An Algorithm for the Numerical Solution of Two-Sided Space-Fractional Partial Differential Equations

Abstract

We introduce an algorithm for solving two-sided space-fractional partial differential equations. The space-fractional derivatives we consider here are left-handed and right-handed Riemann–Liouville fractional derivatives which are expressed by using Hadamard finite-part integrals. We approximate the Hadamard finite-part integrals by using piecewise quadratic interpolation polynomials and obtain a numerical approximation of the space-fractional derivative with convergence order ${O(\Delta x^{3- \alpha })}$, ${1<\alpha <2}$. A shifted implicit finite difference method is applied for solving the two-sided space-fractional partial differential equation and we prove that the order of convergence of the finite difference method is ${O (\Delta t + \Delta x^{\min (3- \alpha , \beta )})}$, ${1< \alpha <2}$, ${\beta >0}$, where ${\Delta t,\Delta x}$ denote the time and space stepsizes, respectively. Numerical examples where the solutions have varying degrees of smoothness are presented and compared with the exact analytical solution to compare the practical performance of the method with the theoretical order of convergence.

An Analysis of the Modified L1 Scheme for Time-Fractional Partial Differential Equations with Nonsmooth Data

“…The L1 scheme may be obtained by the direct approximation of the derivative in the definition of the Caputo fractional derivative, e.g., [25], [24], [16], [26], [37], or by the approximation of the Hadamard finite-part integral, e.g., [9], [10], [13], [14], [15], [23], [41]. Since its first appearance the L1 scheme has been extensively used in practice and currently it is one of the most popular and successful numerical methods for solving the time fractional diffusion equation.…”

An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data

“…The L1 scheme first appeared in the book [41] for the approximation of the Caputo fractional derivative. The L1 scheme may be obtained by direct approximation of the derivative in the definition of the Caputo fractional derivative, e.g., [28], [26], [18], [29], [47], [19], or by the approximation of the Hadamard finite-part integral, e.g., [9], [11], [15], [16], [17], [24], [51].…”

Optimal convergence rates for semidiscrete finite element approximations of linear space-fractional partial differential equations under minimal regularity assumptions

“…Yang et al [39] considered the finite difference methods for solving two-sided fractional partial differential equations. Ford et al [14] studied the finite difference methods for two-sided space-fractional partial differential equations. Ervin and Roop [10] introduced the variational formulation for solving space-fractional advection dispersion equation by using finite element methods, see also [13].…”