A Numerical Method for Caputo Differential Equations and Application of High-Speed Algorithm

Author:

Fukunaga Masataka1,Shimizu Nobuyuki2

Affiliation:

1. College of Engineering, Nihon University, 1-2-35-405, Katahira, Aoba-ku, Sendai, Miyagi 980-0812, Japan e-mail:

2. MotionLabo, Inc., TSK Bld. 302, 9-1, Aza-Sunada, Kanari, Onahama, Iwaki, Fukushima 971-8135, Japan

Abstract

In this paper, a numerical algorithm to solve Caputo differential equations is proposed. The proposed algorithm utilizes the R2 algorithm for fractional integration based on the fact that the Caputo derivative of a function f(t) is defined as the Riemann–Liouville integral of the derivative f(ν)(t). The discretized equations are integer order differential equations, in which the contribution of f(ν)(t) from the past behaves as a time-dependent inhomogeneous term. Therefore, numerical techniques for integer order differential equations can be used to solve these equations. The accuracy of this algorithm is examined by solving linear and nonlinear Caputo differential equations. When large time-steps are necessary to solve fractional differential equations, the high-speed algorithm (HSA) proposed by the present authors (Fukunaga, M., and Shimizu, N., 2013, “A High Speed Algorithm for Computation of Fractional Differentiation and Integration,” Philos. Trans. R. Soc., A, 371(1990), p. 20120152) is employed to reduce the computing time. The introduction of this algorithm does not degrade the accuracy of numerical solutions, if the parameters of HSA are appropriately chosen. Furthermore, it reduces the truncation errors in calculating fractional derivatives by the conventional trapezoidal rule. Thus, the proposed algorithm for Caputo differential equations together with the HSA enables fractional differential equations to be solved with high accuracy and high speed.

Publisher

ASME International

Subject

Applied Mathematics,Mechanical Engineering,Control and Systems Engineering,Applied Mathematics,Mechanical Engineering,Control and Systems Engineering

Reference45 articles.

Cited by 2 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Multiple Fixed Pole-Based Rational Approximation for Fractional Order Systems;Journal of Dynamic Systems, Measurement, and Control;2021-02-04

2. Free Oscillation Solution for Fractional Differential System;Journal of Computational and Nonlinear Dynamics;2019-10-24

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