Fractional PDE: Model, Computation, Theory and Application

Department of Mathematics, University of South Carolina

2017-08-22 10:00-11:00 Science Building No. 1 1418

Abstract: Fractional PDE provides an accurate description of transport processes from many applications, which exhibit anom-alous diffusion and long-range spatial interaction and memory effect. However, FPDE raises mathematical and numerical diff-iculties that have not been encountered in the context of integer-order PDE.

Computationally, because of the nonlocal property of fractional differential operators, the numerical methods for FPDE often generates dense coefficient matrices for which traditional direct solvers were used that have a computational complexity of O(N3) per time step and memory requirement of O(N2) where N is the number of unknowns. This makes numerical simulation of three-dimensional FPDE modeling computationally very expensive.

In addition, FPDE exhibits mathematical properties that have fundamental differences from those of integer-order PDE. We will go over the development of fast numerical methods for FPDE, by exploring the structure of the coefficient matrices. These meth-ods have approximately linear computational complexity per time step and optimal memory requirement.

We will discuss mathematical issues on FPDE such as well posedness and regularity of the problems and their impact on the convergence behavior of numerical methods. We will also describe the application of FPDE model.