scholarly journals Nonlinear Stability and Convergence of Two-Step Runge-Kutta Methods for Neutral Delay Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Haiyan Yuan ◽  
Cheng Song ◽  
Peichen Wang
Keyword(s):  
Differential Equations ◽  
Nonlinear Stability ◽  
Delay Differential Equations ◽  
Stability And Convergence ◽  
Neutral Delay ◽  
Neutral Delay Differential Equations ◽  
Restricted Type ◽  
Runge Kutta ◽  
The Stability ◽  
Delay Differential

This paper is devoted to the stability and convergence analysis of the two-step Runge-Kutta (TSRK) methods with the Lagrange interpolation of the numerical solution for nonlinear neutral delay differential equations. Nonlinear stability and D-convergence are introduced and proved. We discuss theGR(l)-stability,GAR(l)-stability, and the weakGAR(l)-stability on the basis of(k,l)-algebraically stable of the TSRK methods; we also discuss the D-convergence properties of TSRK methods with a restricted type of interpolation procedure.

scholarly journals Stepsize Restrictions for Nonlinear Stability Properties of Neutral Delay Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Wei Gu ◽  
Ming Wang ◽  
Dongfang Li
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Nonlinear Stability ◽  
Delay Differential Equations ◽  
Neutral Delay ◽  
Stability Properties ◽  
Neutral Delay Differential Equations ◽  
Runge Kutta ◽  
Delay Differential ◽  
The Relationship

The present paper is concerned with the relationship between stepsize restriction and nonlinear stability of Runge-Kutta methods for delay differential equations. We obtain a special stepsize condition guaranteeing global and asymptotical stability properties of numerical methods. Some confirmations of the conditions on Runge-Kutta methods are illustrated at last.

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.

Wavelet collocation methods for solving neutral delay differential equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Mo Faheem ◽  
Akmal Raza ◽  
Arshad Khan
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Series Solution ◽  
Collocation Methods ◽  
Proportional Delay ◽  
Neutral Delay ◽  
Neutral Delay Differential Equations ◽  
Runge Kutta ◽  
Delay Differential ◽  
Wavelet Series

Abstract In this paper, we proposed wavelet based collocation methods for solving neutral delay differential equations. We use Legendre wavelet, Hermite wavelet, Chebyshev wavelet and Laguerre wavelet to solve the neutral delay differential equations numerically. We solved five linear and one nonlinear problem to demonstrate the accuracy of wavelet series solution. Wavelet series solution converges fast and gives more accurate results in comparison to other methods present in literature. We compare our results with Runge–Kutta-type methods by Wang et al. (Stability of continuous Runge–Kutta-type methods for nonlinear neutral delay-differential equations,” Appl. Math. Model, vol. 33, no. 8, pp. 3319–3329, 2009) and one-leg θ methods by Wang et al. (Stability of one-leg θ method for nonlinear neutral differential equations with proportional delay,” Appl. Math. Comput., vol. 213, no. 1, pp. 177–183, 2009) and observe that our results are more accurate.

scholarly journals Stability of Runge-Kutta Methods for Neutral Delay Differential Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-8
Author(s):  
Liping Wen ◽  
Xiong Liu ◽  
Yuexin Yu
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Numerical Stability ◽  
Neutral Delay ◽  
Numerical Examples ◽  
Neutral Delay Differential Equations ◽  
Runge Kutta ◽  
Delay Differential ◽  
Stability Results ◽  
Theoretical Results

This paper is concerned with the numerical stability of a class of nonlinear neutral delay differential equations. The numerical stability results are obtained for(k,l)-algebraically stable Runge-Kutta methods when they are applied to this type of problem. Numerical examples are given to confirm our theoretical results.

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