On periodic solutions of fractional-order differential systems with a fixed length of sliding memory

Abstract

The fractional-order derivative of a non-constant periodic function is not periodic with the same period. Consequently, any time-invariant fractional-order systems do not have a non-constant periodic solution. This property limits the applicability of fractional derivatives and makes it unfavorable to model periodic real phenomena.This article introduces a modification to the Caputo and Rieman-Liouville fractional-order operators by fixing their memory length and varying the lower terminal. It is shown that this modified definition of fractional derivative preserves the periodicity. Therefore, periodic solutions can be expected in fractional-order systems in terms of the new fractional derivative operator. To confirm this assertion, one investigates two examples, one linear system for which one gives an exact periodic solution by its analytical expression and another nonlinear system for which one provides exact periodic solutions using qualitative and numerical methods.

A direct integral pseudospectral method for solving a class of infinite-horizon optimal control problems using Gegenbauer polynomials and certain parametric maps

“…The aftermath of this theorem is that autonomous fractionalorder systems whose differential equations contain only a fractionalorder derivative defined based on the Grunwald-Letnikov, RL, or Caputo definitions cannot have nonconstant periodic solutions [9,[12][13][14]. One approach to preserve the periodicity of RL and Caputo fractional-order operators and allow for the existence of periodic solutions of fractional-order models is to fix their memory length and vary their lower terminals, as shown in [10]. In particular, the modified fractional operators, referred to by the RL and Caputo FDs with sliding fixed memory length L > 0, and denoted by MRL L D α t f (t) and MC L D α t f (t), respectively, are defined by…”

Non-polynomial fractional spline method for solving Fredholm integral equations

“…Several fractional derivatives and integral definitions have recently been offered. [28]- [20], [7], and [29]. It also contributes significantly to the progress of other fields of science, such as engineering [31], chemistry [14], physics [6], and biology [15].…”

A high-order embedded domain method combining a Predictor–Corrector-Fourier-Continuation-Gram method with an integral Fourier pseudospectral collocation method for solving linear partial differential equations in complex domains

“…In this study, we provide the first successful attempt in the literature to model FPDEs with periodic solutions using periodic FD operators that can preserve the periodicity of a periodic function and allow for the existence of periodic solutions to FPDEs. In particular, we model FPDEs with periodic solutions using the recently developed periodic FD operator of Elgindy [2], which is a useful reduced form of an earlier periodic FD inaugurated by Bourafa et al [1]. The employed FD operator is a useful modification of the classical RL and Caputo FD operators by fixing their memory length and varying their lower terminals.…”

“…The employed FD operator is a useful modification of the classical RL and Caputo FD operators by fixing their memory length and varying their lower terminals. The reduced FD operator developed in [2] allows accurate computation of the singular integral of the FD formula defined in [1] by removing the singularity prior to numerical integration using a smart change of variables and renders the reduced integral well behaved. In fact, the introduced transformation largely simplifies the problem of calculating the periodic FDs of periodic functions to the problem of evaluating the integral of the first derivatives of their trigonometric Lagrange interpolating polynomials, which can be treated accurately and efficiently using Gegenbauer quadratures.…”

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In this paper, we present a novel pseudospectral (PS) method for solving a new class of initial-value problems (IVPs) of time-dependent one-dimensional fractional partial differential equations (FPDEs) with variable coefficients and periodic solutions. A main ingredient of our work is the use of the recently developed periodic RL/Caputo fractional derivative (FD) operators with sliding positive fixed memory length of Bourafa et al [1] or their reduced forms obtained by Elgindy [2] as the natural FD operators to accurately model FPDEs with periodic solutions. The proposed method converts the IVP into a well-conditioned linear system of equations using the PS method based on Fourier collocations and Gegenbauer quadratures.

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