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\textbf{Finite Difference Method for the Estimation of a Heat Source Dependent}
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\hrule

A. Ashyralyev$^{1}$, \textbf{A.S. Erdogan}$^{1}$ and Z. Cakir$^{2}$

$^{1}$Department of Mathematics, Fatih University, Istanbul, Turkey
$^{2}$Department of Mathematical Engineering, Gumushane University,Gumushane, Turkey
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\textbf{Abstract}

The inverse parabolic problem is of significant importance in mathematical
sciences, applied sciences and engineering [1-3]. In the process of
transportation, diffusion and conduction of natural materials, the parabolic
partial differential equation is induced (see [4]). In this paper, we
present the numerical solutions of the parabolic inverse problem with the
Dirichlet condition. Problem requires finding the temperature $u(x,t)$ and
the unknown right hand side term $p(t)$ satisfying the heat equation

\begin{equation}
u_{t}-u_{xx}+u=p\left( t\right) q\left( x\right) +f\left( t,x\right) ,\text{
{in }}{(x,t)\in (0,L)\times (0,T]}  \label{1}
\end{equation}%
subject to the initial condition

\begin{equation}
u(x,0)=u_{0}(x),{0\leq x\leq L},  \label{2}
\end{equation}%
the Dirichlet boundary condition

\begin{equation}
u(0,t)=u\left( L,t\right) =0,{0<t\leq T},  \label{3}
\end{equation}%
and the overdetermined conditions in an interior point

\begin{equation}
u(x^{\ast },t)=0,{0<t\leq T},\,0<x^{\ast }<L.  \label{4}
\end{equation}

The coercive stability estimates for the solution of first and second orders
of accuracy difference scheme are established in the difference analogue of
a space of smooth functions. The theoretical statements for the solution of
this difference schemes for parabolic inverse problem is supported by the
results of numerical experiments.

\textbf{References}

[1] Isakov V., Inverse Problems for Partial Differential Equations, Applied
Mathematical Sciences, Springer-Verlag, vol. 127, 1997.

[2] Kimura T. and Suzuki T., A parabolic inverse problem arising in a
mathematical model for chromotography, SIAM J. Appl. Math., 53(6),
1747-1761, 1993.

[3] Erdogan A.S., Finite Difference Method for the Estimation of a Heat
Source Dependent, in~Proceedings of the~International Conference on Analysis
and Applied Mathematics, Gumushane, 2012, edited by Ashyralyev A., 1-2, 2012.

[4] Yan L., Fu Chu-Li and Yang Feng-Lian, The method of fundamental
solutions for the inverse heat source problem, Eng. Anal. Bound. Elem., 32,
216-222, 2008.\hrule

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