Inverse problems of time-fractional differential equations : analysis and numerical methods

Abstract

This thesis is devoted to theoretical and numerical analysis of inverse problems of fractional differential equations, which have drawn much attention over the past decades due to the maybe ”mild” ill­-posedness of fractional derivatives. In recent years, some numerical algorithms and mathematical analysis are provided and tested. However, in most of these works, only some convergence results and semidiscrete numerical analysis are analyzed. Then our aim is to give a thorough numerical analysis of inverse problems, where numerical estimates are provided including noise level, regularization and discretization parameters. The numerical estimates provide a balancing way to choose regularization and discretization parameter from the noise level. Therefore, we could use a relevant coarser grid to obtain some optimal convergent results. After a background and preliminary introduction in Chapter 1 and Chapter 2, firstly in Chapter 3 we focus on the backward subdiffusion problem, with the application of quasi-boundary regularization method, piecewise-linear finite element method and convolution quadrature, we show a total error es­timate based on smoothing properties of (discrete) solution operators, and nonstandard error estimate for the direct problem in terms of problem data regularity. Next in Chapter 4 when the backward subdffision model includes a time-dependent coefficient, we use a perturbation argument of freezing the diffusion coefficients. Similarly, we apply a quasi-boundary value method and a fully discrete method consisting of finite element method in space and backward Euler convolution quadrature in time. An a priori error estimate is established. Based on the motivation in subdiffusion we extend our idea to fractional-wave equation in Chapter 5 where we want to simultaneously determine two initial conditions based on two different observations. After a new proposed quasi-boundary value method and a classical fully discrete method in space and time, a conditional a priori error estimate is shown. On the other hand, we focus on the inverse potential problem in Chapter 6, to recover potential in a fractional differential equation, with the severely ill posed nature, we construct a monotone operator one of whose fixed points is the unknown potential. The uniqueness of the identification is theoretically verified. Moreover, we show a conditional stability in Hilbert spaces under some suitable conditions on the problem data. Next, a completely discrete scheme is developed by using Galerkin finite method in space and finite difference method in time. A discrete fixed point iteration is constructed and a thorough numerical analysis is given. Lastly in Chapter 7, we summarize our work and mention possible future research topics. In each chapter, various numerical experiments are provided to support our obtained numerical error estimates. By a balancing choice of parameters, we would obtain an optimal convergence rates, which is strongly supported by our numerical experiments.