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.

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.

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 new stable collocation method for solving a class of nonlinear fractional delay differential equations

2020 ◽  
Vol 85 (4) ◽  
pp. 1123-1153
Author(s):  
Lei Shi ◽  
Zhong Chen ◽  
Xiaohua Ding ◽  
Qiang Ma
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Delay Differential Equations ◽  
Linear Equations ◽  
Approximate Solutions ◽  
Orthonormal Bases ◽  
Fractional Delay ◽  
Nonlinear Condition ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

AbstractIn this paper, a stable collocation method for solving the nonlinear fractional delay differential equations is proposed by constructing a new set of multiscale orthonormal bases of $W^{1}_{2,0}$ W 2 , 0 1 . Error estimations of approximate solutions are given and the highest convergence order can reach four in the sense of the norm of $W_{2,0}^{1}$ W 2 , 0 1 . To overcome the nonlinear condition, we make use of Newton’s method to transform the nonlinear equation into a sequence of linear equations. For the linear equations, a rigorous theory is given for obtaining their ε-approximate solutions by solving a system of equations or searching the minimum value. Stability analysis is also obtained. Some examples are discussed to illustrate the efficiency of the proposed method.

scholarly journals Variational Approximate Solutions of Fractional Delay Differential Equations with Integral Transform

2021 ◽  
pp. 3679-3689
Author(s):  
Eman Mohmmed Namah
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Decomposition Method ◽  
Integral Transform ◽  
Adomian Decomposition Method ◽  
Approximate Solutions ◽  
Adomian Decomposition ◽  
Fractional Delay ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

     The idea of the paper is to consolidate Mahgoub transform and variational iteration method (MTVIM) to solve fractional delay differential equations (FDDEs). The fractional derivative was in Caputo sense. The convergences of approximate solutions to exact solution were quick. The MTVIM is characterized by ease of application in various problems and is capable of simplifying the size of computational operations.  Several non-linear (FDDEs) were analytically solved as illustrative examples and the results were compared numerically. The results for accentuating the efficiency, performance, and activity of suggested method were shown by comparisons with Adomian Decomposition Method (ADM), Laplace Adomian Decomposition Method (LADM), Modified Adomian Decomposition Method (MADM) and Homotopy Analysis Method (HAM).

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.

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