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# A Quasi-Boundary Semi-Analytical Method for Backward in Time Advection-Dispersion Equation

Chein-Shan Liu1, Chih-Wen Chang2, Jiang-Ren Chang2,3
Department of Civil Engineering, National Taiwan University, Taipei 10617, Taiwan
Department of Systems Engineering and Naval Architecture, National Taiwan Ocean University, Keelung 20224, Taiwan
Corresponding author, Tel.: +886-2-24622192x6031. E-mail: cjr@mail.ntou.edu.tw

Computers, Materials & Continua 2009, 9(2), 111-136. https://doi.org/10.3970/cmc.2009.009.111

### Abstract

In this paper, we take the advantage of an analytical method to solve the advection-dispersion equation (ADE) for identifying the contamination problems. First, the Fourier series expansion technique is employed to calculate the concentration field C(x, t) at any time t< T. Then, we consider a direct regularization by adding an extra term αC(x,0) on the ﬁnal condition to carry off a second kind Fredholm integral equation. The termwise separable property of the kernel function permits us to transform itinto a two-point boundary value problem. The uniform convergence and error estimate of the regularized solution Cα(x,t) are provided and a strategy to select the regularized parameter is suggested. The solver used in this work can recover the spatial distribution of the groundwater contaminant concentration. Several numerical examples are examined to show that the new approach can retrieve all past data very well and is good enough to cope with heterogeneous parameters’ problems, even though the ﬁnal data are noised seriously.

### Keywords

Inverse problem, Groundwater contaminant distribution, Advection-dispersion equation, Fredholm integral equation, Two-point boundary value problem

C. . Liu, C. . Chang and J. . Chang, "A quasi-boundary semi-analytical method for backward in time advection-dispersion equation," Computers, Materials & Continua, vol. 9, no.2, pp. 111–136, 2009.

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