Simultaneous determination of time-dependent coefficients in the heat equation. (English) Zbl 1350.65104
Summary: In this paper, the determination of time-dependent leading and lower-order thermal coefficients is investigated. We consider the inverse and ill-posed nonlinear problems of simultaneous identification of a couple of these coefficients in the one-dimensional heat equation from Cauchy boundary data. Unique solvability theorems of these inverse problems are supplied and, in one new case where they were not previously provided, are rigorously proved. However, since the problems are still ill-posed the solution needs to be regularized. Therefore, in order to obtain a stable solution, a regularized nonlinear least-squares objective function is minimized in order to retrieve the unknown coefficients. The stability of numerical results is investigated for several test examples with respect to different noise levels and for various regularization parameters. This study will be significant to researchers working on computational and mathematical methods for solving inverse coefficient identification problems with applications in heat transfer and porous media.
MSC:
| 65M32 | Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs |
| 35R30 | Inverse problems for PDEs |
| 65M30 | Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs |
| 35R25 | Ill-posed problems for PDEs |
| 35K55 | Nonlinear parabolic equations |
| 80A20 | Heat and mass transfer, heat flow (MSC2010) |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 80A23 | Inverse problems in thermodynamics and heat transfer |
Keywords:
inverse problems; thermal properties; nonlinear optimization; ill-posed nonlinear problems; heat equation; Cauchy boundary data; regularized nonlinear least-squares objective function; numerical results; inverse coefficient identification problems; heat transfer; porous mediaReferences:
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