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Source term identification for an axisymmetric inverse heat conduction problem. (English) Zbl 1189.65215
Summary: We consider an inverse heat source problem of determining the heat source term from the final temperature history of a cylinder. This problem is ill-posed. A simplified Tikhonov regularization method is applied to formulate regularized solution, which is stably convergent to the exact one with a logarithmic type error estimate.

MSC:
65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
35R30 Inverse problems for PDEs
65M30 Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs
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[1] Cannon, J.R., Determination of an unknown heat source from overspecified boundary data, SIAM J. numer. anew., 5, 275-286, (1968) · Zbl 0176.15403
[2] Cannon, J.R.; Esteva, S.P., An inverse problem for the heat equation, Inverse problems, 2, 395-403, (1986) · Zbl 0624.35078
[3] Duchateau, P.; Rundell, W., Unicity in an inverse problem for an unknown reaction term in a reation-diffusion-equation, J. differential equations, 59, 155-164, (1985) · Zbl 0564.35097
[4] Cannon, J.R.; Duchateau, P., Structual identification of an unknown source term in a heat equation, Inverse problems, 14, 535-551, (1998) · Zbl 0917.35156
[5] Silva Neto, A.J.; Ă–zisik, M.N., Two-dimensional inverse heat conduction problem of estimating the time-varying strength of a line heat source, J. appl. phys., 71, 5357-5362, (1992)
[6] Hettlich, F.; Rundell, W., Identification of a discontinous source in the heat equation, Inverse problems, 17, 1465-1482, (2001) · Zbl 0986.35129
[7] Hadamard, J., Lectures on Cauchy problem in linear partial differential equations, (1923), Oxford University Press London · JFM 49.0725.04
[8] Cannon, J.R.; Esteva, S.P., Uniqueness and stability of 3D heat source, Inverse problems, 7, 57-62, (1991) · Zbl 0729.35144
[9] Choulli, M.; Yarmamoto, M., Conditional stability in determining a heat source, J. inverse ill-posed probl., 12, 3, 233-243, (2004) · Zbl 1081.35136
[10] Yi, Z.; Murio, D.A., Source term identification in 1-D IHCP, Comput. math. appl., 47, 1921-1933, (2004) · Zbl 1063.65102
[11] Yi, Z.; Murio, D.A., Source term identification in 2-D IHCP, Comput. math. appl., 47, 1517-1533, (2004) · Zbl 1155.65376
[12] Farcas, A.; Lesnic, D., The boundary-element method for the determination of a heat source dependent on one variable, J. eng. math., 54, 247-253, (2006)
[13] Fatullayev, A.G., Numerical solution of the inverse problem of determining an unknown source term in a heat equation, Math. comput. simulation, 8, 2, 161-168, (2002) · Zbl 1050.65086
[14] Dehghan, M.; Tatari, M., Determination of a contral parameter in a one-dimensional parabolic equation using the method of radial basis functions, Math. comput. modelling, 8, 2, 161-168, (2002)
[15] Mahar, P.S.; Datta, B., Optimal identification of ground-water pollution source and parameter estimate, J. water. res. plan. manag., 127, 1, 20-29, (2001)
[16] Constales, D.; Kacur, J.; Van, K.R., Parameter identification by a single injection-extraction well, Inverse problems, 18, 1605-1620, (2002) · Zbl 1023.35093
[17] li, G.S.; Ma, Y.C.; Li, K.T., An inverse parabolic problem for nonlinear source term with nonlinear boundary condition, J. inverse ill-posed probl., 11, 4, 371-387, (2003) · Zbl 1048.35136
[18] Li, G.S.; Tan, Y.G., A conditional stability for an inverse problem arising in groundwater pollution, J. chin. univ., 14, 3, 217-225, (2005) · Zbl 1105.35145
[19] Trong, D.D.; Long, N.T.; Alain, P.N.D., Nonhomogeneous heat equation: identification and regularization for the inhomogeneous term, J. math. anal. appl., 312, 93-104, (2005) · Zbl 1087.35095
[20] Liu, Y., Cylinder functions, (1983), Industry of National Defence Press, (in Chinese)
[21] Carasso, A., Determining surface temperature from interior observations, SIAM J. appl. math., 42, 558-574, (1982) · Zbl 0498.35084
[22] 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
[23] Cheng, W.; Fu, C.L.; Qian, Z., Two regularization methods for a spherically symmetric inverse heat conduction problem, Appl. math. modelling, 32, 4, 432-442, (2008) · Zbl 1387.35615
[24] Cheng, W.; Fu, C.L., Two regularization methods for an axisymmetric inverse heat conduction problem, J. inverse ill-posed probl., 17, 157-170, (2009)
[25] Abramowitz, M.; Stegun, I.A., Handbook of mathematical functions, (1972), Dover Publications, Inc. New York · Zbl 0515.33001
[26] Jiang, L.S.; Chen, Y.J.; Liu, X.H.; Yi, F.K., Teaching materials to equation of mathematical physics, (1995), Higher Education Press, (in Chinese)
[27] Kirsch, A., An introduction to the mathematical theory of inverse problems, (1996), Springer New York · Zbl 0865.35004
[28] Cheng, W.; Fu, C.L.; Qian, Z., A modified Tikhonov regularization method for a spherically symmetric three-dimensional inverse heat conduction problem, Math. comput. simulation, 75, 3-4, 97-112, (2007) · Zbl 1122.65083
[29] Nair, M.T.; Tautenhahn, U., Lavrentiev regularization for linear ill-posed problems under general source conditions, J. anal. appl., 23, 1, 167-185, (2004) · Zbl 1063.65041
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