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Estimation of the conductivity in the one-phase Stefan problem: Numerical results. (English) Zbl 0786.65108
For the one-dimensional diffusion equation, an iterative algorithm is proposed to estimate the transient conductivity coefficient from data obtained through a free boundary problem, the so-called one-phase Stefan problem, namely: \(u_ t = a(t)u_{xx}\), \(0 < t \leq T\), \(0 < s < s(t)\); \(a(t)u_ x(0,t) = g(t)\), \(0 < t \leq T\); \(u(s(t),t) = 0\), \(0 \leq t\leq T\); \(u(x,0) = \phi(x)\), \(0 \leq x \leq b\); \(\dot s(t) = -a(t)u_ x(s(t),t)\), \(0 < t\leq T\), \(s(0) =b\). The conductivity coefficient is obtained by minimizing a cost functional (two of them are proposed) related to an approximated Stefan problem. Stability results for this problem guarantee the algorithm convergence, some numerical results obtained from actual implementations give support to the theoretical results.
The paper is grosso modo self-contained.

MSC:
65Z05 Applications to the sciences
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35R35 Free boundary problems for PDEs
35K05 Heat equation
80A22 Stefan problems, phase changes, etc.
35R30 Inverse problems for PDEs
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References:
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