High-Order Numerical Algorithms for Riesz Derivatives via Constructing New Generating Functions

Abstract

A class of high-order numerical algorithms for Riesz derivatives are established through constructing new generating functions. Such new highorder formulas can be regarded as the modification of the classical (or shifted) Lubich's difference ones, which greatly improve the convergence orders and stability for time-dependent problems with Riesz derivatives. In rapid sequence, we apply the 2nd-order formula to one-dimension Riesz spatial fractional partial differential equations to establish an unconditionally stable finite difference scheme with convergent order O(τ 2 + h 2 ), where τ and h are the temporal and spatial stepsizes, respectively. Finally, some numerical experiments are performed to confirm the theoretical results and testify the effectiveness of the derived numerical algorithms.

A fourth-order compact ADI scheme for two-dimensional Riesz space fractional nonlinear reaction–diffusion equation

“…Ortigueira [14] initially proposed the fractional centered difference method with second-order accuracy for Riesz fractional derivative, and this method was applied in Riesz space fractional partial differential equation [3,15,16,21,24,25,32]. Ding and Li [9] proposed a novel second-order approximation for Riesz derivative via constructing a new generating function, and this second-order approximation was adopted in [2] for two-dimension Riesz space-fractional diffusion equation. Recently, compact difference operator has been focused on the fractional differential equations for increasing the spatial accuracy.…”

A Fourier spectral method for the nonlinear coupled space fractional Klein‐Gordon‐Schrödinger equations

“…In recent years, fractional calculus has been playing more and more important roles in physics and other fields. [9][10][11][12][13][14][15] In [16][17][18][19][20][21][22][23][24][25][26][27], many researchers have made great efforts to studying the theory and computation for fractional equations. Laskin derived the space fractional Schrödinger equations in [28,29].…”

Numerical Solutions of Ordinary Fractional Differential Equations with Singularities

“…When the solution of the two-term equation y ∈ C 3 [0, T ], the numerical solutions which use approximations (1), (2) and (4) [1,4,11,12,19]. When F(t) = 0 the two-term equation (5) has the solution y(t) = E α (−Bt α ).…”