Inverse Problems Of Determining Multi-Point Sources Based On Diffusion Models

Diffusion models can be used to describe many natural phenomena, such as pollutants concentration diffusing through water and atmosphere, conduction of seawater temperature or salinity, electrochemical reactions, contaminant transport, and so on. In this thesis, we will mainly deal with some inverse problems of determining magnitudes of multi-point sources of diffusion equations under the condition that number and location of the point sources are known, including numerical solutions of the direct problems of integer order and fractional order diffusion equations, and numerical inversions for determining magnitudes of multi-point sources.In chapter 1, research background, significance, development and main research work of this thesis are presented.In chapter 2 and 3, numerical solutions of one-dimensional and two-dimensional diffusion equations with multi-point sources are obtained by finite deifference methods, respectively. An optimal perturbation regularization algorithm is introduced and applied to determine the mag-nitudes of multi-point sources under the condition that number and location of point sources are known. Five numerical examples are carried out showing the algorithm's effectiveness and feasibility, and impacts of the regularization parameter, numerical differential step, initial iteration on the inversion algorithm are discussed.In chapter 4, an implicit finite difference scheme for solving fractional advection-diffusion equations (FADE) with Dirichlet boundary conditions is presented, where the time fractional derivative is approximated by Caputo'definition, space fractional derivative is approximated by the shifted Grundwald formula. Numerical inversions for the magnitudes of multi-point sources of FADE are performed also utilizing the optimal perturbation regularization algorithm. Four numerical simulations are carried out showing the algorithm's effectiveness, and impacts of the fractional order, advection velocity on the inversion are also discussed.In chapter 5, a summary of the thesis is given, and some related problems which could be considered for future work are discussed.