Theoretical And Numerical Investigation Of Fractional Partial Differential Equations

The use of fractional partial differential equations (FPDEs) in mathematical modelshas become increasingly popular in recent years. Different models using FPDEs have beenproposed in more and more fields, covering materials, mechanical, and biological systems,and it's found that FPDEs gain the advantage over the classical one in modeling somematerials with memory, heterogeneity or inheritable character. The modeling progresson using FPDEs has led to increasing interest in developing numerical schemes for theirsolutions.In this paper, our work is focused on the theoretical investigation and numericalcomputation of the fractional diffusion equations (FDEs), which are of interest not onlyin their own right, but also in that they constitute the principal parts in many otherFPDEs. The main contribution of this work is threefold:First, we introduce a new family of functional spaces defined by using fractionalderivatives, and prove that these spaces are equivalent to usual Sobolev spaces in thesense that their norms are equivalent. Based on these spaces the variational formulationof the initial boundary value problems of FDEs are developed, and the existence anduniqueness of the weak solution are established by using classical theory for elliptic problems.The obtained results indicate that in the case of Riemann-Liouville definition, theequivalence between FDEs and weak formulation does not require any initial conditions.This contrasts with the case of Caputo definition, in which the initial condition has to beintegrated into the weak formulation in order to establish the equivalence.Second, based on the proposed weak formulation, we investigate the numerical solutionsof the time fractional diffusion equation (TFDE). Essentially, the TFDE differs fromthe standard diffusion equation in the time derivative term. In TFDE, the first-order timederivative is replaced by a fractional derivative, making the problem global in time. Wepropose a spectral method in both temporal and spatial discretizations for this equation.The convergence of the method is proven by providing a priori error estimate. Numericaltests are carried out to confirm the theoretical results. Thanks to the spectral accuracy inboth space and time of the proposed method, the storage requirement due to the "globaltime dependence" can be considerably relaxed, and therefore calculation of the long-time solution becomes possible.Third, we consider the fractional Nernst-Planck equation, which describes theanomalous diffusion in the movement of the ions in neuronal system. A methodcombining finite differences in time and spectral element methods in space is proposed tonumerically solve the underlying problem. The detailed construction and implementationof the method are presented. Our numerical experiences show that the convergence of theproposed method is exponential in space and (2-α)-order (0＜α＜1) in time. Finally,a practical problem with realistic physical parameters is simulated to demonstrate thepotential applicability of the method.