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The asymptotic stability of one-parameter methods for neutral differential equations. (English) Zbl 0814.65078
The paper is concerned with the asymptotic stability of theoretical solutions and numerical methods for systems of neutral differential equations of the form \[ x'(t)= Ax'(t- \tau)+ Bx(t)+ Cx(t- \tau), \] where \(A\), \(B\), \(C\) are constant complex matrices and \(\tau> 0\). For a large class of electrical networks containing lossless transmission lines the describing equations can be reduced to the above ones. A necessary and sufficient condition for asymptotic stability is obtained.
For the numerical solutions, adaptations of the one-parameter method introduced by R. K. Brayton and R. A. Willoughby [J. Math. Analysis Appl., 18, 182-189 (1967; Zbl 0155.473)] are considered. The stability region of the numerical method is compared with the stability region of the equation.
Reviewer: V.Arnautu (Iaşi)

MSC:
65L05 Numerical methods for initial value problems
65L20 Stability and convergence of numerical methods for ordinary differential equations
65L07 Numerical investigation of stability of solutions
34K05 General theory of functional-differential equations
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