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