×

The asymptotic stability of theoretical and numerical solutions for systems of neutral multidelay-differential equations. (English) Zbl 0924.65079

Consider the following problem for a system of linear delayed differential equations: \[ {d\over dt}\Biggl[x(t)- \sum^k_{m=1} N_mx(t- \tau_m)\Biggr]= Lx(t)+ \sum^k_{m= 1}M_mx(t- \tau_m),\quad t>0, \]
\[ x(t)= \varphi(t),\quad -\tau\leq t\leq 0, \] where \(N_m\), \(L\) and \(M_m\) \((1\leq m\leq k)\) are constant complex \(p\times p\)-matrices, \(\tau_m>0\) \((1\leq m\leq k)\) are constant delays and \(\tau= \max\tau_m\).
The authors give a sufficient condition for the asymptotic stability of the exact solution and establish a connection between this stability and the \(A\)-stability of a multistep Runge-Kutta method.

MSC:

65L20 Stability and convergence of numerical methods for ordinary differential equations
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
34K40 Neutral functional-differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
65L07 Numerical investigation of stability of solutions to ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Barwell, V. K., Special stability problem for functional differential equations, BIT, 15, 130-130 (1975) · Zbl 0306.65044 · doi:10.1007/BF01932685
[2] Bickart, T. A., P-stable and P[α, β]-stable integration/interpolation methods in the solution of retarded differential-difference equations, BIT, 22, 464-464 (1982) · Zbl 0531.65044 · doi:10.1007/BF01934410
[3] Liu, M. Z.; Spijker, M. N., The stability of the θ-methods in the numerical solution of delay differential equations, IMA J. Numer. Anal., 10, 31-31 (1990) · Zbl 0693.65056 · doi:10.1093/imanum/10.1.31
[4] In’t Hout, K. J., Stability analysis of Runge-Kutta methods for systems of delay differential equations, IMA J. Numer. Anal., 17, 17-17 (1997) · Zbl 0867.65046 · doi:10.1093/imanum/17.1.17
[5] Tian, H. J.; Kuang, J. X., The stability of the θ-methods in the numerical solution of delay differential equations with several delay terms, J. Comput. Appl. Math., 58, 171-171 (1995) · Zbl 0833.65085 · doi:10.1016/0377-0427(93)E0269-R
[6] Bellen, A.; Jackiewicz, Z.; Zennaro, M., Stability analysis of one-step methods for neutral delay-differential equations, Numer. Math., 52, 606-606 (1988) · Zbl 0644.65049 · doi:10.1007/BF01395814
[7] Kuang, J. X.; Xiang, J. X.; Tian, H. J., The asymptotic stability of one-parameter methods for neutral differential equations, BIT, 34, 400-400 (1994) · Zbl 0814.65078 · doi:10.1007/BF01935649
[8] Koto, T., A stability property of A-stable natural Runge-Kutta methods for systems of delay differential equations, BIT, 34, 262-262 (1994) · Zbl 0805.65083 · doi:10.1007/BF01955873
[9] Hu, G. D.; Mitsui, T., Stability analysis of numerical methods for systems of neutral delay-differential equations, BIT, 35, 504-504 (1995) · Zbl 0841.65062 · doi:10.1007/BF01739823
[10] Hu, G. D.; Hu, G. D.; Liu, M. Z., Estimation of numerically stable stepsize for neutral DDES via spectral radius, J. Comput. Appl. Math., 78, 311-311 (1997) · Zbl 0868.65054 · doi:10.1016/S0377-0427(96)00153-7
[11] Hale, J. K.; Verduyn Lunel, S. M., Introduction tu Functional Differential Equations (1993), New York: Springer-Verlag, New York · Zbl 0787.34002
[12] Lancaster, P.; Tismenetsky, M., The Theory of Matrices (1985), Orlando: Academic Press, Orlando · Zbl 0558.15001
[13] Li, S. F., B-Convergence properties of multistep Runge-Kutta methods, Math. Comput., 62, 565-565 (1994) · Zbl 0799.65082 · doi:10.2307/2153523
[14] Horn, R. A.; Johnson, C. R., Topics in Matrix Analysis (1991), Cambridge: Cambridge University Press, Cambridge · Zbl 0729.15001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.