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The method of simplified Tikhonov regularization for dealing with the inverse time-dependent heat source problem. (English) Zbl 1201.65176
Summary: This paper investigates the inverse problem of determining a heat source using a parabolic equation where data are given at some fixed location. The problem is ill-posed, i.e., the solution (if it exists) does not depend continuously on the data. A simplified Tikhonov regularization method is given and an order optimal stability estimate is obtained. A numerical example shows that the regularization method is effective and stable.

65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
35K10 Second-order parabolic equations
80A22 Stefan problems, phase changes, etc.
80M25 Other numerical methods (thermodynamics) (MSC2010)
Full Text: DOI
[1] Li, G.S.; Tan, Y.J.; Cheng, J.; Wang, X.Q., Determining magnitude of groundwater pollution sources by data compatibility analysis, Inverse probl. sci. eng., 14, 287-300, (2006) · Zbl 1194.76269
[2] Cannon, J.R.; Duchateau, P., Structural identification of an unknown source term in a heat equation, Inverse problems, 14, 535-551, (1998) · Zbl 0917.35156
[3] Savateev, E.G., On problems of determining the source function in a parabolic equation, J. inverse ill-posed probl., 3, 83-102, (1995) · Zbl 0828.35142
[4] Farcas, A.; Lesnic, D., The boundary-element method for the determination of a heat source dependent on one variable, J. eng. math., 54, 375-388, (2006) · Zbl 1146.80007
[5] Johansson, T.; Lesnic, D., Determination of a spacewise dependent heat source, J. comput. appl. math., 209, 66-80, (2007) · Zbl 1135.35097
[6] Nili Ahmadabadi, M.; Arab, M.; Maalek Ghaini, F.M., The method of fundamental solutions for the inverse space-dependent heat source problem, Eng. anal. bound. elem., 33, 1231-1235, (2009) · Zbl 1180.80054
[7] Yan, L.; Fu, C.L.; Yang, F.L., The method of fundamental solutions for the inverse heat source problem, Eng. anal. bound. elem., 32, 216-222, (2008) · Zbl 1244.80026
[8] Liu, C.H., A two-stage LGSM to identify time-dependent heat source through an internal measurement of temperature, Int. J. heat mass transfer, 52, 1635-1642, (2009) · Zbl 1157.80395
[9] Fatullayev, A.G., Numerical solution of the inverse problem of determining an unknown source term in a heat equation, Math. comput. simul., 58, 247-253, (2002) · Zbl 0994.65100
[10] Liu, F.B., A modified genetic algorithm for solving the inverse heat transfer problem of estimating plan heat source, Int. J. heat mass transfer, 51, 3745-3752, (2008) · Zbl 1148.80371
[11] Li, G.S., Data compatibility and conditional stability for an inverse source problem in the heat equation, Appl. math. comput., 173, 566-581, (2006) · Zbl 1105.35144
[12] Yamamoto, M., Conditional stability in determination of force terms of heat equations in a rectangle, Math. comput. modelling, 18, 79-88, (1993) · Zbl 0799.35228
[13] L. Yan, C.L. Fu, F.F. Dou, A computational method for identifying a spacewise-dependent heat source, Comm. Numer. Methods Engrg. doi:10.1002/cnm.1155. · Zbl 1190.65145
[14] Yi, Z.; Murio, D.A., Identification of source terms in 2-D IHCP, Comput. math. appl., 47, 1517-1533, (2004) · Zbl 1155.65376
[15] Yi, Z.; Murio, D.A., Source term identification in 1-D IHCP, Comput. math. appl., 47, 1921-1933, (2004) · Zbl 1063.65102
[16] Yan, L.; Yang, F.L.; Fu, C.L., A meshless method for solving an inverse spacewise-dependent heat source problem, J. comput. phys., 228, 123-136, (2009) · Zbl 1157.65444
[17] Dou, F.F.; Fu, C.L., Determining an unknown source in the heat equation by a wavelet dual least squares method, Appl. math. lett., 22, 661-667, (2009) · Zbl 1172.35511
[18] Dou, F.F.; Fu, C.L.; Yang, F.L., Optimal error bound and Fourier regularization for identifying an unknown source in the heat equation, J. comput. appl. math., 230, 728-737, (2009) · Zbl 1219.65100
[19] Engl, H.W.; Hanke, M.; Neubauer, A., Regularization of inverse problem, (1996), Kluwer Academic Boston, MA · Zbl 0711.34018
[20] Carasso, A., Determining surface temperature from interior observations, SIAM J. appl. math., 42, 558-574, (1982) · Zbl 0498.35084
[21] Fu, C.L., Simplified Tikhonov and Fourier regularization methods on a general sideways parabolic equation, J. comput. appl. math., 167, 449-463, (2004) · Zbl 1055.65106
[22] Cheng, W.; Fu, C.L.; Qian, Z., A modified Tikhonov regularization method for a spherically symmetric three-dimensional inverse heat conduction problem, Math. comput. simul., 75, 97-112, (2007) · Zbl 1122.65083
[23] Cheng, W.; Fu, C.L.; Qian, Z., Two regularization methods for a spherically symmetric inverse heat conduction problem, Appl. math. model., 32, 432-442, (2008) · Zbl 1387.35615
[24] Kirsch, A., An introduction to the mathematical theory of inverse problems, (1996), Springer-Verlag New York · Zbl 0865.35004
[25] Eldén, L.; Berntsson, F.; Regiǹska, T., Wavelet and Fourier methods for solving the sideways heat equation, SIAM J. sci. comput., 21, 2187-2205, (2000) · Zbl 0959.65107
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