Refreshments will be served at 3:30 p.m. in the Math Commons
Title:
Fractional partial differential equations: modeling, numerical method, and analysis
Abstract:
Fractional partial differential equations (FPDEs) provide an adequate and accurate description of transport processes that exhibit anomalous diffusion and long-range spatial interaction and time memory. These processes range from the signaling of biological cells, foraging behavior of animals, finance to subsurface groundwater contaminant transport. However, FPDEs raise mathematical and numerical difficulties that have not been encountered in the context of integer-order PDEs. Computationally, because of the nonlocal property of fractional differential operators, the numerical methods for FPDEs often generate 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.
The significant computational work and memory requirement of these methods makes a numerical simulation of three-dimensional FPDE modeling computationally prohibitively expensive. Mathematically, FPDEs exhibit mathematical properties that have fundamental differences from those of integer-order PDEs. In this talk we go over the development of accurate and efficient numerical methods for FPDEs, by exploring the structure of the coefficient matrices. These methods have approximately linear computational complexity per time step and optimal memory requirement.
Numerical experiments of a three-dimensional FPDE show that the fast numerical method reduces the CPU time from almost three month of CPU time to under 6 seconds. Furthermore, the fast method can simulate much larger problems on the same computational platform. We will also address mathematical issues on FPDEs such as wellposedness and regularity of the problems and their impact on the convergence behavior of numerical methods.