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Segmented tau approximation for test neutral functional differential equations. (English) Zbl 1117.65102

Summary: We use the segmented formulation of the tau method to approximate the solutions of the neutral delay differential equation

$\begin{array}{cc}\hfill {y}^{\text{'}}\left(t\right)& =ay\left(t\right)+by\left(t-\tau \right)+c{y}^{\text{'}}\left(t-\tau \right)+f\left(t\right),\phantom{\rule{4pt}{0ex}}t\ge 0,\hfill \\ \hfill y\left(t\right)& ={\Psi }\left(t\right),\phantom{\rule{4pt}{0ex}}t\le 0,\hfill \end{array}$

which represents, for different values of $a,b,c$ and $\tau$, a family of functional differential equations that some authors have considered as test equations in different numerical experimentations. The tau method introduced by Lanczos is an important example of how to get approximations of functions defined by a differential equation. In the formulation of a step by step tau version is expected that the error is minimized at the matching points of successive steps. Through the study of recent papers it seems to be demonstrated that the step by step tau method is a natural and promising strategy for the numerical solution of functional differential equations. In preliminary experimentation significant improvements have been obtained when compared with the numerical results obtained elsewhere.

##### MSC:
 65L05 Initial value problems for ODE (numerical methods) 34K28 Numerical approximation of solutions of functional-differential equations 34K40 Neutral functional-differential equations
##### Keywords:
numerical examples
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