Date/Time
Date(s) - 07/10/2024
9:30 am - 10:30 am

Via Zoom: https://mcmaster.zoom.us/j/97287926728

Meeting ID: 972 8792 6728

Title: A simple and fast finite difference method for the integral fractional Laplacian of variable order

Speaker: Zhaopeng Hao (Southeast University, China)

Abstract: During the past few decades, the constant-order fractional Laplacians have been extensively studied in the literature as they can accurately describe complex physical phenomena, manifesting in long-range and nonlocal interactions, self-similar structures, and sharp interfaces, which cannot be adequately captured by their integer counterpart. However, the constant-order operator may be insufficient for the heterogeneous effect due to the spatial variability of a complex medium.  To account for heterogeneity, the variable-order operators depending on the spatial location variable have been alternatively proposed. Changing directly the constant order into the variable order may increase not only the model capability but also bring the difficulty of the discretization and the complexity of computation. To address this difficulty, in this talk, we present a simple and fast finite difference method for the integral fractional Laplacian of variable order. We prove that the scheme is of second-order convergence and apply the developed finite difference scheme to solve various equations with the variable-order fractional Laplacian. We also present a fast algorithm for computing variable-order fractional Laplacian. Several numerical examples demonstrate the accuracy and efficiency of our algorithm and verify our theory.