×

Stability analysis for determining a source term in a 1-D advection-dispersion equation. (English) Zbl 1111.35122

Summary: We are concerned with conditional stability for an inverse problem of deciding source terms \(f(x)\) in a 1D advection-dispersion equation \[ \frac{\partial c}{\partial t}-D_L\frac{\partial^2c}{\partial x^2}+u \frac{\partial c}{\partial x}=\alpha(t)f(x) \] by final observations \(c(x,T)=c_T(x)\), \(0\leq x\leq\ell\). The inverse problem here is based on a mathematical model derived from a real world case in a geological region in Shandong province, China. With an integral identity and analysis for a normal Sturm-Liouville problem, conditional stability for the inverse problem is proved.

MSC:

35R30 Inverse problems for PDEs
86A22 Inverse problems in geophysics
35K15 Initial value problems for second-order parabolic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1109/87.880592 · doi:10.1109/87.880592
[2] DOI: 10.1163/1569394042215856 · doi:10.1163/1569394042215856
[3] DOI: 10.1088/0266-5611/18/6/312 · Zbl 1023.35093 · doi:10.1088/0266-5611/18/6/312
[4] R. Courant and D. Hilbert, Methods of Mathematical Physics. I. Interscience, New York, 1953. · Zbl 0051.28802
[5] DOI: 10.1137/S0036141093259257 · Zbl 0849.35146 · doi:10.1137/S0036141093259257
[6] V. Isakov, Inverse Problems for Partial Differential Equations. Springer, New York, 1998. · Zbl 0908.35134
[7] M. K. Klibanov and M. Yamamoto, Lipschitz stability of an inverse problem for an acoustic equation. UTMS, 2004-13. · Zbl 1274.35413
[8] DOI: 10.1061/(ASCE)0733-9496(2001)127:1(20) · doi:10.1061/(ASCE)0733-9496(2001)127:1(20)
[9] N. Z. Sun, Inverse Problem In Groundwater Modeling. Kluwer, Dordrecht, 1994.
[10] N. Z. Sun, Mathematical Model of Groundwater Pollution. Springer, New York, 1996.
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.