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Stability properties of numerical methods for solving delay differential equations. (English) Zbl 0542.65040

The author considers stability properties of numerical methods for delay differential equations \(u'(t)=f(t,u(t),u(t-d(t,u(t))))\).
Reviewer: T.Tadumadze

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
34K05 General theory of functional-differential equations
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[1] Al-Mutib, A. N., Computational methods for solving delay differential equations, (Ph.D. Thesis (1977), Victoria University: Victoria University Manchester, U.K) · Zbl 0542.65040
[2] A.N. Al-Mutib, An explicit one-step method for solving delay differential equations, IMA J. Numer. Anal.; A.N. Al-Mutib, An explicit one-step method for solving delay differential equations, IMA J. Numer. Anal.
[3] A.N. Al-Mutib, One-step implicit methods for solving delay differential equations, BIT; A.N. Al-Mutib, One-step implicit methods for solving delay differential equations, BIT
[4] Barwell, V. K., on the asymptotic behaviour of the solution of a differential difference equation, Utilitas Math., 6, 189-194 (1974)
[5] Barwell, V. K., Special stability problems for functional equations, BIT, 15, 130-135 (1975) · Zbl 0306.65044
[6] Bellman, R. E.; Cooke, K. L., Differential-Difference Equations (1963), Academic Press: Academic Press New York
[7] Brayton, R. K.; Willoughby, R. A., On the numerical integration of a symmetric system of difference-differential equations of neutral type, J. Math. Anal. Appl., 18, 182-189 (1967) · Zbl 0155.47302
[8] Cryer, C. W., Highly-stable multistep methods for retarded differential equations, SIAM J. Numer. Anal., 11, 788-797 (1974) · Zbl 0307.65054
[9] Widerholt, L. F., Numerical integration of delay differential equations, (Ph.D. Thesis (1970), University of Wisconsin: University of Wisconsin Madison, WI)
[10] L.F. Widerholt, Stability of multistep methods for delay differential equations, Math. Comput.30; L.F. Widerholt, Stability of multistep methods for delay differential equations, Math. Comput.30
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