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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.
MSC:
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
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