Higher order numerical schemes for the solution of fractional delay differential equations

2022 ◽  
Vol 402 ◽  
pp. 113810
Author(s):  
Naga Raju Gande ◽  
H. Madduri
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Higher Order ◽  
Numerical Schemes ◽  
Fractional Delay ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

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.

Dependence Analysis of the Solutions on the Parameters of Fractional Delay Differential Equations

2011 ◽  
Vol 3 (5) ◽  
pp. 586-597 ◽  
Author(s):  
Shuiping Yang ◽  
Aiguo Xiao ◽  
Xinyuan Pan
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Initial Function ◽  
Hand Function ◽  
Dependence Analysis ◽  
Numerical Examples ◽  
Fractional Delay ◽  
Delay Differential ◽  
Fractional Delay Differential Equations ◽  
Theoretical Results

AbstractIn this paper, we investigate the dependence of the solutions on the parameters (order, initial function, right-hand function) of fractional delay differential equations (FDDEs) with the Caputo fractional derivative. Some results including an estimate of the solutions of FDDEs are given respectively. Theoretical results are verified by some numerical examples.

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.

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