Numerical procedures for recovering a time dependent coefficient in a parabolic differential equation. (English) Zbl 1055.35132

The authors are concerned with the numerical determination of the conductivity coefficient \(a:[0,T]\to {\mathbb R}\) in the one-dimensional parabolic problem \[ D_tu(x,t)=a(t)D_x^2u(x,t),\quad \text{in\;} Q_T=(0,1)\times (0,T), \tag{1} \]
\[ u(x,0)=\varphi(x),\quad x\in [0,1],\qquad u(j,t)=g_j(t),\quad t\in [0,T],\;j=0,1, \tag{2} \] when they are given the additional information \[ u(x^*,t)=h(t),\quad t\in [0,T], \tag{3} \] where \(x^*\) is a given value in \((0,1)\). Setting \(v=D_x^2u\) and assuming \(v(x^*,t)\neq 0\) for any \(t\in [0,T]\), the classical inverse problem (1)-(3) is reduced to solving the nonlocal parabolic problem \[ D_tu(x,t)=a(t)D_x^2u(x,t),\quad \text{in\;} Q_T=(0,1)\times (0,T), \tag{4} \]
\[ v(x,0)=\varphi''(x),\quad x\in [0,1],\qquad v(j,t)={g_j(t)\over h'(t)}v(x^*,t),\quad t\in [0,T],\;j=0,1. \tag{5} \] The method used to approximate problem (4), (5) makes use of a backward Euler scheme. Under the smoothness assumption \(v\in C^{4,2}({\overline Q}_T)\) the authors show the convergence estimate \[ \max_{i=0,\dots,N,\;n=0,\dots,M}\,| v(i/N,nT/M)-v_i^n| \leq C(N^{-2}+TM^{-1}), \tag{6} \] \(N\) and \(M\) being large enough integers, while \(v_i^n\) denotes the approximated value of \(v\) at \((i/N,nT/M)\).
Then they show that the approximating solution to the direct problem \[ D_x^2u(x,t)=v(x,t),\quad \text{ in } Q_T=(0,1)\times (0,T), \tag{7} \]
\[ u(j,t)=g_j(t),\quad t\in [0,T],\;j=0,1, \tag{8} \] satisfies an estimate like (6), where \((v(i/N,nT/M),v_i^n)\) is replaced with \((u(i/N,nT/M),u_i^n)\) provided \(N\) and \(M\) are large enough.
Similar numerical results are stated for the same identification problem when the Dirichlet boundary conditions are replaced with the Neumann ones.
Finally, the numerical results are tested by some explicit examples.


35R30 Inverse problems for PDEs
35K05 Heat equation
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs