Finite Difference Method for Two-Sided Space-Fractional Partial Differential Equations

摘要

Finite difference methods for solving two-sided space-fractional partial differential equations are studied. The space-fractional derivatives are the left-handed and right-handed Riemann-Liouville fractional derivatives which are expressed by using Hadamard finite-part integrals. The Hadamard finite-part integrals are approximated by using piecewise quadratic interpolation polynomials and a numerical approximation scheme of the space-fractional derivative with convergence order O(Δx~(3-α)), 1 ＜ a ＜ 2 is obtained. A shifted implicit finite difference method is introduced for solving two-sided space-fractional partial differential equation and we prove that the order of convergence of the finite difference method is O(Δt + Δx~(min(3-α,β))), 1 ＜ α ＜ 2, β ＞ 0, where Δt, Δx denote the time and space step sizes, respectively. Numerical examples are presented and compared with the exact analytical solution for its order of convergence.