\documentclass[twoside]{article} \pagestyle{myheadings} \markboth{\hfil Determination of the Source/Sink \hfil}% {\hfil Ping Wang \& Kewang Zheng \hfil} \begin{document} \setcounter{page}{119} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Fourth Mississippi State Conference on Differential Equations\newline and Computational Simulations}, \newline Electronic Journal of Differential Equations, Conference 03, 1999, pp 119--125. \newline URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Determination of the source/sink term in a heat equation \thanks{ {\em Mathematics Subject Classifications:} 35K05, 35R25. \hfil\break\indent {\em Key words:} Heat equation, inverse problem, regularization. \hfil\break\indent \copyright 2000 Southwest Texas State University and University of North Texas. \hfil\break\indent Published July 10, 2000. } } \date{} \author{ Ping Wang \& Kewang Zheng } \maketitle \begin{abstract} In this work, we consider the problem of determining an unknown parameter in a heat equation with ill-posed nature. Applying Tikhonov regularization, we obtain a stable approximation to the unknown parameter from over-specified data. We also present numerical computations that verify the accuracy of our approximation. \end{abstract} \newtheorem{theorem}{Theorem}[section] \section{Introduction} Cannon and Zachmann \cite{kn:Cannon} considered the question of determining an unknown source in the heat equation from over-specified data. More precisely, find the source $f(t)$ in the heat equation \begin{eqnarray} &u_t(x, t) =u_{xx}(x, t)+f(t), \quad 00, $ each $F\in L^2[0, T]$, we define the operator $f_\alpha =R(F, \alpha)$. For an approximate data function $F_\delta$ ($\delta$ measures the error in data), it is important to choose an appropriate parameter $\alpha(\delta)$ so that the according minimizer $f_\alpha(\delta)=R[F_\delta, \alpha(\delta)]$ can be taken as a stable approximate solution of (\ref{1.6}). The following theorem shows how to choose the regularizing parameter $\alpha$. \begin{theorem}\label{T2.2} Let $f_T \in C^1[0, T]$ be the exact solution corresponding to the exact data $F_T$, and $F_\delta$ be approximate datum. Then for every positive $\epsilon$ there exists $\delta(\epsilon)$ such that the inequality $$ \| F_\delta - F_T\|_{L^2[0, T]} \le \delta \le \delta(\epsilon) $$ implies \begin{equation} \label{e2.2} \| f_{\alpha(\delta)} - f_T\|_{C[0, T]} < \epsilon, \end{equation} where $f_{\alpha(\delta)}= R(F_\delta, \alpha(\delta))$ with $\alpha =\alpha(\delta) = \delta^\lambda$ and $0<\lambda \le 2$. \end{theorem} \paragraph{Proof:} By the definition of $f_{\alpha(\delta)}$, we know $$M^{\alpha(\delta)}[f_{\alpha(\delta)}, F_\delta] \le M^{\alpha(\delta)}[f_T, F_\delta].