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Efficient multistep methods for tempered fractional calculus: Algorithms and Simulations
Ling Guo, Fanhai Zeng, Ian Turner +2
In this work, we extend the fractional linear multistep methods in [C. Lubich, SIAM J. Math. Anal., 17 (1986), pp.704--719] to the tempered fractional integral and derivative opera…
Nonlocal flocking dynamics: Learning the fractional order of PDEs from particle simulations
Zhiping Mao, Zhen Li, George Em Karniadakis
Flocking refers to collective behavior of a large number of interacting entities, where the interactions between discrete individuals produce collective motion on the large scale.…
Multi-domain Spectral Collocation Method for Variable-Order Nonlinear Fractional Differential Equations
Tinggang Zhao, Zhiping Mao, George Em Karniadakis
Spectral and spectral element methods using Galerkin type formulations are efficient for solving linear fractional PDEs (FPDEs) of constant order but are not efficient in solving n…
A new class of semi-implicit methods with linear complexity for nonlinear fractional differential equations
Fanhai Zeng, Ian Turner, Kevin Burrage +1
We propose a new class of semi-implicit methods for solving nonlinear fractional differential equations and study their stability. Several versions of our new schemes are proved to…
A fast solver for spectral element approximation applied to fractional differential equations using hierarchical matrix approximation
Xianjuan Li, Zhiping Mao, Fangying Song +2
We develop a fast solver for the spectral element method (SEM) applied to the two-sided fractional diffusion equation on uniform, geometric and graded meshes. By approximating the…
A spectral penalty method for two-sided fractional differential equations with general boundary conditions
Nan Wang, Zhiping Mao, Chengming Huang +1
We consider spectral approximations to the conservative form of the two-sided Riemann-Liouville (R-L) and Caputo fractional differential equations (FDEs) with nonhomogeneous Dirich…
Fractional Gray-Scott Model: Well-posedness, Discretization, and Simulations
Tingting Wang, Fangying Song, Hong Wang +1
The Gray-Scott (GS) model represents the dynamics and steady state pattern formation in reaction-diffusion systems and has been extensively studied in the past. In this paper, we c…
What Is the Fractional Laplacian?
Anna Lischke, Guofei Pang, Mamikon Gulian +8
The fractional Laplacian in R^d has multiple equivalent characterizations. Moreover, in bounded domains, boundary conditions must be incorporated in these characterizations in math…