Source term identification in 1-D IHCP. (English) Zbl 1063.65102

Summary: We propose stable numerical solutions for the simultaneous identification of temperature, temperature gradient, and general source terms in the one-dimensional inverse heat conduction problem (IHCP).
The numerical solution consists of a regularization procedure, based on the mollification method, and a marching scheme for the solution of the stabilized problem. The stability, error analysis and implementation of the algorithm are presented together with a set of numerical results.


65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
35K05 Heat equation
35R30 Inverse problems for PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
80A23 Inverse problems in thermodynamics and heat transfer
80M20 Finite difference methods applied to problems in thermodynamics and heat transfer
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[1] Chavent, G.; Jaffre, J., Mathematical models and finite elements for reservoir simulation, (1986), North Holland Boston, MA · Zbl 0603.76101
[2] ()
[3] Ewing, R.; Lin, T.; Falk, R., (), 483-497
[4] Eldén, L., Solving an inverse heat conduction problem by a “method of lines”, Journal of heat transfer, 119, 406-412, (1997)
[5] Mejfa, C.E.; Murio, D.A., Mollified hyperbolic method for coefficient identification problems, Computers math. applic., 26, 5, 1-12, (1993) · Zbl 0789.65090
[6] Coles, C.; Murio, D.A., Simultaneous space diffusivity and source term reconstruction in 2D IHCP, Computers math. applic., 12, 1549-1564, (2001) · Zbl 1005.65106
[7] Murio, D.A., Mollification and space marching, () · Zbl 1071.65130
[8] Cannon, J.R.; DuChateau, P., Inverse problems for an unknown source in the heat equation, Journal of mathematical analysis and applications, 75, 465-485, (1980) · Zbl 0448.35085
[9] Ewing, R.; Lin, T., Parameter identification problems in single-phase and two-phase flow, (), 85-108
[10] Nanda, A.; Das, P., Determination of the source term in the heat conduction equation, Inverse problems, 12, 325-339, (1996) · Zbl 0851.35135
[11] Isakov, V., Inverse source problems, (1990), American Mathematical Society · Zbl 0721.31002
[12] Z. Yi and D.A. Murio, Source terms identification for the diffusion equation, Proceedings 4^{th} International Conference on Inverse Problems in Engineering (to appear). · Zbl 1155.65376
[13] Murio, D.A.; Mejfa, C.E.; Zhan, S., Discrete mollification and automatic numerical differentiation, Computers math. applic., 35, 5, 1-16, (1998) · Zbl 0910.65010
[14] Anderssen, B.; de Hogg, F.; Hegland, M., A stable finite difference ansatz for higher order differentiation of nonexact data, Bull. austral. math. soc., 58, 223-232, (1998) · Zbl 0918.65017
[15] Murio, D.A., The mollification method and the numerical solution of ill-posed problems, (1993), John Wiley and Sons Providence
[16] de Hoog, F.R.; Hutchinson, M.F., An efficient method for calculating splines using orthogonal transformations, Numer. math., 50, 311-319, (1987) · Zbl 0633.65011
[17] Craven, P.; Wahba, G., Smoothing noisy data with spline functions, Numer. math., 31, 377-403, (1979) · Zbl 0377.65007
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