×

Regularization and stability estimates for an inverse source problem of the radially symmetric parabolic equation. (English) Zbl 1516.35502

Summary: We consider an inverse problem of determining an unknown source term in the radially symmetric parabolic equation from a noisy final data and prove the uniqueness of solution for the problem. Using the Hölder inequality, we obtain a conditional stability for the space-dependent source term. A modified quasi-reversibility method is applied to deal with the ill-posedness of the problem. A Hölder-type error estimate between the approximate solution and the exact solution is provided by introducing some technical inequalities and choosing a suitable regularization parameter.

MSC:

35R30 Inverse problems for PDEs
35B45 A priori estimates in context of PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
65M30 Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adams, RA: Sobolev Spaces. Pure and Applied Mathematics. Academic Press, New York (1975) · Zbl 0314.46030
[2] Savateev, EG: On problems of determining the source function in a parabolic equation. J. Inverse Ill-Posed Probl. 3(1), 83-102 (1995) · Zbl 0828.35142 · doi:10.1515/jiip.1995.3.1.83
[3] Solov’ev, VV: Solvability of the inverse problem of finding a source, using overdetermination on the upper base for a parabolic equation. Differ. Equ. 25, 1114-1119 (1990) · Zbl 0695.35215
[4] Cannon, JR, Pérez, ES: An inverse problem for the heat equation. Inverse Probl. 2(4), 395-403 (1986) · Zbl 0624.35078 · doi:10.1088/0266-5611/2/4/007
[5] Cannon, JR, Duchateau, P: Structural identification of an unknown source term in a heat equation. Inverse Probl. 14, 535-551 (1998) · Zbl 0917.35156 · doi:10.1088/0266-5611/14/3/010
[6] Yamamoto, M: Conditional stability in determination of force terms of heat equations in a rectangle. Math. Comput. Model. 18(1), 79-88 (1993) · Zbl 0799.35228 · doi:10.1016/0895-7177(93)90081-9
[7] Li, GS, Yamamoto, M: Stability analysis for determining a source term in a 1-D advection-dispersion equation. J. Inverse Ill-Posed Probl. 14(2), 147-155 (2006) · Zbl 1111.35122 · doi:10.1515/156939406777571067
[8] Choulli, M, Yamamoto, M: Conditional stability in determining a heat source. J. Inverse Ill-Posed Probl. 12(3), 233-243 (2004) · Zbl 1081.35136 · doi:10.1515/1569394042215856
[9] El Badia, A, Ha-Duong, T: On an inverse source problem for the heat equation. Application to a pollution detection problem. J. Inverse Ill-Posed Probl. 10(6), 585-600 (2002) · Zbl 1028.35164 · doi:10.1515/jiip.2002.10.6.585
[10] Li, GS: Data compatibility and conditional stability for an inverse source problem in the heat equation. Appl. Math. Comput. 173(1), 566-581 (2006) · Zbl 1105.35144 · doi:10.1016/j.amc.2005.04.053
[11] Burykin, AA, Denisov, AM: Determination of the unknown sources in the heat-conduction equation. Comput. Math. Model. 8(4), 309-313 (1997) · Zbl 0901.65083 · doi:10.1007/BF02404048
[12] Johansson, T, Lesnic, D: Determination of a spacewise dependent heat source. J. Comput. Appl. Math. 209(1), 66-80 (2007) · Zbl 1135.35097 · doi:10.1016/j.cam.2006.10.026
[13] Yan, L, Yang, FL, Fu, CL: A meshless method for solving an inverse spacewise-dependent heat source problem. J. Comput. Phys. 228(1), 123-136 (2009) · Zbl 1157.65444 · doi:10.1016/j.jcp.2008.09.001
