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Numerical procedures for the determination of an unknown coefficient in semi-linear parabolic differential equations. (English) Zbl 0805.65133

The authors study a finite difference method for approximating the unknown source parameter $p=p\left(t\right)$ and $u=u\left(x,y,t\right)$ of the following inverse problem: find $u=u\left(x,y,t\right)$ and $p=p\left(t\right)$ which satisfy (1) ${u}_{t}={\Delta }u+p\left(t\right)u+f\left(x,y,t\right)$ in ${Q}_{T}$; $u\left(x,y,0\right)=\phi \left(x,y\right)$, $\left(x,y\right)\in {\Omega }$; $u\left(x,y,t\right)=g\left(x,y,t\right)$ on $\partial {\Omega }×\left[0,T\right]$; subject to the additional specification (2) $u\left({x}^{*},{y}^{*},t\right)=E\left(t\right)$, $\left({x}^{*},{y}^{*}\right)\in {\Omega }$, $0\le t\le T$, where ${Q}_{T}={\Omega }×\left(0,T\right]$, $T>0$, ${\Omega }=\left(0,1\right)×\left(0,1\right)$, $f$, $\phi$, $g$ and $E\ne 0$ are known functions, and $\left({x}^{*},{y}^{*}\right)$ is a fixed prescribed interior point in ${\Omega }$ whose boundary is denoted by $\partial {\Omega }$. If $u$ represents the temperature then the problem $\left(1\right)-\left(2\right)$ can be viewed as a control problem of finding the control $p=p\left(t\right)$ such that the internal constraint (2) is satisfied.

The backward Euler scheme is studied and its convergence is proved via an application of the discrete maximum principle for a transformed problem. The approximation of $u$ and $p$ in terms of the approximation obtained for the transformed problem is discussed. Finally the paper contains some numerical computations for several examples which support the theoretical analysis.

##### MSC:
 65Z05 Applications of numerical analysis to physics 65M30 Improperly posed problems (IVP of PDE, numerical methods) 65M06 Finite difference methods (IVP of PDE) 35K15 Second order parabolic equations, initial value problems 35R30 Inverse problems for PDE