Research On Inverse Problems For The Multi-Term Time Fractional Diffusion Equations

The fractional integrals and derivatives are more and more used in modeling the so-called anomalous phenomena and in the theory of complex systems during the last few decades, and the research on fractional diffusion problems have become a hot topic in the fields of mathematics and engineering. The multi-term time fractional diffusion equation is one kind of fractional anomalous models, which arises from the multi-scale time fractional diffusion overlay. The objective of this thesis is to deal with the solutions and inverse problems for such kind of complicated anomalous diffusion equations.In this paper, we first consider the solutions of forward problem in the multi-term time fractional anomalous diffusion equations. For homogeneous linear problems, utilizing the method of separating variables, Laplace transform and the eigenfunction expansion, we can get the analytical solution based on the multivariate Mittag-Leffler function to the problem. In addition, we deal with the numerical solutions to the nonhomogeneous anomalous diffusion equations by using finite difference method, including 1-D variable coefficient diffusion and the 2-D variable coefficient diffusion equations. By estimating the spectral radius of the coefficient matrix to the difference schemes, we can prove the noncondition stability and convergence of above schemes. From the numerical results, we find the numerical solutions are good approximations to the exact solutions.For research on inverse problems for the multi-term time fractional diffusion equations, we consider three kinds of inverse problems including a backward problem of determining the initial function using final observations, the identifying problem for the multiple fractional orders with finite data at one interior point in the space domain, an inverse problem of simultaneously identifying the space-dependent diffusion coefficient and the source function in Chapter 4, Chapter 5 and Chapter 6, respectively. For the backward problem, based on the solution’s expression for the forward problem and estimation to the multivariate Mittag-Leffler function, we prove the uniqueness and instability. For the inverse problem of indentifying the multi-term fractional orders, the uniqueness is proved by using Laplace transform and the eigenfunction expansion. For the simultaneous inversion for a space-dependent diffusion coefficient and a source function, we give the Lipschitz continuity of the solution operator with the unknowns utilizing the solution’s expression, and we prove the existence of minimum for the error functional corresponding to the inverse problem by using the Sobolev embedding theorem, which is also the throretical basis for the construction of our inverse algorithm.At the same time, for the above three kinds of inverse problems, the homotopy regularization algorithm is applied to carry out the numerical inversion. From view point of optimality, solving the invese problems are transformed to minimizing the corresponding error functional, and the homotopy regularization algorithm is an effective method for solving above minimal problems based on regularization strategy and the homtopy idea, using finite dimensional approximation, iterative and linear gradient approximation. In this paper, a number of numerical inversion examples(including one and two dimensional cases) are given, and the numerical inversion of the inverse algorithm is discussed in detail with different parameter values. The inversion solutions give good approximations to the exact solutions with stability and adaptivity demonstrating that the presented algorim is efficient for the considered inverse problems in the multi-term time fractional diffusion model.