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Volume 19, Issue 2
The Stability of Linear Multistep Methods for Linear Systems of Neutral Differential Equations

Hong-Jiong Tian, Jiao-Xun Kuang & Lin Qiu

J. Comp. Math., 19 (2001), pp. 125-130.

Published online: 2001-04

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  • Abstract

This paper deals with the numerical solution of initial value problems for systems of neutral differential equations $$y'(t)=f(t,y(t),y(t- \tau ),y'(t- \tau )), t > 0, $$ $$y(t) = φ(t) \ t<0,$$ where $\tau> 0, f$ and φ denote given vector-valued functions. The numerical stability of a linear multistep method is investigated by analysing the solution of the test equations $y'(t)=Ay(t) + By(t-\tau) + Cy'(t-\tau),$ where $A, B$ and $C$ denote constant complex $N \times N$-matrices, and $\tau > 0$. We investigate the properties of adaptation of the linear multistep method and the characterization of the stability region. It is proved that the linear multistep method is NGP-stable if and only if it is A-stable for ordinary differential equations.  

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@Article{JCM-19-125, author = {Tian , Hong-JiongKuang , Jiao-Xun and Qiu , Lin}, title = {The Stability of Linear Multistep Methods for Linear Systems of Neutral Differential Equations}, journal = {Journal of Computational Mathematics}, year = {2001}, volume = {19}, number = {2}, pages = {125--130}, abstract = {

This paper deals with the numerical solution of initial value problems for systems of neutral differential equations $$y'(t)=f(t,y(t),y(t- \tau ),y'(t- \tau )), t > 0, $$ $$y(t) = φ(t) \ t<0,$$ where $\tau> 0, f$ and φ denote given vector-valued functions. The numerical stability of a linear multistep method is investigated by analysing the solution of the test equations $y'(t)=Ay(t) + By(t-\tau) + Cy'(t-\tau),$ where $A, B$ and $C$ denote constant complex $N \times N$-matrices, and $\tau > 0$. We investigate the properties of adaptation of the linear multistep method and the characterization of the stability region. It is proved that the linear multistep method is NGP-stable if and only if it is A-stable for ordinary differential equations.  

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8963.html} }
TY - JOUR T1 - The Stability of Linear Multistep Methods for Linear Systems of Neutral Differential Equations AU - Tian , Hong-Jiong AU - Kuang , Jiao-Xun AU - Qiu , Lin JO - Journal of Computational Mathematics VL - 2 SP - 125 EP - 130 PY - 2001 DA - 2001/04 SN - 19 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8963.html KW - Numerical stability, Linear multistep method, Delay differential equations. AB -

This paper deals with the numerical solution of initial value problems for systems of neutral differential equations $$y'(t)=f(t,y(t),y(t- \tau ),y'(t- \tau )), t > 0, $$ $$y(t) = φ(t) \ t<0,$$ where $\tau> 0, f$ and φ denote given vector-valued functions. The numerical stability of a linear multistep method is investigated by analysing the solution of the test equations $y'(t)=Ay(t) + By(t-\tau) + Cy'(t-\tau),$ where $A, B$ and $C$ denote constant complex $N \times N$-matrices, and $\tau > 0$. We investigate the properties of adaptation of the linear multistep method and the characterization of the stability region. It is proved that the linear multistep method is NGP-stable if and only if it is A-stable for ordinary differential equations.  

Hong-Jiong Tian, Jiao-Xun Kuang & Lin Qiu. (1970). The Stability of Linear Multistep Methods for Linear Systems of Neutral Differential Equations. Journal of Computational Mathematics. 19 (2). 125-130. doi:
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