Geodesic
A simple regularization method for the ill-posed evolution equation
Czechoslovak Mathematical Journal, Tome 61 (2011) no. 1, pp. 85-95
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

The nonhomogeneous backward Cauchy problem $$u_t +Au(t) = f(t),\quad u(T) = \varphi$$, where $A$ is a positive self-adjoint unbounded operator which has continuous spectrum and $f$ is a given function being given is regularized by the well-posed problem. New error estimates of the regularized solution are obtained. This work extends earlier results by N. Boussetila and by M. Denche and S. Djezzar.
The nonhomogeneous backward Cauchy problem $$u_t +Au(t) = f(t),\quad u(T) = \varphi$$, where $A$ is a positive self-adjoint unbounded operator which has continuous spectrum and $f$ is a given function being given is regularized by the well-posed problem. New error estimates of the regularized solution are obtained. This work extends earlier results by N. Boussetila and by M. Denche and S. Djezzar.
DOI : 10.1007/s10587-011-0019-9
Classification : 35K05, 35K99, 47H10, 47J06
Keywords: nonlinear parabolic problem; backward problem; semigroup of operators; ill-posed problem; contraction principle 
@article{10_1007_s10587_011_0019_9,
     author = {Tuan, Nguyen Huy and Trong, Dang Duc},
     title = {A simple regularization method for the ill-posed evolution equation},
     journal = {Czechoslovak Mathematical Journal},
     pages = {85--95},
     year = {2011},
     volume = {61},
     number = {1},
     doi = {10.1007/s10587-011-0019-9},
     mrnumber = {2782761},
     zbl = {1224.35165},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10587-011-0019-9/}
}
TY  - JOUR
AU  - Tuan, Nguyen Huy
AU  - Trong, Dang Duc
TI  - A simple regularization method for the ill-posed evolution equation
JO  - Czechoslovak Mathematical Journal
PY  - 2011
SP  - 85
EP  - 95
VL  - 61
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.1007/s10587-011-0019-9/
DO  - 10.1007/s10587-011-0019-9
LA  - en
ID  - 10_1007_s10587_011_0019_9
ER  - 
%0 Journal Article
%A Tuan, Nguyen Huy
%A Trong, Dang Duc
%T A simple regularization method for the ill-posed evolution equation
%J Czechoslovak Mathematical Journal
%D 2011
%P 85-95
%V 61
%N 1
%U http://geodesic.mathdoc.fr/articles/10.1007/s10587-011-0019-9/
%R 10.1007/s10587-011-0019-9
%G en
%F 10_1007_s10587_011_0019_9
Tuan, Nguyen Huy; Trong, Dang Duc. A simple regularization method for the ill-posed evolution equation. Czechoslovak Mathematical Journal, Tome 61 (2011) no. 1, pp. 85-95. doi: 10.1007/s10587-011-0019-9

[1] Ames, K. A., Hughes, R. J.: Structural stability for ill-posed problems in Banach space. Semigroup Forum 70 (2005), 127-145. | DOI | MR | Zbl

[2] Boussetila, N., Rebbani, F.: Optimal regularization method for ill-posed Cauchy problems. Electron. J. Differ. Equ. 147 (2006), 1-15. | MR | Zbl

[3] Clark, G. W., Oppenheimer, S. F.: Quasireversibility methods for non-well posed problems. Electron. J. Diff. Eqns. 1994 (1994), 1-9. | MR | Zbl

[4] Denche, M., Bessila, K.: A modified quasi-boundary value method for ill-posed problems. J. Math. Anal. Appl. 301 (2005), 419-426. | DOI | MR | Zbl

[5] Denche, M., Djezzar, S.: A modified quasi-boundary value method for a class of abstract parabolic ill-posed problems. Bound. Value Probl. 2006, Article ID 37524 (2006), 1-8. | MR | Zbl

[6] Eldén, L., Berntsson, F., Reginska, T.: Wavelet and Fourier methods for solving the sideways heat equation. SIAM J. Sci. Comput. 21 (2000), 2187-2205. | DOI | MR

[7] Fu, C.-L., Xiong, X.-T., Fu, P.: Fourier regularization method for solving the surface heat flux from interior observations. Math. Comput. Modelling 42 (2005), 489-498. | DOI | MR | Zbl

[8] Fu, C.-L.: Simplified Tikhonov and Fourier regularization methods on a general sideways parabolic equation. J. Comput. Appl. Math. 167 (2004), 449-463. | DOI | MR | Zbl

[9] Fu, C.-L., Xiang, X.-T., Qian, Z.: Fourier regularization for a backward heat equation. J. Math. Anal. Appl. 331 (2007), 472-480. | DOI | MR

[10] Gajewski, H., Zaccharias, K.: Zur regularisierung einer klass nichtkorrekter probleme bei evolutiongleichungen. J. Math. Anal. Appl. 38 (1972), 784-789. | DOI | MR

[11] Hào, D. N., Duc, N. Van, Sahli, H.: A non-local boundary value problem method for parabolic equations backward in time. J. Math. Anal. Appl. 345 (2008), 805-815. | DOI | MR

[12] Huang, Y., Zheng, Q.: Regularization for a class of ill-posed Cauchy problems. Proc. Am. Math. Soc. 133 (2005), 3005-3012. | DOI | MR | Zbl

[13] Lattès, R., Lions, J.-L.: Méthode de Quasi-réversibilité et Applications. Dunod Paris (1967), French. | MR

[14] Long, N. T., Ding, A. Pham Ngoc: Approximation of a parabolic nonlinear evolution equation backwards in time. Inverse Probl. 10 (1994), 905-914. | MR

[15] Mel'nikova, I. V., Filinkov, A. I.: Abstract Cauchy problems: Three approaches. Monograph and Surveys in Pure and Applied Mathematics, Vol. 120. Chapman & Hall/CRC London-New York/Boca Raton (2001). | MR

[16] Miller, K.: Stabilized quasi-reversibility and other nearly-best-possible methods for non-well posed problems. Sympos. non-well posed probl. logarithmic convexity. Lect. Notes Math. Vol. 316 Springer Berlin (1973), 161-176. | DOI | MR

[17] Payne, L. E.: Improperly Posed Problems in Partial Differential Equations. SIAM Philadelphia (1975). | MR | Zbl

[18] Pazy, A.: Semigroups of Linear Operators and Application to Partial Differential Equations. Springer New York (1983). | MR

[19] Showalter, R. E.: The final value problem for evolution equations. J. Math. Anal. Appl. 47 (1974), 563-572. | DOI | MR | Zbl

[20] Showalter, R. E.: Quasi-reversibility of first and second order parabolic evolution equations. Improp. Posed Bound. Value Probl. (Conf. Albuquerque, 1974). Res. Notes in Math., No. 1 Pitman London (1975), 76-84. | MR

[21] Tautenhahn, U., Schröter, T.: On optimal regularization methods for the backward heat equation. Z. Anal. Anwend. 15 (1996), 475-493. | DOI | MR

[22] Tautenhahn, U.: Optimality for ill-posed problems under general source conditions. Numer. Funct. Anal. Optimization 19 (1998), 377-398. | DOI | MR | Zbl

[23] Trong, D. D., Tuan, N. H.: Stabilized quasi-reversibility method for a class of nonlinear ill-posed problems. Electron. J. Differ. Equ. No 84 (2008). | MR | Zbl

Cité par Sources :