A Hybrided Method for Temporal Variable-Order Fractional Partial Differential Equations with Fractional Laplace Operator

Abstract

In this paper, we present a more general approach based on a Picard integral scheme for nonlinear partial differential equations with a variable time-fractional derivative of order α(x,t)∈(1,2) and space-fractional order s∈(0,1), where v=u′(t) is introduced as the new unknown function and u is recovered using the quadrature. In order to get rid of the constraints of traditional plans considering the half-time situation, integration by parts and the regularity process are introduced on the variable v. The convergence order can reach O(τ2+h2), different from the common L1,2−α schemes with convergence rate O(τ2,3−α(x,t)) under the infinite norm. In each integer time step, the stability, solvability and convergence of this scheme are proved. Several error results and convergence rates are calculated using numerical simulations to evidence the theoretical values of the proposed method.

Keywords

• time–space fractional advection–diffusion equation
• variable fractional order
• nonlinearity
• stability
• second order