scholarly journals Stability of delay differential equations in the sense of Ulam on unbounded intervals

2019 ◽  
Vol 9 (2) ◽  
pp. 125-131 ◽  
Author(s):  
Süleyman Öğrekçi
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Stability Problem ◽  
The Stability ◽  
Delay Differential ◽  
Unbounded Intervals

In this paper, we consider the stability problem of delay differential equations in the sense of Hyers-Ulam-Rassias. Recently this problem has been solved for bounded intervals, our result extends and improve the literature by obtaining stability in unbounded intervals. An illustrative example is also given to compare these results and visualize the improvement.

Galerkin Approximations for Stability of Delay Differential Equations With Time Periodic Coefficients

2015 ◽  
Vol 10 (2) ◽  
Author(s):  
Anwar Sadath ◽  
C. P. Vyasarayani
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Approximate Model ◽  
Periodic Coefficients ◽  
Galerkin Approximations ◽  
Approximate System ◽  
The Stability ◽  
Delay Differential ◽  
Time Periodic ◽  
Least Squares Approach

A numerical method to determine the stability of delay differential equations (DDEs) with time periodic coefficients is proposed. The DDE is converted into an equivalent partial differential equation (PDE) with a time periodic boundary condition (BC). The PDE, along with its BC, is then converted into a system of ordinary differential equations (ODEs) with time periodic coefficients using the Galerkin least squares approach. In the Galerkin approach, shifted Legendre polynomials are used as basis functions, allowing us to obtain explicit expressions for the approximate system of ODEs. We analyze the stability of the discretized ODEs, which represent an approximate model of the DDEs, using Floquet theory. We use numerical examples to show that the stability charts obtained with our method are in excellent agreement with those existing in the literature and those obtained from direct numerical simulation.

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 Numerical Stability Test of Neutral Delay Differential Equations

2008 ◽  
Vol 2008 ◽  
pp. 1-10 ◽  
Author(s):  
Z. H. Wang
Keyword(s):  
Characteristic Function ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Characteristic Root ◽  
Initial Guess ◽  
Lambert W Function ◽  
Neutral Delay ◽  
Neutral Delay Differential Equations ◽  
The Stability ◽  
Delay Differential

The stability of a delay differential equation can be investigated on the basis of the root location of the characteristic function. Though a number of stability criteria are available, they usually do not provide any information about the characteristic root with maximal real part, which is useful in justifying the stability and in understanding the system performances. Because the characteristic function is a transcendental function that has an infinite number of roots with no closed form, the roots can be found out numerically only. While some iterative methods work effectively in finding a root of a nonlinear equation for a properly chosen initial guess, they do not work in finding the rightmost root directly from the characteristic function. On the basis of Lambert W function, this paper presents an effective iterative algorithm for the calculation of the rightmost roots of neutral delay differential equations so that the stability of the delay equations can be determined directly, illustrated with two examples.

scholarly journals A New Multi-Step Method for Solving Delay Differential Equations using Lagrange Interpolation

2021 ◽  
pp. 159-164
Author(s):  
V. J. Shaalini ◽  
S. E. Fadugba
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Lagrange Interpolation ◽  
Search Algorithm ◽  
Step Method ◽  
Delay Argument ◽  
State Dependent ◽  
Constant Delay Time ◽  
The Stability ◽  
Delay Differential

This paper presents 2-step p-th order (p = 2,3,4) multi-step methods that are based on the combination of both polynomial and exponential functions for the solution of Delay Differential Equations (DDEs). Furthermore, the delay argument is approximated using the Lagrange interpolation. The local truncation errors and stability polynomials for each order are derived. The Local Grid Search Algorithm (LGSA) is used to determine the stability regions of the method. Moreover, applicability and suitability of the method have been demonstrated by some numerical examples of DDEs with constant delay, time dependent and state dependent delays. The numerical results are compared with the theoretical solution as well as the existing Rational Multi-step Method2 (RMM2). 

scholarly journals A Method for Stability Analysis of Periodic Delay Differential Equations with Multiple Time-Periodic Delays

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Gang Jin ◽  
Houjun Qi ◽  
Zhanjie Li ◽  
Jianxin Han ◽  
Hua Li
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Spindle Speed ◽  
Multiple Time ◽  
Variable Pitch ◽  
Periodic Delay ◽  
The Stability ◽  
Delay Differential ◽  
Time Periodic ◽  
Periodic Delays

Delay differential equations (DDEs) are widely utilized as the mathematical models in engineering fields. In this paper, a method is proposed to analyze the stability characteristics of periodic DDEs with multiple time-periodic delays. Stability charts are produced for two typical examples of time-periodic DDEs about milling chatter, including the variable-spindle speed milling system with one-time-periodic delay and variable pitch cutter milling system with multiple delays. The simulations show that the results gained by the proposed method are in close agreement with those existing in the past literature. This indicates the effectiveness of our method in terms of time-periodic DDEs with multiple time-periodic delays. Moreover, for milling processes, the proposed method further provides a generalized algorithm, which possesses a good capability to predict the stability lobes for milling operations with variable pitch cutter or variable-spindle speed.

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