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The backward problem of parabolic equations with the measurements on a discrete set. (English) Zbl 1431.65152
Summary: The backward problems of parabolic equations are of interest in the study of both mathematics and engineering. In this paper, we consider a backward problem for the one-dimensional heat conduction equation with the measurements on a discrete set. The uniqueness for recovering the initial value is proved by the analytic continuation method. We discretize this inverse problem by a finite element method to deduce a severely ill-conditioned linear system of algebra equations. In order to overcome the ill-posedness, we apply the discrete Tikhonov regularization with the generalized cross validation rule to obtain a stable numerical approximation to the initial value. Numerical results for three examples are provided to show the effect of the measurement data.
65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
35R25 Ill-posed problems for PDEs
65F22 Ill-posedness and regularization problems in numerical linear algebra
35R30 Inverse problems for PDEs
Full Text: DOI
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