[14] Yan, L, Fu, CL, Dou, FF: A computational method for identifying a spacewise-dependent heat source. Int. J. Numer. Methods Biomed. Eng. 26, 597-608 (2010) · Zbl 1190.65145
[15] Shidfar, A, Babaei, A, Molabahrami, A: Solving the inverse problem of identifying an unknown source term in a parabolic equation. Comput. Math. Appl. 60, 1209-1213 (2010) · Zbl 1201.65175 · doi:10.1016/j.camwa.2010.06.002
[16] Ma, YJ, Fu, CL, Zhang, YX: Identification of an unknown source depending on both time and space variables by a variational method. Appl. Math. Model. 26(10), 1209-1213 (2012)
[17] Wang, ZW, Liu, JJ: Identification of the pollution source from one-dimensional parabolic equation. Appl. Math. Comput. 219(8), 3400-3413 (2012) · Zbl 1311.35333
[18] Yang, F, Fu, CL: A mollification regularization for the spatial-dependent heat source problem. J. Comput. Appl. Math. 255(1), 555-567 (2014) · Zbl 1291.80010 · doi:10.1016/j.cam.2013.06.012
[19] Wei, T, Wang, JG: A modified quasi-boundary value method for an inverse source problem of the time-fractional diffusion equation. Appl. Numer. Math. 78, 95-111 (2014) · Zbl 1282.65141 · doi:10.1016/j.apnum.2013.12.002
[20] Cheng, W, Ma, YJ, Fu, CL: Identifying an unknown source term in radial heat conduction. Inverse Probl. Sci. Eng. 20(3), 335-349 (2012) · Zbl 1258.65085 · doi:10.1080/17415977.2011.624616
[21] Yang, F, Zhang, M, Li, XX: A quasi-boundary value regularization method for identifying an unknown source in the Poisson equation. J. Inequal. Appl. 2014, 117 (2014) · Zbl 1432.35262 · doi:10.1186/1029-242X-2014-117
[22] Lattès, R, Lions, JL: The Method of Quasi-Reversibility: Applications to Partial Differential Equations. Elsevier, New York (1969) · Zbl 1220.65002
[23] Eldén, L: Approximations for a Cauchy problem for the heat equation. Inverse Probl. 3(2), 263-273 (1987) · Zbl 0645.35094 · doi:10.1088/0266-5611/3/2/009
[24] Qian, Z, Fu, CL, Xiong, XT: A modified method for a nonstandard inverse heart conduction problem. Appl. Math. Comput. 180(2), 453-468 (2006) · Zbl 1105.65097 · doi:10.1016/j.amc.2005.12.033
[25] Dang, DT, Nguyen, NI: Regularization and error estimates for nonhomogeneous backward heat problems. Electron. J. Differ. Equ. 2006, 4 (2006) · Zbl 1091.35114
[26] Klibanov, MV, Santosa, F: A computational quasi-reversibility method for Cauchy problems for Laplace’s equation. SIAM J. Appl. Math. 51(6), 1653-1675 (1991) · Zbl 0769.35005 · doi:10.1137/0151085
[27] Qian, Z, Fu, CL, Xiong, XT: Fourth-order modified method for the Cauchy problem for the Laplace equation. J. Comput. Appl. Math. 192(2), 205-218 (2006) · Zbl 1093.65107 · doi:10.1016/j.cam.2005.04.031
[28] Qin, HH, Wei, T: Quasi-reversibility and truncation methods to solve a Cauchy problem for the modified Helmholtz equation. Math. Comput. Simul. 80, 352-366 (2009) · Zbl 1185.65172 · doi:10.1016/j.matcom.2009.07.005
[29] Abramowitz, M, Stegun, IA: Handbook of Mathematical Functions. Dover, New York (1972) · Zbl 0543.33001
[30] Engl, HW, Hanke, M, Neubauer, A: Regularization of Inverse Problems. Mathematics and Its Applications, vol. 375. Kluwer Academic, Dordrecht (1996) · Zbl 0859.65054 · doi:10.1007/978-94-009-1740-8
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.