Recent advances in the numerical analysis of Volterra functional differential equations with variable delays. (English) Zbl 1170.65103
The author considers initial value problems of a linear (ordinary) delay differential equation involving Volterra integral operators, i.e., a Volterra functional integro-differential equation (VFIDE). Thereby, the delay is a predetermined nonlinear time-dependent function, which may be equal to zero at the beginning (vanishing delay). Omitting the integral operators, the linear delay differential equation reduces to the pantograph equation, [cf. M. Arnold and B. Simeon [Appl. Numer. Math. 34, 345–362 (2000; Zbl 0964.65101)].
Considering the VFIDE, a collocation method is constructed straightforward using piecewise polynomials of degree not exceeding \(m\) on a uniform mesh. In case of non-vanishing delay, previous work shows an optimal global (uniform) superconvergence of order \(m+1\), whereas the optimal local superconvergence exhibits the order \(2m\) in the collocation points. In case of vanishing delay, the author proves an optimal global (uniform) convergence of order \(m\) and an optimal local superconvergence of order \(m+2\) for \(m \geq 3\). The generalisation of the results to a VFIDE of second order is outlined.
Finally, the author specifies some open problems in this context, for example, the asymptotic behaviour of the exact solutions or of the numerical solutions for VFIDEs. Numerical results of test examples are not within the scope of the paper. Since the author wants to provide a survey on recent results, a huge number of references is given.
Considering the VFIDE, a collocation method is constructed straightforward using piecewise polynomials of degree not exceeding \(m\) on a uniform mesh. In case of non-vanishing delay, previous work shows an optimal global (uniform) superconvergence of order \(m+1\), whereas the optimal local superconvergence exhibits the order \(2m\) in the collocation points. In case of vanishing delay, the author proves an optimal global (uniform) convergence of order \(m\) and an optimal local superconvergence of order \(m+2\) for \(m \geq 3\). The generalisation of the results to a VFIDE of second order is outlined.
Finally, the author specifies some open problems in this context, for example, the asymptotic behaviour of the exact solutions or of the numerical solutions for VFIDEs. Numerical results of test examples are not within the scope of the paper. Since the author wants to provide a survey on recent results, a huge number of references is given.
Reviewer: Roland Pulch (Wuppertal)
MSC:
| 65R20 | Numerical methods for integral equations |
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 65L20 | Stability and convergence of numerical methods for ordinary differential equations |
| 65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |
| 65L70 | Error bounds for numerical methods for ordinary differential equations |
| 34K06 | Linear functional-differential equations |
| 34K28 | Numerical approximation of solutions of functional-differential equations (MSC2010) |
| 45J05 | Integro-ordinary differential equations |
| 45G10 | Other nonlinear integral equations |
Keywords:
Volterra functional integro-differential equations; delay differential equations; vanishing delays; Volterra integral operator; pantograph equation; collocation method; superconvergence; error analysis; uniform meshCitations:
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