Related papers: A numerical method for the fractional Zakharov-Kuz…
We analyze a Fourier spectral Galerkin method for the fractional Camassa-Holm (fCH) equation involving a fractional Laplacian of exponent $\alpha \in [1,2]$ with periodic boundary conditions. The semi-discrete scheme preserves both mass and…
We present a fully discrete Crank-Nicolson Fourier-spectral-Galerkin (FSG) scheme for approximating solutions of the fractional Korteweg-de Vries (KdV) equation, which involves a fractional Laplacian with exponent $\alpha \in [1,2]$ and a…
We study a numerical reconstruction strategy for the potential in the fractional Calder\'on problem from a single partial exterior measurement. The forward model is the fractional Schr\"odinger equation in a bounded domain, with prescribed…
This paper develops and analyses semi-discrete numerical method for two dimensional Vlasov-Stokes' system with periodic boundary condition. The method is based on coupling of semi-discrete discontinuous Galerkin method for the Vlasov…
In this paper, a non-polynomial spectral Petrov-Galerkin method and associated collocation method for substantial fractional differential equations (FDEs) are proposed, analyzed, and tested. We extend a class of generalized Laguerre…
We present and analyze a space-time Petrov-Galerkin finite element method for a time-fractional diffusion equation involving a Riemann-Liouville fractional derivative of order $\alpha\in(0,1)$ in time and zero initial data. We derive a…
In this paper, we investigate a spectral Petrov-Galerkin method for fractional initial value problems. Singularities of the solution at the origin inherited from the weakly singular kernel of the fractional derivative are considered, and…
This article focuses on an energy-conservation Galerkin finite element method (FEM) for the generalized Klein-Gordon-Zakharov (KGZ) equations. This method combines the bilinear finite element method for spatial discretization with the…
In this paper, the periodic initial-value problem for the fractional nonlinear Schr\"odinger (fNLS) equation is discretized in space by a Fourier spectral Galerkin method and in time by diagonally implicit, high-order Runge-Kutta schemes,…
Anomalous diffusions are ubiquitous in nature, whose functional distributions are governed by the backward Feynman-Kac equation. In this paper, the local discontinuous Galerkin (LDG) method is used to solve the 2D backward Feynman-Kac…
We propose a local discontinuous Galerkin (LDG) method for the fractional Korteweg-de Vries (KdV) equation, involving the fractional Laplacian with exponent $\alpha \in (1,2)$ in one and multiple space dimensions. By decomposing the…
We study two schemes for a time-fractional Fokker-Planck equation with space- and time-dependent forcing in one space dimension. The first scheme is continuous in time and is discretized in space using a piecewise-linear Galerkin finite…
In this paper a finite difference/local discontinuous Galerkin method for the fractional diffusion-wave equation is presented and analyzed. We first propose a new finite difference method to approximate the time fractional derivatives, and…
A direct discontinuous Galerkin (DDG) finite element method is developed for solving fractional convection-diffusion and Schr\"{o}dinger type equations with a fractional Laplacian operator of order $\alpha$ $(1<\alpha<2)$. The fractional…
We study regularity and numerical methods for two-sided fractional diffusion equations with a lower-order term. We show that the regularity of the solution in weighted Sobolev spaces can be greatly improved compared to that in standard…
We present the stability and error analysis of the unified Petrov-Galerkin spectral method, developed in \cite{samiee2017Unified}, for linear fractional partial differential equations with two-sided derivatives and constant coefficients in…
We present a novel numerical framework for studying nonlinear dispersive equations in higher-dimensional settings, specifically designed for solutions featuring traveling waves along a preferred axis (or field-aligned traveling waves).…
We propose and analyze a time-stepping discontinuous Petrov-Galerkin method combined with the continuous conforming finite element method in space for the numerical solution of time-fractional subdiffusion problems. We prove the existence,…
The tempered fractional diffusion equation could be recognized as the generalization of the classic fractional diffusion equation that the truncation effects are included in the bounded domains. This paper focuses on designing the high…
In this paper, in order to improve the spatial accuracy, the exponential integrator Fourier Galerkin method (EIFG) is proposed for solving semilinear parabolic equations in rectangular domains. In this proposed method, the spatial…