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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:
 [1] P. M. ANSELONE, 1965, Convergence and Error Bounds for Approximate Solutions of Integral and Operator Equations, Error in Digital Computation, Vol. II (L. B. Rall, eds.), Wiley and Sons, New York, pp.231-252. Zbl0158.34102 MR189277 · Zbl 0158.34102 [2] K. ATKINSON, 1967, The Numerical Solution of Fredholm Integral Equations of the Second Kind, SIAM J. Numerical Anal. 4, 337-348. Zbl0155.47404 MR224314 · Zbl 0155.47404 [3] [3] K. ATKINSON, 1972, The Numerical Solution of Fredholm Integral Equations of the Second Kind with Singular Kernels, Numer. Math. 19, 248-259. Zbl0258.65117 MR307512 · Zbl 0258.65117 [4] H. T. BANKS and K. KUNISCH, 1989, Estimation Technique for Distributed Systems, Birkhäuser, Boston. Zbl0695.93020 MR1045629 · Zbl 0695.93020 [5] J. R. CANNON, 1984, The One-Dimensional Heat Equation, Addison-Wesley, Menlo Park, CA. Zbl0567.35001 MR747979 · Zbl 0567.35001 [6] A. FRIEDMAN, 1959, Free Boundary Problems for Parabolic Equations I. Melting of Solids, J. Math. and Mech. 8, 499-517. Zbl0089.07801 MR144078 · Zbl 0089.07801 [7] K.-H. HOFFMANN and H.-J. KORNSTAEDT, 1982, Ein numerisches Verfahren zur Lösung eines Identifizierungsproblems bei der Wärmeleitungsgleichung, Numerical treatment of Free Boundary Value Prolbems (J. Albrecht, L. Collatz, and K.-H. Hoffmann, eds.), Birkhäuser Verlag, Boston, pp. 108-126. Zbl0473.65080 MR680238 · Zbl 0473.65080 [8] K. KUNISCH, K. MURPHY, G. PEICHL, 1991, Estimation of the Conductivity in the One-Phase Stefan Problem I: Basic Results, Tech. Univ. Graz/Univ. Graz, Inst. für Math. Technical Report No. 184-1991. Zbl0848.35140 · Zbl 0848.35140
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