zbMATH — the first resource for mathematics

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.

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
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.