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)


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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