Abstract
In this paper, we consider a multi-term variable-order fractional diffusion equation on a finite domain, which involves the Caputo variable-order time fractional derivative of order alpha(x,t) in(0,1) and the Riesz variable-order space fractional derivatives of order beta(x,t) in (0,1), gamma(x,t)in(1,2). Approximating the temporal direction derivative by L1-algorithm and the spatial direction derivative by the standard and shifted Grünwald method, respectively, a characteristic finite difference scheme is proposed. The stability and convergence of the difference schemes are analyzed via mathematical induction. Some numerical experiments are provided to show the efficiency of the proposed difference schemes.
Highlights
As far as we are concerned, the theory of fractional partial differential equations (FPDE), as a new and effective mathematical tool, is very popular and important in many scientific and engineering problems
The multi-term FPDEs have been employed to some models for describing the processes in practice, such as the oxygen delivery through a capillary to tissues [6], the underlying processes with loss [7], the anomalous diffusion in highly heterogeneous aquifers and complex viscoelastic materials [8], and so on
We prove the convergence of the scheme by using errors estimation method, and the convergence rate of order (τ + h) is obtained
Summary
As far as we are concerned, the theory of fractional partial differential equations (FPDE), as a new and effective mathematical tool, is very popular and important in many scientific and engineering problems. The development of numerical methods to solve variable-order fractional differential equations is an actual and important problem. Lin et al [20] investigated the stability and convergence of an explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation. Zhuang et al [21] proposed explicit and implicit Euler approximations for the variable-order fractional advection-diffusion equation with a nonlinear source term. Zhuang et al [23] proposed an implicit Euler approximation for the time and space variable fractional-order advectiondispersion model with first-order temporal and spatial accuracy. In the existing literature, there is little work on higher-order numerical methods for the multi-term time-space variable-order fractional differential equations because more numerical analysis is involved. We consider the following multi-term time-space variable-order fractional diffusion equations with initial-boundary value problem:.
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