varying laser pulse to obtain desired temperature at the material surface Optics and Laser Technology 2012 44 6 1675 1680 10.1016/j.optlastec.2012.01.024 2-s2.0-84859163511 Saitoh S. Tuan V. K. Yamamoto M. Reverse convolution inequalities and applications to inverse heat source problems Journal of Inequalities in Pure and Applied Mathematics 2002 3 5, article 80 MR1966515 Itô S. Diffusion Equations 1992 Providence, RI, USA AMS MR1195786