An explicit hybrid method for multi-term fractional differential equations based on Adams and Runge-Kutta schemes

Abstract

An explicit hybrid numerical algorithm is proposed by combining Adams-type and Runge-Kutta (RK) methods to solve fractional differential equations (FDEs). A general kind of multi-term FDEs is transformed into combined systems containing both FDEs and ordinary differential equations (ODEs). An explicit Adams-type scheme is constructed to integrate the FDEs. With the obtained solutions of FDEs, the ODEs are explicitly solved by the classical fourth-order RK scheme. Several types of FDEs are presented to validate the proposed method. Compared with an extended predictor corrector method, the presented method can provide more accurate results by an order of magnitude at the expense of one half of computational resources. With such high precision and efficiency, the presented method is applied in numerical simulations of long-term dynamic responses such as asymptotic periodic or limit cycle solutions of fractional dynamic systems.