Theoretical And Numerical Method Studies On The Direct And Inverse Problems Of Fractional Diffusion Equations

In this thesis,we mainly consider theoretical and numerical methods of direct and inverse problems for time fractional diffusion equations:two L1 schemes on graded meshes for time fractional Feynman-Kac equation;two L2-1σ schemes on graded meshes for time fractional Feynman-Kac equation;The problem of space dependent source term identification for multi-time fractional diffusion equation;recovering a time-dependent potential function in a multi-term time fractional diffusion equation by using a nonlinear condition.The first part of this thesis,we mainly consider the time fractional Feynman-Kac equation is numerically calculated by two kinds of L1 scheme,which approximates the first time derivative based on piecewise linear interpolation.Its convergence order shall drop down to O(τmin{2-α,rα})by the implicit L1 scheme on graded meshes.This motivates us to design the implicit-explicit L1 scheme,which reaches the optimal convergence order O(τmin{2-α,rα})on graded meshes.We prove the stability and convergence of two L1 schemes.Finally,the feasibility and effectiveness of the proposed method are verified by one and two dimensional examples.In the second part of this thesis,we mainly consider the time fractional FeynmanKac equation is numerically calculated by two kinds of L2-1σ scheme,which approximates the first time derivative based on piecewise quadratic interpolation.We design the implicit L2-1σ scheme and modified implicit L2-1σ schemes,their order of convergence is O(τmin{1,rα}).This motivates us to design another modified implicit L2-1σ scheme,which reach the optimal convergence order O(τmin{2,rα})on graded meshes.We give the stability analysis of the modified implicit L2-1σmethod.Finally,the feasibility and effectiveness of the proposed method are verified by two dimensional example.In the third part of this thesis,we mainly consider recovering the space-dependent source for a multi-term time fractional diffusion equation from noisy final data.Firstly,we proved that the direct problem has a unique solution by the properties of the Caputo derivative operator and elliptic operator.Secondly,using the properties of the Caputo derivative operator and Fourier transform,we proved the uniqueness of the space-dependent source term identification problem.Finally,we apply a nonstationary iterative Tikhonov regularization method combined with a finite dimensional approximation numerical solution.Four different examples are presented to show the feasibility and efficiency of the proposed method.In the fourth part of this thesis,we focus on the reconstruction of the timedependent potential function in a multi-term time fractional diffusion equation from an additional measurement in the form of an integral over the space domain.Firstly,we prove the existence,uniqueness and regularity of the solution for the direct problem by using the fixed point theorem.Secondly,we prove the uniqueness of the inverse problem by using the property of Caputo derivative.Finally,we use LevenbergMarquardt method to simulate the potential function.Four examples show that the proposed numerical method is feasible and effective.