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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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##### References:
 [1] V. K. Barwell,Special stability problems for functional differential equations BIT, 15 (1975), pp. 130–135. · Zbl 0306.65044 · doi:10.1007/BF01932685 [2] R. Bellman and K. L. Cooke,Differential-Difference Equations Academic Press, New York, 1963. [3] R. K. Brayton and R. A. Willoughby,On the numerical integration of a symmetric system of difference-differential equations of neutral type J. Math. An. Appl., 18 (1967), pp. 182–189. · Zbl 0155.47302 · doi:10.1016/0022-247X(67)90191-6 [4] J. K. Hale,Theory of Functional Differential Equations Springer Verlag, New York, 1977. · Zbl 0352.34001 [5] P. Henrici,Applied and Computational Complex Analysis. Volume 1, John Wiley & Sons, New York, 1974. · Zbl 0313.30001 [6] K. J. In’t Hout and M. N. Spijker,Stability analysis of numerical methods for delay differential equations Numer. Math., 59 (1991), pp. 807–814. · Zbl 0737.65073 · doi:10.1007/BF01385811 [7] K. J. In’t Hout,The stability of $$\theta$$-methods for systems of delay differential equations, Report Series No. 282, March 1993, ISSN 0112-4021, Univ. Auckland, 1993. [8] K. Knopp,Funktionen Theorie II Gruyter, Berlin, 1955. [9] P. Lancaster and M. Tismenetsky,The Theory of Matrices second edition, Academic Press, New York, 1985. · Zbl 0558.15001 [10] M. Z. Liu and M. N. Spijker,The stability of the $$\theta$$-methods in the numerical solution of delay differential equations IMA J. Numer. Anal., 10 (1991), pp. 31–48. · Zbl 0693.65056 · doi:10.1093/imanum/10.1.31 [11] L. H. Lu,The stability of the block $$\theta$$-methods IMA J. Numer. Anal., 13 (1993), pp. 101–114. · Zbl 0764.65039 · doi:10.1093/imanum/13.1.101 [12] W. L. Miranker,The wave equation with a nonlinear interface condition IBM J. Res. Developm., 25 (1961), pp. 2–24. · Zbl 0148.08405 · doi:10.1147/rd.51.0002 [13] W. L. Miranker,Existence, uniqueness, and stability of solutions of systems of nonlinear difference-differential equations J. Math. Mech., 11 (1962), pp. 101–108. · Zbl 0114.04201 [14] W. Snow,Existence, uniqueness, and stability for nonlinear differential-difference equations in the neutral case, N.Y.U., Courant Inst. Math. Sci. Rep. IMM-NYU 328, 1965.
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