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Recovering the initial condition in the one-phase Stefan problem. (English) Zbl 1487.35446

Summary: We consider the problem of recovering the initial condition in the one-dimensional one-phase Stefan problem for the heat equation from the knowledge of the position of the melting point. We first recall some properties of the free boundary solution. Then we study the uniqueness and stability of the inversion. The principal contribution of the paper is a new logarithmic type stability estimate that shows that the inversion may be severely ill-posed. The proof is based on integral equations representation techniques, and the unique continuation property for parabolic type solutions. We also present few numerical examples operating with noisy synthetic data.

MSC:

35R30 Inverse problems for PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35R35 Free boundary problems for PDEs
80A22 Stefan problems, phase changes, etc.
45Q05 Inverse problems for integral equations
65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs

References:

[1] K. Ammari; A. Bchatnia; K. El Mufti, A remark on observability of the wave equation with moving boundary, J. Appl. Anal., 23, 43-51 (2017) · Zbl 1365.35088 · doi:10.1515/jaa-2017-0007
[2] K. Ammari; F. Triki, On weak observability for evolution systems with skew-adjoint generators, SIAM J. Math. Anal., 52, 1884-1902 (2020) · Zbl 1445.35319 · doi:10.1137/19M1241830
[3] G. Bruckner; J. Cheng, Tikhonov regularization for an integral equation of the first kind with logarithmic kernel, J. Inverse Ill-Posed Probl., 8, 665-675 (2000) · Zbl 0982.65142 · doi:10.1515/jiip.2000.8.6.665
[4] J. R. Cannon; J. Douglas Jr., The Cauchy problem for the heat equation, SIAM J. Numer. Anal., 4, 317-336 (1967) · Zbl 0154.36502 · doi:10.1137/0704028
[5] J. R. Cannon; J. Douglas Jr., The stability of the boundary in a Stefan problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 21, 83-91 (1967) · Zbl 0154.36402
[6] J. R. Cannon; C. D. Hill, Existence, uniqueness, stability, and monotone dependence in a Stefan problem for the heat equation, J. Math. Mech., 17, 1-19 (1967) · Zbl 0154.36403 · doi:10.1512/iumj.1968.17.17001
[7] J. R. Cannon; M. Primicerio, Remarks on the one-phase Stefan problem for the heat equation with the flux prescribed on the fixed boundary, J. Math. Anal. Appl., 35, 361-373 (1971) · Zbl 0221.35035 · doi:10.1016/0022-247X(71)90223-X
[8] M. Choulli, Various stability estimates for the problem of determining an initial heat distribution from a single measurement, Riv. Math. Univ. Parma (N.S.), 7, 279-307 (2016) · Zbl 1379.35349
[9] M. Choulli and M. Yamamoto, Logarithmic stability of parabolic Cauchy problems, preprint, arXiv: 1702.06299v4. · Zbl 1481.35387
[10] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Mathematics and its Applications, 375, Kluwer Academic Publishers Group, Dordrecht, 1996. · Zbl 0859.65054
[11] E. Fernández-Cara; F. Hernández; J. Límaco, Local null controllability of a 1D Stefan problem, Bull. Braz. Math. Soc. (N.S.), 50, 745-769 (2019) · Zbl 1425.93039 · doi:10.1007/s00574-018-0093-9
[12] E. Fernández-Cara; J. Limaco; S. B. de Menezes, On the controllability of a free-boundary problem for the 1D heat equation, Systems Control Lett., 87, 29-35 (2016) · Zbl 1327.93076 · doi:10.1016/j.sysconle.2015.10.011
[13] A. Friedman, Variational Principles and Free Boundary Problems, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982. · Zbl 0564.49002
