Analysis of a system of fractional differential equations

Abstract

We prove existence and uniqueness theorems for the initial value problem for the system of. , x n (t)] t and A is a square matrix. The unique solution to this initial value problem turns out to be E α (t α A)x 0 , where E α denotes the Mittag-Leffler function generalized for matrix arguments. Further we analyze the system D α [x(t) −x(0)] =f (t,x),x(0) =x 0 , 0 < α < 1, and investigate dependence of the solutions on the initial conditions.  2004 Elsevier Inc. All rights reserved.

Fractional differential repetitive processes with Riemann–Liouville and Caputo derivatives

“…They are used, among others things, to the modeling of capacitor (Westerlund and Ekstam 1994;Dzielinski et al 2011Dzielinski et al , 2010, fluid flow through porous material (Langlands 2006), viscoelastic materials (Bagley and Torvik 1986). Fractional systems are investigated in many aspects: existence and uniqueness of solutions (Daftardar-Gejji and Babakhani 2004;Delbosco and Rodino 1996;Hayek et al 1999;Idczak and Kamocki 2011;Kilbas et al 2006), stability and positivity (Buslowicz 2008(Buslowicz , 2011Kaczorek 2011a, b), calculus of variations (Agrawal 2002;Baleanu and Muslih 2005;El-Nabulusi and Torres 2007;Idczak and Majewski 2012;Riewe 1996Riewe , 1997, controllability (Kaczorek 2011c) and optimal control (Agrawal 2004;Jelicic and Petrovacki 2009;Kamocki 2012;Tricaud and Chen 2010).…”

A Numerical Method for Delayed Fractional-Order Differential Equations: Based on G-L Definition

“…Several numerical methods based on Caputo or Riemann-Liouville definition have been proposed and analyzed [13]- [32]. For instance, based on the predictor-corrector scheme, Diethelm et al introduced Adams-Bashforth-Moulton algorithm [17,18,19], and meanwhile some error analyses and an extension of Richardson extrapolation were also presented to improve the numerical accuracy.…”

Study for System of Nonlinear Differential Equations with Riemann-Liouville Fractional Derivative

“…The existence of solution of initial value problems for fractional order differential equations have been studied in many literatures such as [1][2][3][4]. In this paper, we present the analysis of the system of fractional differential equations…”