zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Numerical approximation of solution of nonhomogeneous backward heat conduction problem in bounded region. (English) Zbl 1166.65048
The authors consider the backward problem for the 1D heat equation with constant coefficients subject to homogeneous Dirichlet boundary conditions. The numerical method is based on the explicit solution of the forward problem which can be obtained by the Fourier method, i.e., the method of separation of variables. As approximate inverse the inverse of a truncated Fourier series is used which is regularized with the Tikhonov method. An error estimate for perturbed data is given as well as various numerical examples.
MSC:
65M30Improperly posed problems (IVP of PDE, numerical methods)
35K05Heat equation
35R25Improperly posed problems for PDE
65M70Spectral, collocation and related methods (IVP of PDE)
References:
[1]Engl, H. W.; Hanke, M.; Neubauer, A.: Regularization of inverse problem, (1996)
[2]Groetsch, C. W.: The theory of Tikhonov regularization for Fredholm equation of the first kind, (1984)
[3]Hadamard, J.: Lectures on Cauchy problem in linear partial differential equations, (1923) · Zbl 49.0725.04
[4]Han, H.; Ingham, D. B.; Yuan, Y.: The boundary element method for the asolution of the backward heat conduction equation, J. comput. Phys. 116, 292-299 (1995) · Zbl 0821.65064 · doi:10.1006/jcph.1995.1028
[5]Hào, D. N.: A mollification method for ill-posed problems, Numer. math. 68, 469-506 (1994) · Zbl 0817.65041 · doi:10.1007/s002110050073
[6]Iijima, K.: Numerical solution of backward heat conduction problems by a high order lattice-free finite difference method, J. chin. Inst. eng. 27, 611-620 (2004)
[7]Jourhmane, M.; Mera, N. S.: An iterative algorithm for the backward heat conduction problem based on variable relaxation factors, Inverse probl. Eng. 10, 293-308 (2002)
[8]Kirkup, S. M.; Wadsworth, M.: Solution of inverse diffusion problems by operator-splitting methods, Appl. math. Modelling 26, 1003-1018 (2002) · Zbl 1014.65095 · doi:10.1016/S0307-904X(02)00053-7
[9]Kirsch, A.: An introduction to the mathematical theory of inverse problems, (1996)
[10]Lattès, R.; Lions, J. L.: The method of quasi-reversibility applications to partial differential equations, (1967)
[11]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)
[12]Liu, C. S.: Group preserving scheme for backward heat conduction problems, Int. J. Heat mass transfer 47, 2567-2576 (2004) · Zbl 1100.80005 · doi:10.1016/j.ijheatmasstransfer.2003.12.019
[13]Mera, N. S.: The method of fundamental solutions for the backward heat conduction problem, Oxford ISSN 0969 – 6016, Inverse probl. Sci. eng. 13, 79-98 (2005) · Zbl 1194.80107 · doi:10.1080/10682760410001710141
[14]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 · doi:10.1016/S0017-9310(00)00235-0
[15]Mera, N. S.; Elliott, L.; Ingham, D. B.; Lesnic, D.: An inversion method with decreasing regularization for the backward heat conduction problem, Numer. heat transfer part B 42, 215-230 (2002)
[16]K. Miller, Stabilized quasi-reversibility and other nearly best possible methods for non-well-posed problems, in: Symposium on Non-Well-Posed Problems and Logarithmic Convexity, Lecture Notes in Mathematics, vol. 316, Springer-Verlag, Berlin, 1973, pp. 161 – 176. · Zbl 0279.35004
[17]L.E. Payne, Improperly posed problems in partial differential equations, in: Reional Conference Series in Applied Mathematics SIAM, Philadelphia, PA, 1975. · Zbl 0302.35003
[18]Ramm, A. G.; Smirnova, A. B.: On stable numerical differentiation, Math. comp. 70, 1131-1153 (2001) · Zbl 0973.65015 · doi:10.1090/S0025-5718-01-01307-2
[19]Schröter, T.; Tautenhahn, U.: On optimal regularization methods for the backward heat equation, Z. anal. Anw. 15, 475-493 (1996) · Zbl 0848.65044
[20]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 · doi:10.1016/S0096-3003(98)10010-3
[21]Yildiz, B.; Yetis, H.: A stability estimate on the regularized solution of the backward heat equation, Appl. math. Comput. 135, 561-567 (2003) · Zbl 1135.35368 · doi:10.1016/S0096-3003(02)00069-3