Three-point combined compact difference schemes for time-fractional advection–diffusion equations with smooth solutions

A Semidiscrete Finite Element Approximation of a Time-Fractional Fokker--Planck Equation with NonSmooth Initial Data

“…here, the lower-order terms are included to allow for a zero-flux boundary condition (10), in which case the functions in S h do not have to vanish on ∂ Ω and so the Poincaré inequality is not applicable. Since the Galerkin finite element method is quasi-optimal in H 1 (Ω ), we know that v − R h v 1 ≤ Ch v 2 for v ∈ H 2 (Ω ).…”

“…where the first term is just the Caputo fractional derivative of order α. For a one-or two-dimensional spatial domain Ω , numerical methods applicable to (3) have been widely studied [2][3][4][5]7,[9][10][11]14,[19][20][21][22]. In all of these works, the solution u was assumed to be sufficiently regular, including at t = 0.…”

“…By employing a different approach that based on novel energy arguments, we are able to relax significantly the regularity requirements on u, in addition to permitting d ≥ 1, 0 < α < 1, and zero-flux (10) as well as Dirichlet boundary conditions. This new approach leads to an error bound of optimal order in L 2 (Ω ) at each fixed t > 0, even for non-smooth initial data u 0 .…”

Three-point compact finite difference scheme on non-uniform meshes for the time-fractional Black-Scholes equation

“…In the next section we construct a compact finite-difference scheme for the TFBSD equation using the fourth-order compact approximation (5) on a three-point stencil of the Tavella-Randal and the quadratic non-uniform meshes and the L1-approximation for the Caputo fractional derivative defined as [10] when U (s, t) is a twice continuously differentiable function [8,10]. From the properties of the Caputo derivative, the TFBSD equation has a natural singularity at t = 0.…”

Three-Point Combined Compact Alternating Direction Implicit Difference Schemes for Two-Dimensional Time-Fractional Advection-Diffusion Equations

“…As the authors' known, the applications of the CCD technique into the fractional differential equations have never been found except our first attempt to discuss the onedimensional time-fractional advection-diffusion equations (TFADEs) by the CCD method in [32], where the effective numerical schemes were proposed and strict theoretical analysis of the resultant scheme for constant coefficient case with periodic boundary conditions was also given. Now this paper aims to extend the idea to solve the two-dimensional time-fractional advection-diffusion equations.…”