Delay-dependent stability of linear multistep methods for differential systems with distributed delays

doi:10.1007/s10483-018-2392-9

Abstract

Abstract: This paper deals with the stability of linear multistep methods for multidimensional differential systems with distributed delays. The delay-dependent stability of linear multistep methods with compound quadrature rules is studied. Several new sufficient criteria of delay-dependent stability are obtained by means of the argument principle. An algorithm is provided to check delay-dependent stability. An example that illustrates the effectiveness of the derived theoretical results is given.

Key words: Newton-Raphson method, penalty function method, nonlineareffect, drill string, delay-dependent stability, linear multistep method, argument principle, differential system with distributed delays

2010 MSC Number:

Yanpei WANG, Yuhao CONG, Guangda HU. Delay-dependent stability of linear multistep methods for differential systems with distributed delays. Applied Mathematics and Mechanics (English Edition), 2018, 39(12): 1837-1844.

TrendMD

References

[1] BELLEN, A. and ZENNARO, M. Numerical Methods for Delay Differential Equations, Oxford University Press, Oxford, 152-182(2003)
[2] HALE, J. K. and VERDUYN LUNEL, S. M. Introduction to Functional Differential Equations, Springer, Berlin/Heidelberg/New York, 36-56(1993)
[3] HUANG, C. M. and VANDEWALLE, S. Stability of Runge-Kutta-Pouzet methods for Volterra integro-differential equations with delays. Frontiers of Mathematics in China, 4(1), 63-87(2009)
[4] KIM, A. V. and IVANOV, A. V. Systems with Delays Analysis, Control and Computations, Scrivener Publishing LLC, Beverly, MA, 89-110(2015)
[5] LUBICH, C. Runge-Kutta theory for Volterra integrodifferential equations. Numerische Mathematik, 40(1), 119-135(1982)
[6] ZHANG, C. J. and VANDEWALLE, S. Stability analysis of Runge-Kutta methods for nonlinear Volterra delay integro-differential equations. IMA Journal of Numerical Analysis, 24(2), 193-214(2004)
[7] BAKER, C. T. H. and FORD, N. J. Stability properties of a scheme for the approximate solution of a delay-integro-differential equation. Applied Numerical Mathematics, 9(3), 357-370(1992)
[8] LUZYANINA, T., ENGELBORGHS, K., and ROOSE, D. Computing stability of differential equations with bounded distributed delays. Numerical Algorithm, 34(1), 41-66(2003)
[9] HUANG, C. M. Stability of linear multistep methods for delay integro-differential equations. Computers and Mathematics with Applications, 55(12), 2830-2838(2008)
[10] CHEN, H. and ZHANG, C. J. Convergence and stability of extended block boundary value methods for Volterra delay integro-differential equations. Applied Numerical Mathematics, 62(2), 141-154(2012)
[11] KOTO, T. Stability of Runge-Kutta methods for delay integro-differential equations. Journal of Computational and Applied Mathematics, 145(2), 483-492(2002)
[12] KOTO, T. Stability of θ-methods for delay integro-differential equations. Journal of Computational and Applied Mathematics, 161(2), 393-404(2003)
[13] QIN, T. T. and ZHANG, C. J. Stable solutions of one-leg methods for a class of nonlinear functional-integro-differential equations. Applied Mathematics and Computation, 250(11), 47-57(2015)
[14] ZHANG, C. J. and VANDEWALLE, S. Stability analysis of Volterra delay-integro-differential equations and their backward differentiation time discretization. Journal of Computational and Applied Mathematics, 164-165, 797-814(2004)
[15] ZHANG, C. J. and VADEWALLE, S. General linear methods for Volterra integro-differential equations with memory. SIAM Journal on Scientific Computing, 27(6), 2010-2031(2006)
[16] HUANG, C. M. and VANDEWALLE, S. An analysis of delay-dependent stability for ordinary and partial differential equations with fixed and distributed delays. SIAM Journal on Scientific Computing, 25(5), 1608-1632(2004)
[17] WU, S. F. and GAN, S. Q. Analytical and numerical stability of neutral delay integro-differential equations and neutral delay partial differential equations. Computer and Mathematics with Application, 55(11), 2426-2443(2008)
[18] XU, Y. and ZHAO, J. J. Analysis of delay-dependent stability of linear θ-methods for linear delay-integro differential equations. IMA Journal of Applied Mathematics, 74(6), 851-869(2009)
[19] ZHAO, J. J., FAN, Y., and XU, Y. Delay-dependent stability analysis of symmetric boundary value methods for linear delay integro-differential equations. Numerical Algorithm, 65(1), 125-151(2014)
[20] GUGLIELMI, N. Asymptotic stability barriers for natural Runge-Kutta processes for delay equations. SIAM Journal on Numerical Analysis, 39(3), 763-783(2001)
[21] HU, G. D. and MITSUI, T. Delay-dependent stability of numerical methods for delay differential systems of neutral type. BIT Numerical Mathematics, 57(3), 1-22(2017)
[22] HU, G. D. Computable stability criterion of linear neutral systems with unstable difference operators. Journal of Computational and Applied Mathematics, 271(12), 223-232(2014)
[23] BROWNN, J. W. and CHURCHILL, R. V. Complex Variables and Applications, McGraw-Hill Companies, Inc. and China Machine Press, Beijing, 291-294(2004)
[24] LMABERT, J. D. Numerical Methods for Ordinary Differential Equations, 2nd ed., John Wiley and Sons, New York, 45-101(1991)
[25] BRUNNER, H. and VAN DER HOUWEN, P. J. The Numerical Solution of Volterra Equations, North-Holland Publishing Co., Amsterdam, 20-150(1986)
[26] VAN DER HOUWEN, P. J. and TE RIELE, H. J. J. Linear multistep methods for Volterra integral and integro-differential equations. Mathematics of Computation, 45(172), 439-461(1985)