Stability Analysis of Fractional Delay Differential Equations by Chebyshev Polynomial

2012 ◽  
Vol 500 ◽  
pp. 586-590
Author(s):  
Xiang Mei Zhang ◽  
Xian Zhou Guo ◽  
Anping Xu
Keyword(s):  
Differential Equations ◽  
Chebyshev Polynomial ◽  
Delay Differential Equations ◽  
Fractional Derivatives ◽  
Mathieu Equation ◽  
Smooth Coefficients ◽  
Fractional Delay ◽  
The Stability ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

The paper is devoted to the numerical stability of fractional delay differential equations with non-smooth coefficients using the Chebyshev collocation method. In this paper, based on the Grunwald-Letnikov fractional derivatives, we discuss the approximation of fractional differentiation by the Chebyshev polynomial of the first kind. Then we solve the stability of the fractional delay differential equations. Finally, the stability of the delayed Mathieu equation of fractional order is examined for a set of case studies that contain the complexities of periodic coefficients, delays and discontinuities.

An Extended Predictor–Corrector Algorithm for Variable-Order Fractional Delay Differential Equations

2016 ◽  
Vol 11 (6) ◽  
Author(s):  
B. Parsa Moghaddam ◽  
Sh. Yaghoobi ◽  
J. A. Tenreiro Machado
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Fractional Derivatives ◽  
Variable Order ◽  
The Novel ◽  
Fractional Delay ◽  
Novel Method ◽  
Delay Differential ◽  
Predictor Corrector ◽  
Fractional Delay Differential Equations

This article presents a numerical method based on the Adams–Bashforth–Moulton scheme to solve variable-order fractional delay differential equations (VFDDEs). In these equations, the variable-order (VO) fractional derivatives are described in the Caputo sense. The existence and uniqueness of the solutions are proved under Lipschitz condition. Numerical examples are presented showing the applicability and efficiency of the novel method.

scholarly journals Hypergeometric fractional derivatives formula of shifted Chebyshev polynomials: tau algorithm for a type of fractional delay differential equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Waleed M. Abd-Elhameed ◽  
José A. Tenreiro Machado ◽  
Youssri H. Youssri
Keyword(s):  
Differential Equations ◽  
Chebyshev Polynomials ◽  
Delay Differential Equations ◽  
Fractional Derivatives ◽  
Approximate Solutions ◽  
Tau Method ◽  
Fractional Delay ◽  
Delay Differential ◽  
Derivatives Of ◽  
Fractional Delay Differential Equations

Abstract This paper presents an explicit formula that approximates the fractional derivatives of Chebyshev polynomials of the first-kind in the Caputo sense. The new expression is given in terms of a terminating hypergeometric function of the type 4 F 3(1). The integer derivatives of Chebyshev polynomials of the first-kind are deduced as a special case of the fractional ones. The formula will be applied for obtaining a spectral solution of a certain type of fractional delay differential equations with the aid of an explicit Chebyshev tau method. The shifted Chebyshev polynomials of the first-kind are selected as basis functions and the spectral tau method is employed for obtaining the desired approximate solutions. The convergence and error analysis are discussed. Numerical results are presented illustrating the efficiency and accuracy of the proposed algorithm.

scholarly journals Numerical Study for Time Delay Multistrain Tuberculosis Model of Fractional Order

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-14 ◽  
Author(s):  
Nasser Hassan Sweilam ◽  
Seham Mahyoub Al-Mekhlafi ◽  
Taghreed Abdul Rahman Assiri
Keyword(s):  
Time Delay ◽  
Differential Equations ◽  
Fractional Order ◽  
Delay Differential Equations ◽  
Numerical Study ◽  
Equilibrium Points ◽  
Fractional Delay ◽  
The Stability ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

A novel mathematical fractional model of multistrain tuberculosis with time delay memory is presented. The proposed model is governed by a system of fractional delay differential equations, where the fractional derivative is defined in the sense of the Grünwald–Letinkov definition. Modified parameters are introduced to account for the fractional order. The stability of the equilibrium points is investigated for any time delay. Nonstandard finite deference method is proposed to solve the resulting system of fractional-order delay differential equations. Numerical simulations show that nonstandard finite difference method can be applied to solve such fractional delay differential equations simply and effectively.

Stability Analysis of Fractional Delay Differential Equations by Lagrange Polynomial

2012 ◽  
Vol 500 ◽  
pp. 591-595
Author(s):  
Xiang Mei Zhang ◽  
An Ping Xu ◽  
Xian Zhou Guo
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Collocation Method ◽  
Delay Differential Equations ◽  
Numerical Stability ◽  
Mathieu Equation ◽  
Lagrange Polynomial ◽  
Fractional Delay ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

The paper deals with the numerical stability analysis of fractional delay differential equations with non-smooth coefficients using the Lagrange collocation method. In this paper, based on the Grunwald-Letnikov fractional derivatives, we discuss the approximation of fractional differentiation by the Lagrange polynomial. Then we study the numerical stability of the fractional delay differential equations. Finally, the stability of the delayed Mathieu equation of fractional order is studied and examined by Lagrange collocation method.

scholarly journals Modified finite difference method for solving fractional delay differential equations

2017 ◽  
Vol 35 (2) ◽  
pp. 49-58 ◽  
Author(s):  
Behrouz Parsa Moghaddam ◽  
Zeynab Salamat Mostaghim
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Fractional Derivatives ◽  
Scientific Models ◽  
Initial Value ◽  
Caputo Fractional Derivatives ◽  
Fractional Delay ◽  
Delay Differential ◽  
T Tau ◽  
Fractional Delay Differential Equations

In this paper we present and discuss a new numerical scheme for solving fractional delay differential equations of the generalform:$$D^{\beta}_{*}y(t)=f(t,y(t),y(t-\tau),D^{\alpha}_{*}y(t),D^{\alpha}_{*}y(t-\tau))$$on $a\leq t\leq b$,$0<\alpha\leq1$,$1<\beta\leq2$ and under the following interval and boundary conditions:\\$y(t)=\varphi(t) \qquad\qquad -\tau \leq t \leq a,$\\$y(b)=\gamma$\\where $D^{\beta}_{*}y(t)$,$D^{\alpha}_{*}y(t)$ and $D^{\alpha}_{*}y(t-\tau)$ are the standard Caputo fractional derivatives, $\varphi$ is the initial value and $\gamma$ is a smooth function.\\We also provide this method for solving some scientific models. The obtained results show that the propose method is veryeffective and convenient.

Numerical Simulation of Fractional Delay Differential Equations Using the Operational Matrix of Fractional Integration for Fractional-Order Taylor Basis

2021 ◽  
Vol 6 (1) ◽  
pp. 10
Author(s):  
İbrahim Avcı 
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Numerical Solutions ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Absolute Error ◽  
Algebraic Equations ◽  
Fractional Delay ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

In this paper, we consider numerical solutions for a general form of fractional delay differential equations (FDDEs) with fractional derivatives defined in the Caputo sense. A fractional integration operational matrix, created using a fractional Taylor basis, is applied to solve these FDDEs. The main characteristic of this approach is, by utilizing the operational matrix of fractional integration, to reduce the given differential equation to a set of algebraic equations with unknown coefficients. This equation system can be solved efficiently using a computer algorithm. A bound on the error for the best approximation and fractional integration are also given. Several examples are given to illustrate the validity and applicability of the technique. The efficiency of the presented method is revealed by comparing results with some existing solutions, the findings of some other approaches from the literature and by plotting absolute error figures.

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