I study fractional-order partial differential equations (FPDEs).
FPDEs govern the mascroscopic properties of systems undergoing
anomalous diffusion. They also model nonlocality, memory effects (nonlocality in time), and can also be used to continuously interpolate
between standard models/PDEs.
I am working on:
Machine Learning of FPDEs from Data. In the past decades, a vast number of systems in hydrology, telecommunications, finance, viscoelastics, biology, and plasma physics have been discovered to exhibit heavy-tailed stochastics or anomalous diffusion, suggesting they are governed in some way by FPDEs. Machine Learning can allow us to automatically obtain FPDEs from data, bypassing the analytical difficulties of Fractional Calculus.
Path Integral Monte Carlo Methods for Fractional Quantum Mechanics. We have developed a fractional PIMC method, allowing us to study for the first time realistic systems governed by the fractional Schrodinger equation.
Boundary Conditions for fractional Laplacians on Bounded Domains, especially the Stochastic (Levy) Processes that underlie such Boundary Value Problems. This is both a way to understand boundary conditions, and to solve fractional elliptic and parabolic BVPs using Monte Carlo Feynman-Kac formulas.
Fast-convolution methods for nonlinear time-fractional PDEs. Fast convolution/Laplace transform methods drastically decrease (by almost an order of magnitude in the number of time-steps) the memory requirements and complexity of time-fractional PDEs.
I work in the FPDE group of my advisor,
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