Stability of \(\vartheta\)-methods for delay integro-differential equations. (English) Zbl 1042.65108

Koto, Toshiyuki

J. Comput. Appl. Math. 161, No. 2, 393-404 (2003).

This paper is concerned with the stability behaviour of the \(\theta\)-methods for the numerical solution of delay integro-differential equations: \[ u'(t)= f \left(t, u(t), u(t- \tau), \int_{t- \tau}^t g(t,s,u(s)) \,ds\right) \] where \( \tau\) is a constant delay. Here the delayed argument \(u(t- \tau)\) is approximated by linear interpolation between steps and the integral term is approximated by the trapezoidal rule.
Considering the linear test equation \(u'(t)= \lambda u(t) + \mu u(t- \tau) + \kappa \int_{t-\tau}^t u(s) \,ds\) with \(\lambda,\mu,\kappa\) real o complex constants and taking into account the conditions on this coefficients for the asymptotic stability of the zero solution obtained by the author in a previous paper [ibid. 145, No. 2, 483-492 (2002; Zbl 1002.65148)], the author studies the stability behaviour of the \(\theta\)–methods when the step size \(h\) is such that \( \tau = n h \) for some positive integer \(n\) and also in the case of arbitrary step sizes.
In the first case, it is found that for all \( \theta \in [1/2,1]\), i.e. for A-stable \(\theta\)–methods, the discretization preserves the asymptotic behaviour of the exact solutions, i.e. for all values of the parameters \( \lambda\), \(\mu\), \(\kappa\) such that the exact solution \( u(t) \to 0\) when \( t \to \infty\) the numerical solution exhibits also the same property. Moreover it is proved this property does not hold for arbitrary step sizes and an example is constructed to show this fact.

Reviewer: Manuel Calvo (Zaragoza)

MSC:

65R20	Numerical methods for integral equations
45J05	Integro-ordinary differential equations
45G10	Other nonlinear integral equations

Keywords:

delay integro-differential equations; stability; \(\theta\)-methods; linear test equations; numerical example

Citations:

Zbl 1002.65148

References:

[1]	Baker, C.T.H., Retarded differential equations, J. comput. appl. math., 125, 309-335, (2000) · Zbl 0970.65079
[2]	Baker, C.T.H.; Ford, N.J., Stability properties of a scheme for the approximate solution of a delay-integro-differential equation, Appl. numer. math., 9, 357-370, (1992) · Zbl 0754.65111
[3]	Baker, C.T.H.; Makroglou, A.; Short, E., Regions of stability in the numerical treatment of Volterra integro-differential equations, SIAM J. numer. anal., 16, 890-910, (1979) · Zbl 0428.65066
[4]	Barwell, V.K., Special stability problems for functional differential equations, Bit, 15, 130-135, (1975) · Zbl 0306.65044
[5]	Brunner, H.; Hairer, E.; Nørsett, S.P., Runge – kutta theory for Volterra integral equations of the second kind, Math. comput., 39, 147-163, (1982) · Zbl 0496.65066
[6]	Calvo, M.; Grande, T., On the asymptotic stability of θ-methods for delay differential equations, Numer. math., 54, 257-269, (1988) · Zbl 0631.65076
[7]	Crisci, M.R.; Russo, E.; Vecchio, A., Stability analysis of the de hoog and Weiss implicit runge – kutta methods for the Volterra integral and integrodifferential equations, J. comput. appl. math., 29, 329-341, (1990) · Zbl 0699.65089
[8]	Guglielmi, N., Delay dependent stability regions of θ-methods for delay differential equations, IMA J. numer. anal., 18, 399-418, (1998) · Zbl 0909.65050
[9]	K.J. in ’t Hout, The stability of θ-methods for systems of delay differential equations, Ann. Numer. Math. 1 (1994) 323-334. · Zbl 0837.65087
[10]	Jackiewicz, Z., Asymptotic stability analysis of θ-methods for functional differential equations, Numer. math., 43, 389-396, (1984) · Zbl 0557.65047
[11]	Koto, T., Stability of runge – kutta methods for delay integro-differential equations, J. comput. appl. math., 145, 483-492, (2002) · Zbl 1002.65148
[12]	Liu, Y., Stability analysis of θ-methods for neutral functional-differential equations, Numer. math., 70, 473-485, (1995) · Zbl 0824.65081
[13]	Liu, Y., On the θ-method for delay differential equations with infinite lag, J. comput. appl. math., 71, 177-190, (1996) · Zbl 0853.65076
[14]	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, 31-48, (1990) · Zbl 0693.65056
[15]	Lu, L.H., Numerical stability of the θ-methods for systems of differential equations with several delay terms, J. comput. appl. math., 34, 291-304, (1991) · Zbl 0731.65072
[16]	Pólya, G.; Szegö, G., Problems and theorems in analysis I, (1972), Springer Berlin · Zbl 0236.00003
[17]	Qiu, L.; Mitsui, T.; Kuang, J., The numerical stability of the θ-method for delay differential equations with many variable delays, J. comput. math., 17, 523-532, (1999) · Zbl 0942.65087
[18]	van den Heuvel, E.G., Using resolvent conditions to obtain new stability results for θ-methods for delay differential equations, IMA J. numer. anal., 21, 421-438, (2001) · Zbl 0976.65076
[19]	Vermiglio, R., On the stability of runge – kutta methods for delay integral equations, Numer. math., 61, 561-577, (1992) · Zbl 0723.65139
[20]	Watanabe, D.S.; Roth, M.G., The stability of difference formulas for delay differential equations, SIAM J. numer. anal., 22, 132-145, (1985) · Zbl 0571.65075
[21]	Zennaro, M., P-stability properties of runge – kutta methods for delay differential equations, Numer. math., 49, 305-318, (1986) · Zbl 0598.65056
[22]	Zennaro, M., Delay differential equations: theory and numerics, (), 291-333 · Zbl 0847.34072
[23]	Zubik-Kowal, B., Stability in the numerical solution of linear parabolic equations with a delay term, Bit, 41, 191-206, (2001) · Zbl 0984.65091