Article
MSC:
35K05,
35K15,
35K58,
35K99,
35R30,
47J06 | MR 3233554 | Zbl 06362238 | DOI: 10.1007/s10492-014-0066-2
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
nonlinear heat problem; ill-posed problem; Fourier transform; time-dependent coefficient
nonlinear heat problem; ill-posed problem; Fourier transform; time-dependent coefficient
Summary:
In this paper, a nonlinear backward heat problem with time-dependent coefficient in the unbounded domain is investigated. A modified regularization method is established to solve it. New error estimates for the regularized solution are given under some assumptions on the exact solution.
References:
[1] Cheng, W., L.-Fu, C.: A spectral method for an axisymmetric backward heat equation. Inverse Probl. Sci. Eng. 17 (2009), 1085-1093. DOI 10.1080/17415970903063193 | MR 2573817
[2] Clark, G. W., Oppenheimer, S. F.: Quasireversibility methods for non-well-posed problems. Electron. J. Differ. Equ. 1994 (1994), 1-9 (electronic). MR 1302574 | Zbl 0811.35157
[3] Denche, M., Bessila, K.: A modified quasi-boundary value method for ill-posed problems. J. Math. Anal. Appl. 301 (2005), 419-426. DOI 10.1016/j.jmaa.2004.08.001 | MR 2105682 | Zbl 1084.34536
[4] Fu, C.-L., Xiong, X.-T., Qian, Z.: Fourier regularization for a backward heat equation. J. Math. Anal. Appl. 331 (2007), 472-480. DOI 10.1016/j.jmaa.2006.08.040 | MR 2306017 | Zbl 1146.35420
[5] Lattès, R., Lions, J.-L.: Méthode de Quasi-Réversibilité et Applications. French Travaux et Recherches Mathématiques 15 Dunod, Paris (1967). MR 0232549 | Zbl 0159.20803
[6] Liu, C.-S.: Group preserving scheme for backward heat conduction problems. Int. J. Heat Mass Transfer 47 (2004), 2567-2576. DOI 10.1016/j.ijheatmasstransfer.2003.12.019 | Zbl 1100.80005
[7] Payne, L.: Improperly Posed Problems in Partial Differential Equations. CBMS-NSF Regional Conference Series in Applied Mathematics 22 SIAM, Philadelphia (1975). MR 0463736 | Zbl 0302.35003
[8] Qian, Z., Fu, C.-L., Shi, R.: A modified method for a backward heat conduction problem. Appl. Math. Comput. 185 (2007), 564-573. DOI 10.1016/j.amc.2006.07.055 | MR 2297827 | Zbl 1112.65090
[9] Quan, P. H., Trong, D. D.: A nonlinearly backward heat problem: uniqueness, regularization and error estimate. Appl. Anal. 85 (2006), 641-657. DOI 10.1080/00036810500474671 | MR 2232412 | Zbl 1099.35045
[10] Quan, P. H., Trong, D. D., Triet, L. M., Tuan, N. H.: A modified quasi-boundary value method for regularizing of a backward problem with time-dependent coefficient. Inverse Probl. Sci. Eng. 19 (2011), 409-423. DOI 10.1080/17415977.2011.552111 | MR 2795221 | Zbl 1227.65089
[11] Seidman, T. I.: Optimal filtering for the backward heat equation. SIAM J. Numer. Anal. 33 (1996), 162-170. DOI 10.1137/0733010 | MR 1377249 | Zbl 0851.65066
[12] Showalter, R. E.: The final value problem for evolution equations. J. Math. Anal. Appl. 47 (1974), 563-572. DOI 10.1016/0022-247X(74)90008-0 | MR 0352644 | Zbl 0296.34059
[13] Tautenhahn, U., Schröter, T.: On optimal regularization methods for the backward heat equation. Z. Anal. Anwend. 15 (1996), 475-493. DOI 10.4171/ZAA/711 | MR 1394439
[14] Trong, D. D., Tuan, N. H.: Regularization and error estimate for the nonlinear backward heat problem using a method of integral equation. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71 (2009), 4167-4176. DOI 10.1016/j.na.2009.02.092 | MR 2536322 | Zbl 1172.35517
[15] Wang, J. R.: Shannon wavelet regularization methods for a backward heat equation. J. Comput. Appl. Math. 235 (2011), 3079-3085. DOI 10.1016/j.cam.2011.01.001 | MR 2771288 | Zbl 1233.65068
[16] z, B. Yıldı, Yetişkin, H., Sever, A.: A stability estimate on the regularized solution of the backward heat equation. Appl. Math. Comput. 135 (2003), 561-567. DOI 10.1016/S0096-3003(02)00069-3 | MR 1937275
Search
Partner of