$$ That is \begin{eqnarray*} \|A f_{\alpha(\delta)}-F_\delta\|_{L^2}^2 + \alpha(\delta) \| f_{\alpha(\delta)} \|_{W_2^1}^2 &\le & \|A f_T-F_\delta\|_{L^2}^2 + \alpha(\delta) \| f_T \|_{W_2^1}^2 \\ & \le & \delta^2 + \delta^\lambda \|f_T\|_{W_2^1}^2\\ & \le & \delta^\lambda d^2, \quad d=(1+ \|f_T\|_{W_2^1}^2)^{1/2}. \end{eqnarray*} Hence, $\| f_{\alpha(\delta)} \|_{W_2^1} \le d$ and $\|A f_{\alpha(\delta)} -F_\delta\|_{L^2} \le d \delta^{\lambda/2}$. It is easy to see that both $f_{\alpha(\delta)}$ and $f_T$ belong to the set $E=\{f: \|f\|_{W_2^1[0, T]} \le d \}$, which is a compact subset of space $C[0, T]$. The continuity of $A^{-1}$ on $A\,E$ implies that \begin{eqnarray*} \|f_{\alpha(\delta)} - f_T\|_{C[0, T]} & \le & \|A^{-1}\|\cdot \|A f_{\alpha(\delta)}-Af_T\|_{L^2} \\ &\le &\|A^{-1}\|( \|A f_{\alpha(\delta)}-F_\delta\|_{L^2} +\|Af_T -F_\delta\|_{L^2} ) \\ &\le & \|A^{-1}\|(d \delta^{\lambda/2}+\delta)\\ &\le & \delta^{\lambda/2}\|A^{-1}\|(1+d)\,. \end{eqnarray*} By setting $$ \delta(\epsilon)=\Big[\frac{\epsilon}{\|A^{-1}\|(1+d)}\Big]^{2/\lambda}$$ we obtain (\ref{e2.2}) and the proof is complete. \hfill $\diamondsuit$ \medskip Next, we show that $F$ depends continuously on the initial data $\phi, \psi, g$. \begin{theorem}\label{T2.3} Suppose that exact data $F_T, \phi_T, \psi_T, g_T$ satisfy (\ref{1.7}), and that the appximate data $F_\delta, \phi_\delta, \psi_\delta, g_\delta$ also satisfy (\ref{1.7}). Then inequalities $\|\phi_T-\phi_\delta\|_{L^2} \le \delta,\|\psi_T -\psi_\delta\|_{L^2} \le \delta$ and $\|g_T-g_\delta\|_{L^2} \le \delta$ imply $$ \|F_T-F_\delta\|_{L^2} \le D\delta, D=\Big[6(1+\frac{2T}{\pi} +\sqrt{\frac{T}{2\pi}})\Big]^{1/2}$$ \end{theorem} \paragraph{Proof:} Applying Cauchy's inequality, we have \begin{eqnarray*} \|F_\delta - F_T\|_{L^2}^2 & \le & 3 ( \int_0^T[\phi_T-\phi_\delta]^2 dt +\frac{1}{\pi}\int_0^T[\int_0^t\frac{\psi_\delta(\tau) - \psi_T(\tau)}{\sqrt{t-\tau}}]^2d\tau \\ && + \frac{1}{\pi}\int_0^T\frac{1}{t}[\int_0^\infty (g_\delta(x)-g_T(x))e^{-x^2/(4t)}dx]^2 dt ) \\ &\le&3(\delta^2+\frac{2}{\pi}\int_0^T[\phi_\delta(\tau) -\phi_T(\tau)]^2\int_\tau ^T\sqrt{\frac{t}{t-\tau}}\,dt \,d\tau \\ && +\frac{\delta^2}{\pi}\int_0^T\frac{1}{t}\int_0^\infty e^{-x^2/(2t)}\,dx\,dt) \\ &\le& 3\delta^2(1+\frac{4T}{\pi}+\sqrt{\frac{2T}{\pi}})\\ &<& D^2\delta^2. \end{eqnarray*} \hfill $\diamondsuit$ \medskip Combining Theorems \ref{T2.2}, \ref{T2.3}, we obtain the following stability theorem. \begin{theorem}\label{T2.4} Suppose $f_T$ is the exact solution of (\ref{1.6}) corresponding to data functions $\phi_T, \psi_T, g_T$. For any $\epsilon>0$ and approximate data $\phi_\delta, \psi_\delta, g_\delta$, there exists a $\delta(\epsilon)$ and an $\alpha(\delta)$such that inequalities $\|\phi_T-\phi_\delta\|_{L^2} \le \delta,\|\psi_T-\psi_\delta\|_{L^2} \le \delta$ and $\|g_T-g_\delta\|_{L^2} \le \delta$ imply that \begin{equation} \| f_{\alpha(\delta)} - f_T\|_{C[0, T]} < \epsilon, \end{equation} where $f_{\alpha(\delta)}= R(F_\delta, \alpha(\delta))$. \end{theorem} The above result shows that, for carefully chosen $\alpha$, $f_{\alpha(\delta)}$, the minimizer of functional (\ref{2.1}), can be taken as a stable approximate solution of (\ref{1.1}). \section{Numerical Verification} We will study a concrete overdetermined system in this section to numerically test the applicability of the regularization approach discussed in Section 2. For $T=1$, we take \begin{eqnarray*} \phi_T(t)&=&\frac{t^3}{3}-\frac{t^4}{2}+0.0002\sqrt{ \frac{t}{\pi}}(1+4t(e^{-\frac{1}{4t}}-1)),\\ \psi_T(t)&=&\frac{256t^{4.5}} {315\sqrt{\pi}},\\ g_T(x)&=&0.0001x(1-x^2). \end{eqnarray*} Then $F_T(t)=\frac{t^3}{3}-\frac{t^4}{2}+\frac{t^5}{5}$. The corresponding exact solution of (\ref{1.6}) is $$f_T(t)=t^2(t-1)^2.