Stability Concepts of Numerical Solutions of Delay Differential Equations

2021 ◽  
pp. 51-67
Author(s):  
Fathalla A. Rihan
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Numerical Solutions ◽  
Delay Differential ◽  
Stability Concepts

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.

scholarly journals Stability of Analytical and Numerical Solutions for Nonlinear Stochastic Delay Differential Equations with Jumps

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Qiyong Li ◽  
Siqing Gan
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Numerical Solutions ◽  
Mean Square ◽  
Stochastic Delay Differential Equations ◽  
Stochastic Delay ◽  
Analytical And Numerical Solutions ◽  
Mean Square Stability ◽  
Delay Differential

This paper is concerned with the stability of analytical and numerical solutions fornonlinearstochastic delay differential equations (SDDEs) with jumps. A sufficient condition for mean-square exponential stability of the exact solution is derived. Then, mean-square stability of the numerical solution is investigated. It is shown that the compensated stochastic θ methods inherit stability property of the exact solution. More precisely, the methods are mean-square stable for any stepsizeΔt=τ/mwhen1/2≤θ≤1, and they are exponentially mean-square stable if the stepsizeΔt∈(0,Δt0)when0≤θ<1. Finally, some numerical experiments are given to illustrate the theoretical results.

scholarly journals Stability Analysis of Analytical and Numerical Solutions to Nonlinear Delay Differential Equations with Variable Impulses

2017 ◽  
Vol 2017 ◽  
pp. 1-8
Author(s):  
X. Liu ◽  
Y. M. Zeng
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Numerical Scheme ◽  
Numerical Solutions ◽  
Analytical And Numerical Solutions ◽  
Impulsive Delay Differential Equations ◽  
Nonlinear Delay Differential Equations ◽  
Impulsive Delay ◽  
Delay Differential ◽  
Nonlinear Delay

A stability theory of nonlinear impulsive delay differential equations (IDDEs) is established. Existing algorithm may not converge when the impulses are variable. A convergent numerical scheme is established for nonlinear delay differential equations with variable impulses. Some stability conditions of analytical and numerical solutions to IDDEs are given by the properties of delay differential equations without impulsive perturbations.

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