A Compact Difference Scheme for Fractional Sub-diffusion Equations with the Spatially Variable Coefficient Under Neumann Boundary Conditions

Abstract

In this paper, a compact finite difference scheme with global convergence order O τ 2−α + h 4 is derived for fractional sub-diffusion equations with the spatially variable coefficient subject to Neumann boundary conditions. The difficulty caused by the variable coefficient and the Neumann boundary conditions is overcome by subtle decomposition of the coefficient matrices. The stability and convergence of the proposed scheme are studied using its matrix form by the energy method. The theoretical results are supported by numerical experiments.

A high-order compact difference method for fractional sub-diffusion equations with variable coefficients and nonhomogeneous Neumann boundary conditions

“…By the same argument used in the proof of Lemma 2.2 of Vong et al (2016) (see the proof of (9) in Vong et al 2016), we have…”

“…Proof In fact, the matrices A and A −1 1 are just the matricesH and A in Vong et al (2016). Therefore, the results of the lemma follows from Lemmas 3.1 and 3.2 in Vong et al (2016).…”

A class of intrinsic parallel difference methods for time-space fractional Black–Scholes equation

“…To verify whether the space-convergent orders and the time-convergent orders are consistent with theoretical analysis, we define L 2 as errors, Order1 as space-convergent orders, and Order2 as time-convergent orders [31]. The PASI-E scheme is similar to the PASE-I scheme.…”

Numerical analysis of two new finite difference methods for time-fractional telegraph equation

“…Let E-I scheme, I-E scheme and implicit scheme numerical solutionũ j i be the perturbation solution, the exact solution u j i be the control solution. The definition of E 2 (h, τ ), Order1 and Order2 are as follows [28,30]: First verify the spatial accuracy of the implicit scheme, E-I scheme and I-E scheme. Let M = 6, 12, τ = h 2 (N = M 2 ) and α = 0.6, 0.7, 0.8, 0.9.…”