$$ It remains to be seen how well the equation (\ref{2.3}) recovers the value of $f_T$ with the following altered initial data \begin{eqnarray*} \phi_\delta(t)&=&\phi_T(t) +\delta\sin (50\pi t), \\ \psi_\delta(t)&=&\psi_T(t)+\delta\sin (50\pi t),\\ g_\delta(x)&=&g_T(x)+\delta(1-x)(1-(1-x)^2). \end{eqnarray*} First of all, (\ref{2.3}) is replaced by its finite difference approximation on a uniform grid with step $h=T/(n+1)$. Thus we obtain the following system of linear equations in which the coefficient matrix is of five diagonal form: $$A^hf^h=h^3DF^h,$$ where $$A^h=\left(\begin{array}{ccccccc} \tilde{a} & \tilde{b} & \alpha & & & & \\ \tilde{b} & a & b & \alpha & & & \\ \alpha & b & a & b & \alpha & & \\ & \cdot & \cdot& \cdot& \cdot & \cdot & \\ & & \alpha & b & a & b &\alpha \\ & & &\alpha &b & a & \tilde{b}\\ & & & & \alpha & b & \tilde{\tilde{a}} \end{array} \right), \quad D=\left(\begin{array}{ccccccc} 1 & & & & & & \\ -1 & 1 & & & & & \\ & -1 & 1 & & & & \\ & & & & & & \\ & & & & & & \\ & & & & -1 & 1 & \\ & & & & & -1 & 1 \end{array} \right), $$ $a=h^4+2\alpha(h^2+3)$, $\tilde{a}=h^4+\alpha(h^2+2)$, $\tilde{\tilde{a}} =h^4+\alpha(2h^2+3)$, $b=-\alpha(h^2+4)$, $\tilde{b}=-\alpha(h^2+3)$. $F^h=(F_1^h, \cdots, F_n^h)$, $f^h=(f_1^h, \cdots, f_n^h)$ are difference functions ($f_0^h=f_1^h, f_{n+1}^h=f_n^h$). The difference scheme for (\ref{1.7}) is $$F_i^h=\phi_i^h+\frac{1}{\sqrt{\pi}}\sum_{j=1}^ib_{ij} \psi_j^h - \sum_{j=1}^nc_{ij}g_i^h, \quad i=1, 2, \cdots, n\,. $$ where $$b_{ij}=\left\{ \begin{array}{ll} 2\sqrt{h}(\sqrt{i-j+1}-\sqrt{i-j})&\,\, j\le i \\ 0& \,\,j>i \end{array} \right., $$ $$c_{ij}=\mbox{erf}(\frac{j+1}{2}\sqrt{\frac{h}{i}})- \mbox{erf}(\frac{j}{2}\sqrt{\frac{h}{i}}),\, i, j=1, 2, \ldots, n,$$ and $$\phi_i^h=\phi(ih), \psi_j^h=\psi(jh), g_j^h=g(jh).$$ The results of the numerical simulation are shown in the following table ($\delta=0.0001$ and $\lambda=1.2$. $f_{\alpha1}, f_{\alpha2}, f_{\alpha3}$ are approximate solutions corresponding to $n=39, 79, 159$ respectively). \begin{table} \label{tbl1} \begin{center} \begin{tabular}{|c|c|c|c|c|} \hline $t$ & $f_T(t)$ & $f_{\alpha1}(t)$ & $f_{\alpha2}(t)$ & $f_{\alpha3}(t)$\\ \hline 0.025 & 0.000594 & 0.001808 & 0.003578 & 0.004813 \\ \hline 0.05 & 0.002256 & 0.003663 & 0.005205 & 0.006199 \\ \hline 0.1 & 0.008100 & 0.011018 & 0.011220 & 0.011334 \\ \hline 0.2 & 0.025600 & 0.031311 & 0.028778 & 0.027295 \\ \hline 0.3 & 0.044100 & 0.049948 & 0.046540 & 0.044596 \\ \hline 0.4 & 0.057600 & 0.062285 & 0.059168 & 0.057404 \\ \hline 0.5 & 0.062500 & 0.066572 & 0.063701 & 0.062082 \\ \hline \end{tabular} \end{center} \caption{Exact and approximate solutions} \end{table} One