Two Linearized Schemes for One-Dimensional Time and Space Fractional Differential Equations

Abstract

This paper investigates the solution to one-dimensional fractional differential equations with two types of fractional derivative operators of orders in the range of (1,2). Two linearized schemes of the numerical method are constructed. The considered FDEs are equivalently transformed by the Riemann–Liouville integral into their integro-partial differential problems to reduce the requirement for smoothness in time. The analysis of stability and convergence is rigorously discussed. Finally, numerical experiments are described, which confirm the obtained theoretical analysis.

Keywords

• time and space fractional differential equations
• linearized schemes
• integro-differential equation
• stability
• convergence
• weighted and shifted Grünwald difference operator