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Inverse problem of time-dependent heat sources numerical reconstruction. (English) Zbl 1219.65103
Authors’ abstract: We study the inverse problem of reconstructing a time-dependent heat source in the heat conduction equation using the temperature measurement specified at an internal point. Problems of this type have important applications in several fields of applied science. By the Green function method, the inverse problem is reduced to an operator equation of the first kind which is known to be ill-posed. The uniqueness of the solution for the inverse problem is obtained by the contraction mapping principle. A numerical algorithm on the basis of the Landweber iteration is designed to deal with the operator equation and some typical numerical experiments are also performed in the paper. The numerical results show that the proposed method is stable and the unknown heat source is recovered very well.
65M32Inverse problems (IVP of PDE, numerical methods)
35K05Heat equation
35R30Inverse problems for PDE
65M38Boundary element methods (IVP of PDE)
[1]Adams, R. A.: Sobolev spaces, (1975)
[2]Beck, V.; Blackwell, B.; Clair, St.C.R.: Inverse heat conduction, ill-posed problems, (1985) · Zbl 0633.73120
[3]Cannon, J. R.: Determination of an unknown heat source from overspecified boundary data, SIAM J. Numer. anal. 5, 275-286 (1986) · Zbl 0176.15403 · doi:10.1137/0705024
[4]Cannon, J. R.; Lin, Y.: An inverse problem of finding a parameter in a semilinear heat equation, J. math. Anal. appl. 145, 470-484 (1990) · Zbl 0727.35137 · doi:10.1016/0022-247X(90)90414-B
[5]Cannon, J. R.: The one-dimensional heat equation, (1984) · Zbl 0567.35001
[6]Dehghan, M.: An inverse problems of finding a source parameter in a semilinear parabolic equation, Appl. math. Model. 25, 743-754 (2001) · Zbl 0995.65098 · doi:10.1016/S0307-904X(01)00010-5 · doi:http://www.elsevier.com/gej-ng/10/10/8/47/39/29/abstract.html
[7]Dehghan, M.: Determination of a control function in three-dimensional parabolic equations, Math. comput. Simul. 61, 89-100 (2003) · Zbl 1014.65097 · doi:10.1016/S0378-4754(01)00434-7
[8]Dehghan, M.: Determination of a control parameter in the two-dimensional diffusion equation, Appl. numer. Math. 37, 489-502 (2001) · Zbl 0982.65103 · doi:10.1016/S0168-9274(00)00057-X
[9]Deng, Z. C.; Yu, J. N.; Yang, L.: Optimization method for an evolutional type inverse heat conduction problem, J. phys. A: math. Theor. 41, 035201 (2008) · Zbl 1149.35079 · doi:10.1088/1751-8113/41/3/035201
[10]Deng, Z. C.; Yu, J. N.; Yang, L.: Identifying the coefficient of first-order in parabolic equation from final measurement data, Math. comput. Simul. 77, 421-435 (2008) · Zbl 1141.65073 · doi:10.1016/j.matcom.2008.01.002
[11]Engl, H. W.; Hanke, M.; Neubauer, A.: Regularization of inverse problems, (1996)
[12]Friedman, A.: Partial differential equations of parabolic type, (1964) · Zbl 0144.34903
[13]Hon, Y. C.; Wei, T.: A fundamental solution method for inverse heat conduction problem, Eng. anal. Boundary elem. 28, 489-495 (2004) · Zbl 1073.80002 · doi:10.1016/S0955-7997(03)00102-4
[14]Isakov, V.: Inverse problems for partial differential equations, (1998) · Zbl 0906.35111 · doi:emis:journals/DMJDMV/xvol-icm/10/10.html
[15]Johansson, T.; Lesnic, D.: Determination of a spacewise dependent heat source, J. comput. Appl. math. 209, 66-80 (2007) · Zbl 1135.35097 · doi:10.1016/j.cam.2006.10.026
[16]Johansson, T.; Lesnic, D.: A procedure for determining a spacewise dependent heat source and the initial temperature, Appl. anal. 87, 265-276 (2008) · Zbl 1133.35436 · doi:10.1080/00036810701858193
[17]Kirsch, A.: An introduction to the mathematical theory of inverse problem, (1999)
[18]Ladyzenskaya, O.; Solonnikov, V.; Ural’ceva, N.: Linear and quasilinear equations of parabolic type, (1968) · Zbl 0174.15403
[19]Niliot, C. L.; Callet, P.: Infrared thermography applied to the resolution of inverse heat conduction problems: recovery of heat line sources and boundary conditions, Rev. gén. Therm. 37, 629-643 (1998)
[20]Onyango, T. T. M.; Ingham, D. B.; Lesnic, D.; Slodička, M.: Determination of a time-dependent heat transfer coefficient from non-standard boundary measurements, Math. comput. Simul. 79, 1577-1584 (2009) · Zbl 1169.65091 · doi:10.1016/j.matcom.2008.07.014
[21]Özisik, M. N.: Heat conduction, (1993)
[22]Prilepko, A. I.; Orlovsky, D. G.; Vasin, I. A.: Methods for solving inverse problems in mathematical physics, (2000)
[23]Rundell, W.: The determination of a parabolic equation from initial and final data, Proc. am. Math. soc. 99, 637-642 (1987) · Zbl 0644.35093 · doi:10.2307/2046467
[24]Neto, A. J. Silva; Özisik, M. N.: Inverse problem of simultaneously estimating the timewise-varying strengths of two plane heat sources, J. appl. Phys. 73, No. 5, 2132-2137 (1993)
[25]Neto, A. J. Silva; Özisik, 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)
[26]Stoer, J.; Bulirsch, R.: Introduction to numerical analysis, (1980) · Zbl 0553.65004
[27]Tikhonov, A.; Arsenin, V.: Solutions of ill-posed problems, (1979)
[28]Wang, Z. W.; Liu, J. J.: Identification of the pollution source from one-dimensional parabolic equation models, Appl. math. Comput. (2008)
[29]Yan, L.; Fu, C. L.; Yang, F. L.: The method of fundamental solutions for the inverse heat source problem, Eng. anal. Boundary elem. 32, 216-222 (2008)
[30]Yang, L.; Deng, Z. C.; Yu, J. N.; Luo, G. W.: Optimization method for the inverse problem of reconstructing the source term in a parabolic equation, Math. comput. Simul. 80, 314-326 (2009) · Zbl 1183.65118 · doi:10.1016/j.matcom.2009.06.031
[31]Yang, L.; Yu, J. N.; Deng, Z. C.: An inverse problem of identifying the coefficient of parabolic equation, Appl. math. Modelling 32, No. 10, 1984-1995 (2008) · Zbl 1145.35468 · doi:10.1016/j.apm.2007.06.025
[32]Yi, Z.; Murio, D. A.: Source term identification in 1-D IHCP, Comput. math. Appl. 47, 1921-1933 (2004) · Zbl 1063.65102 · doi:10.1016/j.camwa.2002.11.025