Volume 22, Issue 4
Asymptotic Stability of Runge-Kutta Methods for the Pantograph Equations
DOI:

J. Comp. Math., 22 (2004), pp. 523-534

Published online: 2004-08

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• Abstract

This paper considers the asymptotic stability analysis of both exact and numerical solutions of the following neutral delay differential equation with pantograph delay.$x'(t)+Bx(t)+Cx'(qt)+Dx(qt)=0,t>0,x(0)=x_0$,where $B,C,D\in C^{d\times d},q\in (0,1)$,and B is regular. After transforming the above equation to non-automatic neutral equation with constant delay, we determine sufficient conditions for the asymptotic stability of the zero solution. Furthermore, we focus on the asymptotic stability behavior of Runge-Kutta method with variable stepsize. It is proved that a L- stable Runge-Kutta method can preserve the above-mentioned stability properties.

• Keywords

Neutral delay differential equations Pantograph delay Asymptotic stability Runge-Kutta methods L-stable

@Article{JCM-22-523, author = {}, title = {Asymptotic Stability of Runge-Kutta Methods for the Pantograph Equations}, journal = {Journal of Computational Mathematics}, year = {2004}, volume = {22}, number = {4}, pages = {523--534}, abstract = { This paper considers the asymptotic stability analysis of both exact and numerical solutions of the following neutral delay differential equation with pantograph delay.$x'(t)+Bx(t)+Cx'(qt)+Dx(qt)=0,t>0,x(0)=x_0$,where $B,C,D\in C^{d\times d},q\in (0,1)$,and B is regular. After transforming the above equation to non-automatic neutral equation with constant delay, we determine sufficient conditions for the asymptotic stability of the zero solution. Furthermore, we focus on the asymptotic stability behavior of Runge-Kutta method with variable stepsize. It is proved that a L- stable Runge-Kutta method can preserve the above-mentioned stability properties. }, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8861.html} }
TY - JOUR T1 - Asymptotic Stability of Runge-Kutta Methods for the Pantograph Equations JO - Journal of Computational Mathematics VL - 4 SP - 523 EP - 534 PY - 2004 DA - 2004/08 SN - 22 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8861.html KW - Neutral delay differential equations KW - Pantograph delay KW - Asymptotic stability KW - Runge-Kutta methods KW - L-stable AB - This paper considers the asymptotic stability analysis of both exact and numerical solutions of the following neutral delay differential equation with pantograph delay.$x'(t)+Bx(t)+Cx'(qt)+Dx(qt)=0,t>0,x(0)=x_0$,where $B,C,D\in C^{d\times d},q\in (0,1)$,and B is regular. After transforming the above equation to non-automatic neutral equation with constant delay, we determine sufficient conditions for the asymptotic stability of the zero solution. Furthermore, we focus on the asymptotic stability behavior of Runge-Kutta method with variable stepsize. It is proved that a L- stable Runge-Kutta method can preserve the above-mentioned stability properties.