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.

A novel method to solve variable-order fractional delay differential equations based in lagrange interpolations

2019 ◽  
Vol 126 ◽  
pp. 266-282 ◽  
Author(s):  
C.J. Zúñiga-Aguilar ◽  
J.F. Gómez-Aguilar ◽  
R.F. Escobar-Jiménez ◽  
H.M. Romero-Ugalde
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Variable Order ◽  
Fractional Delay ◽  
Novel Method ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

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.

scholarly journals A collocation approach for multiterm variable-order fractional delay-differential equations using shifted Chebyshev polynomials

2021 ◽  
Author(s):  
Khalid K. Ali ◽  
Emad M.H. Mohamed ◽  
Mohamed A. Abd El salam ◽  
Kottakkaran Sooppy Nisar ◽  
M. Motawi Khashan ◽  
...  
Keyword(s):  
Differential Equations ◽  
Chebyshev Polynomials ◽  
Delay Differential Equations ◽  
Variable Order ◽  
Fractional Delay ◽  
Collocation Approach ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

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 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.

scholarly journals Neural minimization methods (NMM) for solving variable order fractional delay differential equations (FDDEs) with simulated annealing (SA)

PLoS ONE ◽  
2019 ◽  
Vol 14 (10) ◽  
pp. e0223476
Author(s):  
Amber Shaikh ◽  
M. Asif Jamal ◽  
Fozia Hanif ◽  
M. Sadiq Ali Khan ◽  
Syed Inayatullah
Keyword(s):  
Simulated Annealing ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Variable Order ◽  
Minimization Methods ◽  
Fractional Delay ◽  
Delay Differential ◽  
Fractional Delay Differential Equations
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