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Determination of a spacewise dependent heat source. (English) Zbl 1135.35097
Summary: This paper investigates the inverse problem of determining a spacewise dependent heat source in the parabolic heat equation using the usual conditions of the direct problem and information from a supplementary temperature measurement at a given single instant of time. The spacewise dependent temperature measurement ensures that the inverse problem has a unique solution, but this solution is unstable, hence the problem is ill-posed. For this inverse problem, we propose an iterative algorithm based on a sequence of well-posed direct problems which are solved at each iteration step using the boundary element method (BEM). The instability is overcome by stopping the iterations at the first iteration for which the discrepancy principle is satisfied. Numerical results are presented for various typical benchmark test examples which have the input measured data perturbed by increasing amounts of random noise.

35R30Inverse problems for PDE
35K05Heat equation
65M32Inverse problems (IVP of PDE, numerical methods)
65M60Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE)
78M15Boundary element methods (optics)
Full Text: DOI
[1] Cannon, J. R.: Determination of an unknown heat source from overspecified boundary data. SIAM J. Numer. anal. 5, 275-286 (1968) · Zbl 0176.15403
[2] A. Farcas, D. Lesnic, The boundary-element method for the determination of a heat source dependent on one variable, J. Eng. Math. 2006, in press. · Zbl 1146.80007
[3] Goldstein, J. A.: Semigroups of linear operators and applications. (1985) · Zbl 0592.47034
[4] Engl, H. W.; Hanke, M.; Neubauer, A.: Regularization of inverse problems. (1996) · Zbl 0859.65054
[5] Lions, J. -L.; Magenes, E.: Non-homogeneous boundary value problems and applications. (1972) · Zbl 0223.35039
[6] Morozov, V. A.: On the solution of functional equations by the method of regularization. Dokl. akad. Nauk SSSR 167, 510-512 (1966) · Zbl 0187.12203
[7] Prilepko, A. I.; Solov’ev, V. V.: Solvability theorems and rothe’s method for inverse problems for a parabolic equation. Differential equations 23, 1341-1349 (1988) · Zbl 0683.35090
[8] Rundell, W.: Determination of an unknown non-homogeneous term in a linear partial differential equation from overspecified boundary data. Appl. anal. 10, 231-242 (1980) · Zbl 0454.35045
[9] Solov’ev, V. V.: Solvability of the inverse problems of finding a source, using overdetermination on the upper base for a parabolic equation. Differential equations 25, 1114-1119 (1990)