## 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 $$p=p(t)$$ and $$u=u(x,y,t)$$ of the following inverse problem: find $$u=u(x,y,t)$$ and $$p=p(t)$$ which satisfy (1) $$u_ t=\Delta u+p(t)u+f(x,y,t)$$ in $$Q_ T$$; $$u(x,y,0)=\phi(x,y)$$, $$(x,y)\in\Omega$$; $$u(x,y,t)=g(x,y,t)$$ on $$\partial\Omega\times[0,T]$$; subject to the additional specification (2) $$u(x^*,y^*,t)=E(t)$$, $$(x^*,y^*)\in\Omega$$, $$0\leq t\leq T$$, where $$Q_ T=\Omega\times(0,T]$$, $$T>0$$, $$\Omega=(0,1)\times(0,1)$$, $$f$$, $$\phi$$, $$g$$ and $$E\neq 0$$ are known functions, and $$(x^*,y^*)$$ is a fixed prescribed interior point in $$\Omega$$ whose boundary is denoted by $$\partial\Omega$$. If $$u$$ represents the temperature then the problem $$(1)-(2)$$ can be viewed as a control problem of finding the control $$p=p(t)$$ 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.
Reviewer: G.Dimitriu (Iaşi)

### MSC:

 65Z05 Applications to the sciences 65M30 Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35K15 Initial value problems for second-order parabolic equations 35R30 Inverse problems for PDEs
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