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A modified method for a backward heat conduction problem. (English) Zbl 1112.65090
The authors introduce a new method to consider the ill-posed problem of the backward heat conduction. Several theorems for convergence and error analysis are developed, unfortunately, asymptotic error estimate could not be proved. The developed method is illustrated by numercal experiments.

##### MSC:
 65M30 Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65M15 Error bounds 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 35K05 Heat equation
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##### References:
 [1] Hào, D.N., A mollification method for ill-posed problems, Numer. math., 68, 469-506, (1994) · Zbl 0817.65041 [2] Payne, L.E., Improperly posed problems in partial differential equations, Regional conference series in applied mathematics, (1975), SIAM Philadelphia, PA · Zbl 0302.35003 [3] Hadamard, J., Lectures on Cauchy problem in linear partial differential equations, (1923), Yale University Press New Haven, CT · JFM 49.0725.04 [4] Kirsch, A., An introduction to the mathematical theory of inverse problems, (1999), Springer-Verlag Berlin [5] Lesnic, D.; Elliott, L.; Ingham, D.B., An iterative boundary element method for solving the backward heat conduction problem using an elliptic approximation, Inverse probl. eng., 6, 255-279, (1998) [6] Han, H.; Ingham, D.B.; Yuan, Y., The boundary element method for the solution of the backward heat conduction equation, J. comput. phys., 116, 292-299, (1995) · Zbl 0821.65064 [7] Mera, N.S.; Elliott, L.; Ingham, D.B.; Lesnic, D., An iterative boundary element method for solving the one dimensional backward heat conduction problem, Int. J. heat mass transfer, 44, 1937-1946, (2001) · Zbl 0979.80008 [8] Mera, N.S.; Elliott, L.; Ingham, D.B., An inversion method with decreasing regularization for the backward heat conduction problem, Numer. heat transfer, part B, 42, 215-230, (2002) [9] Liu, C.-S., Group preserving scheme for backward heat conduction problems, Int. J. heat mass transfer, 47, 2567-2576, (2004) · Zbl 1100.80005 [10] Coleman, B.D.; Duffin, R.J.; Mizel, V.J., Instability, uniqueness and non-existence theorems for the equation ut=uxx−uxtx on a strip, Arch. rational mech. anal., 19, 100-116, (1965) [11] Yildiz, B.; Özdemir, M., Stability of the solution of backward heat equation on a weak compactum, Appl. math. comput., 111, 1-6, (2000) · Zbl 1021.35041 [12] Yildiz, B.; Yetiş, H.; Sever, A., A stability estimate on the regularized solution of the backward heat equation, Appl. math. comput., 135, 561-567, (2003) · Zbl 1135.35368 [13] Eldén, L., Approximations for a Cauchy problem for the heat equation, Inverse probl., 3, 263-273, (1987) · Zbl 0645.35094 [14] Chen, P.J.; Gurtin, M.E., On a theory of heat conduction involving two temperatures, Zamp, 19, 614-627, (1968) · Zbl 0159.15103 [15] Barenblatt, G.I.; Zheltov, Iu.P.; Kochina, I.N., Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks (strata), Priklad. mat. mekh., 24, 852-864, (1960) · Zbl 0104.21702 [16] Coleman, B.D.; Noll, W., An approximation theorem for functionals, with application in continuum mechanics, Arch. rational mech. anal., 6, 355-370, (1960) · Zbl 0097.16403
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