can see from the data in the Table~\ref{tbl1} that the numbers generated through the computation show that the approximate solutions and the exact solution match better as $n$ becomes larger. The numbers also show that the approximation when $t$ is very close to 0 is not nearly as good as the approximation elsewhere. We think it is because we know little about $f(0)$ in advance. The only assumption on $f$ at 0 is $f'(0)=0$ (see (\ref{2.3})). Therefore we do not have much control of $f$ when $t$ is very very small. Overall, the table shows that, for large $n$, our regularization approach i s a reliable way of recovering unknown source or sink term in a heat equation from non-smooth overspecified data. \begin{thebibliography}{10} \bibitem{kn:cjr1} Cannon, J. R., \newblock Determination of an unknown coefficient in a parabolic differential equation, \newblock {\em Duke Math. J.}, 30, pp 313-323, (1963). \bibitem{kn:cjr2} Cannon, J. R., \newblock Determination of the unknown coefficient $k(u)$ in the equation $\gamma \cdot k(u)\gamma u =0$ from overspecified boundary data, \newblock {\em J. Math. Anal. Appl,}, 18, (1967). \bibitem{kn:cjr3} Cannon, J. R. and Lin, Y.,. \newblock An inverse problem of finding a parameter in a semi-linear heat equation, \newblock {\em J. Math. Anal. Appl.}, Vol. 145, No. 2, 1990. \bibitem{kn:Cannon} Cannon, J. R. and Zachmann, D., \newblock Parameter determination in parabolic partial differential equations from overspecified boundary data, \newblock {\em Int. J. Engineering,} 20 (6), pp. 779-788, (1982). \bibitem{kn:rw} Rundell, W., \newblock An inverse problem for a parabolic partial differential equation, \newblock {\em Rocky Mountain J. Math.,} 13, pp. 667-688, (1983). \bibitem{kn:say} Shcheblov, A. Yu., \newblock A method for the approximate solution of an inverse problem for the heat-conduction equation, \newblock {\em Comp. Math. and Math. Phys.}, Vol. 32, No. 6, 1992. \bibitem{kn:tan} Tikhonov, A. N. and Arsenin, V. Y., \newblock {\em Solution of Ill-posed Problems}, \newblock John Wiley \& Sons, (1977). \bibitem{kn:wz2} Wang, P. and Zheng, K., \newblock Regularization of an Abel equation, \newblock {\em Integr. Equ. Oper. Theory.,} 29, pp. 243 - 249, (1997). \bibitem{kn:wz1} Zheng, K. and Wang, P., \newblock Numerical Approximation of an unknown boundary term in a heat equation, \newblock {\em Neur. Parall. Sci. Comp.,} 2, pp. 451 - 458, (1994). \bibitem{kn:wz3} Wang, P., Zheng, K., \newblock Reconstruction of heat sources in heat conduction equations, \newblock to appear in {\em Comput. and Appl. Math.} \bibitem{kn:wz4} Wang, P., Zheng, K., \newblock Determination of conductivity in a heat equation, \newblock to appear in {\em Int. J. of Math. and Math. Sci.} \end{thebibliography} \medskip \noindent{\sc Ping Wang } \\ Department of Mathematics, Pennsylvania State University \\ Schuylkill Haven, PA 17972, USA \\ e-mail: pxw10@psu.edu \medskip \noindent{\sc Kewang Zheng }\\ Department of Mathematics \\ Hebei University of Science and Technology,\\ China \end{document}