×

A backward problem for the time-fractional diffusion equation. (English) Zbl 1204.35177

Summary: We consider a backward problem in time for a time-fractional partial differential equation in one-dimensional case, which describes the diffusion process in porous media related with the continuous time random walk problem. Such a backward problem is of practically great importance because we often do not know the initial density of substance, but we can observe the density at a positive moment. The backward problem is ill-posed and we propose a regularizing scheme by the quasi-reversibility with fully theoretical analysis and test its numerical performance. With the help of the memory effect of the fractional derivative, it is found that the property of the initial status of the medium can be recovered in an efficient way. Since our solution is established on the eigenfunction expansion of elliptic operator, the method proposed in this article can be used to higher dimensional case with variable coefficients.

MSC:

35R30 Inverse problems for PDEs
35R11 Fractional partial differential equations
35R25 Ill-posed problems for PDEs
65M30 Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs
76S05 Flows in porous media; filtration; seepage
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ames KA, Non-standard and Improperly Posed Problems (1997)
[2] Cannon JR, The One-dimensional Heat Equation (1984)
[3] Isakov V, Inverse Problems for Partial Differential Equations (1998) · doi:10.1007/978-1-4899-0030-2
[4] Payne, LE.Inproperly Posed Problems in Partial Differential Equations, SIAM, Regional Series Conferences in Applied Mathematics, Philadelphia, 1975
[5] Duc TD, Electron. J. Diff. Eqns. 2006 pp 1– (2006)
[6] DOI: 10.1088/0266-5611/7/2/008 · Zbl 0731.65108 · doi:10.1088/0266-5611/7/2/008
[7] DOI: 10.1088/0266-5611/16/1/314 · Zbl 0972.35187 · doi:10.1088/0266-5611/16/1/314
[8] DOI: 10.1006/jmaa.1998.6243 · Zbl 0922.35189 · doi:10.1006/jmaa.1998.6243
[9] Liu JJ, Comm. Korean Math. Soc. 16 pp 385– (2001)
[10] DOI: 10.1016/S0377-0427(01)00595-7 · Zbl 1005.65107 · doi:10.1016/S0377-0427(01)00595-7
[11] DOI: 10.1142/9789812704924_0012 · doi:10.1142/9789812704924_0012
[12] DOI: 10.1142/S0252959903000049 · Zbl 1038.35150 · doi:10.1142/S0252959903000049
[13] Murio DA, The Mollification Method and the Numerical Solution of Ill-Posed Problems (1993)
[14] Shidfar A, Far East J. Appl. Math. 10 pp 145– (2003)
[15] DOI: 10.1016/0022-247X(74)90008-0 · Zbl 0296.34059 · doi:10.1016/0022-247X(74)90008-0
[16] DOI: 10.1016/S0378-4371(99)00503-8 · doi:10.1016/S0378-4371(99)00503-8
[17] Podlubny I, Fractional Differential Equations (1999)
[18] DOI: 10.1016/j.jcp.2007.02.001 · Zbl 1126.65121 · doi:10.1016/j.jcp.2007.02.001
[19] DOI: 10.1137/030602666 · Zbl 1119.65379 · doi:10.1137/030602666
[20] Sakamoto, K and Yamamoto, M. Initial value/boundary value problems for fractional diffusion-wave equations and inverse problems, preprint · Zbl 1219.35367
[21] Sakamoto, K.Inverse source problems for diffusion equations and fractional diffusion equations, Doctoral dissertation, Graduate School of Mathematical Sciences, The University of Tokyo, 2009
[22] Lattés R, The Method of Quasireversibility: Applications to Partial Differential Equations (1969)
[23] Engl HW, Regularization of Inverse Problems (1996) · 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.