Numerical methods for fractional partial differential equations
In this review paper, we are mainly concerned with the finite difference methods, the
Galerkin finite element methods, and the spectral methods for fractional partial differential …
Galerkin finite element methods, and the spectral methods for fractional partial differential …
Fractional modeling in action: A survey of nonlocal models for subsurface transport, turbulent flows, and anomalous materials
Modeling of phenomena such as anomalous transport via fractional-order differential
equations has been established as an effective alternative to partial differential equations …
equations has been established as an effective alternative to partial differential equations …
A second-order and nonuniform time-stepping maximum-principle preserving scheme for time-fractional Allen-Cahn equations
In this work, we present a second-order nonuniform time-stepping scheme for the time-
fractional Allen-Cahn equation. We show that the proposed scheme preserves the discrete …
fractional Allen-Cahn equation. We show that the proposed scheme preserves the discrete …
Mathematical analysis and numerical methods for Caputo-Hadamard fractional diffusion-wave equations
In this paper, mathematical analysis and numerical methods for Caputo-Hadamard fractional
diffusion-wave equations with initial singularity are investigated. By adopting the modified …
diffusion-wave equations with initial singularity are investigated. By adopting the modified …
Fast algorithm based on TT-M FE system for space fractional Allen–Cahn equations with smooth and non-smooth solutions
In this article, a fast algorithm based on time two-mesh (TT-M) finite element (FE) scheme,
which aims at solving nonlinear problems quickly, is considered to numerically solve the …
which aims at solving nonlinear problems quickly, is considered to numerically solve the …
Good (and not so good) practices in computational methods for fractional calculus
The solution of fractional-order differential problems requires in the majority of cases the use
of some computational approach. In general, the numerical treatment of fractional differential …
of some computational approach. In general, the numerical treatment of fractional differential …
A fast high order method for the time-fractional diffusion equation
H Zhu, C Xu - SIAM Journal on Numerical Analysis, 2019 - SIAM
In this paper, we present a fast (3-α)-order numerical method for the Caputo fractional
derivative based on the L2 scheme and the sum-of-exponentials (SOE) approximation to the …
derivative based on the L2 scheme and the sum-of-exponentials (SOE) approximation to the …
Efficient multistep methods for tempered fractional calculus: Algorithms and simulations
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 operators in …
Anal., 17 (1986), pp. 704--719 to the tempered fractional integral and derivative operators in …
[PDF][PDF] Fast finite difference schemes for time-fractional diffusion equations with a weak singularity at initial time
J Shen, Z Sun, R Du - East Asian J. Appl. Math, 2018 - researchgate.net
A sharp estimate for the L1 formula on graded meshes, which approximates the Caputo
derivatives of functions with a weak singularity at t= 0 is obtained. Combining such …
derivatives of functions with a weak singularity at t= 0 is obtained. Combining such …
Fitted schemes for Caputo-Hadamard fractional differential equations
In the present paper, the regularity and finite difference methods for Caputo-Hadamard
fractional differential equations with initial value singularity are taken into consideration. To …
fractional differential equations with initial value singularity are taken into consideration. To …