Regularization methods for a Cauchy problem for a parabolic equation in multiple dimensions. (English) Zbl 1186.35243
Summary: We study a Cauchy problem for a parabolic equation in multiple dimensions, which is naturally a generalization of some one-dimensional and two-dimensional inverse heat conduction problems. This is a severely ill-posed problem, i.e., the solution (if it exists) does not depend continuously on the data. After simply analyzing the ill-posedness of the Cauchy problem in the frequency space, from a new viewpoint we propose two regularization methods: Tikhonov method and Fourier truncation method. We give and prove the convergence estimate between the exact solution and its regularized approximation. We also discuss the relationship of these two and other regularization methods. At last, we employ some numerical examples to illustrate the behavior of the proposed methods.
MSC:
35R30 | Inverse problems for PDEs |
65M30 | Numerical methods for ill-posed problems 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 |
65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
35R25 | Ill-posed problems for PDEs |
80A23 | Inverse problems in thermodynamics and heat transfer |
Keywords:
inverse heat conduction problem; Tikhonov regularization; Fourier truncation; error estimateReferences:
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