Refreshments will be served at 3:30 p.m. in the Math Commons
Fractional partial differential equations: modeling, numerical
method, and analysis
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
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.