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Inverse time-dependent source problems for the heat equation with nonlocal boundary conditions. (English) Zbl 1428.80011

Summary: In this paper, we consider inverse problems of finding the time-dependent source function for the population model with population density nonlocal boundary conditions and an integral over-determination measurement. These problems arise in mathematical biology and have never been investigated in the literature in the forms proposed, although related studies do exist. The unique solvability of the inverse problems are rigorously proved using generalized Fourier series and the theory of Volterra integral equations. Continuous dependence on smooth input data also holds but, as in reality noisy errors are random and non-smooth, the inverse problems are still practically ill-posed. The degree of ill-posedness is characterised by the numerical differentiation of a noisy function. In the numerical process, the boundary element method together with either a smoothing spline regularization or the first-order Tikhonov regularization are employed with various choices of regularization parameter. One is based on the discrepancy principle and another one is the generalized cross-validation criterion. Numerical results for some benchmark test examples are presented and discussed in order to illustrate the accuracy and stability of the numerical inversion.

MSC:

80A23 Inverse problems in thermodynamics and heat transfer
35K20 Initial-boundary value problems for second-order parabolic equations
35K91 Semilinear parabolic equations with Laplacian, bi-Laplacian or poly-Laplacian
35R30 Inverse problems for PDEs
65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
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References:

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