[14] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1964. · Zbl 0144.34903
[15] G. C. Garcia; A. Osses; M. Tapia, A heat source reconstruction formula from single internal measurements using a family of null controls, J. Inverse Ill-Posed Probl., 21, 755-779 (2013) · Zbl 1278.35266 · doi:10.1515/jip-2011-0001
[16] B. Geshkovski; E. Zuazua, Controllability of one-dimensional viscous free boundary flows, SIAM J. Control Optim., 59, 1830-1850 (2021) · Zbl 1467.93034 · doi:10.1137/19M1285354
[17] C. Ghanmi, S. Mani-Aouadi and F. Triki, Identification of a Boundary Influx Condition in A One-Phase Stefan Problem, Appl. Anal., to appear. · Zbl 1498.35615
[18] N. L Gol’dman, Inverse Stefan Problems, Springer Science & Business Media, 2012.
[19] A. Hajiollow; Y. Lotfi; K. Parand; A. H. Hadian; K. Rashedi; J. A. Rad, Recovering a moving boundary from Cauchy data in an inverse problem which arises in modeling brain tumor treatment: The (quasi) linearization idea combined with radial basis functions (RBFs) approximation, Engineering with Computers, 37, 1735-1749 (2021) · doi:10.1007/s00366-019-00909-8
[20] M. Hanke; A. Neubauer; O. Scherzer, A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numer. Math., 72, 21-37 (1995) · Zbl 0840.65049 · doi:10.1007/s002110050158
[21] P. Jochum, The numerical solution of the inverse Stefan problem, Numer. Math., 34, 411-429 (1980) · Zbl 0411.65058 · doi:10.1007/BF01403678
[22] B. T. Johansson; D. Lesnic; T. Reeve, A method of fundamental solutions for the one-dimensional inverse Stefan problem, Appl. Math. Model., 35, 4367-4378 (2011) · Zbl 1225.80007 · doi:10.1016/j.apm.2011.03.005
[23] P. Knabner, Control of Stefan problems by means of linear-quadratic defect minimization, Numer. Math., 46, 429-442 (1985) · Zbl 0568.65085 · doi:10.1007/BF01389495
[24] W. T. Kyner, An existence and uniqueness theorem for a nonlinear Stefan problem, J. Math. Mech., 8, 483-498 (1959) · Zbl 0087.09301 · doi:10.1512/iumj.1959.8.58035
[25] O. A. Ladyzhenskaia, V. A. Solonnikov and N. N. Ural’tseva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Soc., 1968. · Zbl 0174.15403
[26] L. Landweber, An iteration formula for Fredholm integral equations of the first kind, Amer. J. Math., 73, 615-624 (1951) · Zbl 0043.10602 · doi:10.2307/2372313
[27] J. Li; M. Yamamoto; J. Zou, Conditional stability and numerical reconstruction of initial temperature, Commun. Pure Appl. Anal., 8, 361-382 (2009) · Zbl 1181.65121 · doi:10.3934/cpaa.2009.8.361
[28] R. Nevanlinna, H. Behnke, L. V. Grauert, H. Ahlfors, D. C. Spencer, L. Bers, K. Kodaira, M. Heins and J. A. Jenkins, Analytic Functions, Berlin, Springer, 1970.
[29] R. Reemtsen; A. Kirsch, A method for the numerical solution of the one-dimensional inverse Stefan problem, Numer. Math., 45, 253-273 (1984) · Zbl 0591.65086 · doi:10.1007/BF01389470
[30] L. I. Rubenšteǐn, The Stefan Problem, Translations of Mathematical Monographs, 27, American Mathematical Society, Providence, RI, 1971. · Zbl 0219.35043
[31] W. Rudin, Real and Complex Analysis, \(2^{nd}\) edition, McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974. · Zbl 0278.26001
[32] T. Wei; M. Yamamoto, Reconstruction of a moving boundary from Cauchy data in one-dimensional heat equation, Inverse Probl. Sci. Eng., 17, 551-567 (2009) · Zbl 1165.65387 · doi:10.1080/17415970802231610
[33] L. C. Wrobel, A boundary element solution to Stefan’s problem, Boundary Elements V, (1983). · Zbl 0541.73150
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