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

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.

### MSC:

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

Zbl 1002.65148
Full Text:

### References:

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