Numerical Inversion Of A Class Of Solute Transport Equations And Source Term Inversion Of Groundwater Pollution

Inverse problems in mathematical physics develop rapidly,it is not just involved inverse problems in mathematics and physics. According to the development of science and technology and the expansion of the research field, many subjects such as environment, economics, finance, Oil exploration and so on have proposed inverse problems which search the "reason" based on the "result".As we all know, numerical inversion of Laplace transform is an ill-posed prob-lem, that is to say the solution does not depend continuously on the change of the initial measurement data. It is clear that one has no choice but to go to the regu-larization methods proven to be effective ways of treating ill-posed problem. In this paper, we proposed a new method to deal with two-dimensional solute transport in an aquifer-aquitard system.By using the finite difference method, we obtained the numerical solutions of concentrations in the aquifer and upper or lower aquitard which are in the Laplace domain. Then by variable substitution, we translated numerical inversion of Laplace transform into Hausdorff moment problem. Finally, through regularization approximate solution of the moment problem, we obtained numerical inversion of Laplace transform by N+1 data of F(s), which is also the solution in physical space.Source term inversion in groundwater pollution is a class of inverse problems in groundwater modeling,and is also a primary field in the research of inverse problems in mathematical physics. Many experts have done much work and obtained many results.In paper[17], a new gradient regularization algorithm was proposed and ap- plied to solve an inverse problem of determining source terms in one-dimensional advection-dispersion equation with final observations.By mass conservation law, so-lute transport can be described by mathematical models of advection-dispersion-reaction equations. In this paper, we will introduce the conception of fractional derivatives into source term inversion of groundwater pollution and study how to determine source terms. Zhang Yang[10] proposed a practical implicit method to solve a class of initial-boundary value space-time fractional convection-diffusion equations with variable coefficients. we continue study on the basis of his work. In this paper, we obtain the numerical difference scheme in source term identification with Dirichlet boundary and the numerical difference scheme in source coefficients identification with both Dirichlet boundary and Robbins boundary. But we focus on the problem of source term identification. When solving the forward problem, we take use of the numerical scheme derived in this paper, and when solving the inverse problem, we take use of the gradient regularization method which is based on Tikhonov regularization strategy. We not only prove the stability and conver-gence of the numerical scheme, but also verify the practicality and effectiveness of the algorithm. Then we apply the algorithm into the practical problem in [24] and obtain reasonable result.