135 TMCB

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.