PSU Mark
Eberly College of Science Mathematics Department

Meeting Details

For more information about this meeting, contact Changhe Qiao, Fei Wang, Lu Wang.

Title:Fractional Sturm-Liouville Eigen-Problems: Theory and Applications to Fractional PDEs in High Dimensions
Seminar:CCMA PDEs and Numerical Methods Seminar Series
Speaker:Mohsen Zayernouri, Brown University
Abstract:
Fractional Calculus is unifying theory that generalizes the standard (integer-order) calculus to real (fractional-order) differentiation and integration. The corresponding non-local operators of fractional order (in time and/or space) are shown to appear in a wide rang of applications. In continuum-time random walk (CTRW), which allows to incorporate waiting times and/or non-Gaussian jump distributions, the continuous limit leads to the notion of fractional calculus yielding fractional in time and/or space diffusion equation, governing the PDF of the Levy processes. In Bioengineering, there is an extensive amount of experimental evidence that highlights such processes occur in critical applications such as diffusion processes in human brain, cardiac tissue electrode interface, viscoelastic response of human red-blood-cells, wall-arteries, lung tissues, etc. In such power-law attributed phenomena, fractional PDEs naturally appear as the right mathematical framework. extension of existing numrical methods to FPDEs is not trivial because of their non-local and history-dependent nature. To this end, we first present a fractional (regular and singular) Sturm-Liouville eigen-problems, recently developed by Zayernouri & Karniadakis JCP 2013, which serves as a fundamental theory that provides explicit (non-polynomial) eigen-functions, namely as Jacobi Poly-fractonomials. These eigen-functions extend the well-known family of Jacobi polynomials to their fractional counterparts, where we show that they enjoy many attractive properties such as orthogonality, recurrence relations, exact fractional derivatives/integrations, in addition to their spectral properties. Based upon this base fractional theory, we develop a unified Petrov-Galerkin spectral method for solving the whole family of parabolic, elliptic, and hyperbolic (time- and space-) fractional PDEs in any (d+1)-dimensions in a hypercube. While the existing numerical methods can take days to run a One-Dimensional problem, we also formulate a unified fast linear solver for our scheme, which allows one to run a "Ten-Dimensional" FPDE on a laptop within a fractional of an hour!

Room Reservation Information

Room Number:MB315
Date:10 / 31 / 2014
Time:03:30pm - 05:00pm