On the convergence of Jacobi‐Gauss collocation method for linear fractional delay differential equations

2020 ◽  
Vol 44 (2) ◽  
pp. 2237-2253
Author(s):  
N. Peykrayegan ◽  
M. Ghovatmand ◽  
M. H. Noori Skandari
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Delay Differential Equations ◽  
Fractional Delay ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

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.

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.

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.

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