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Zbl 1119.65121
Brunner, Hermann
High-order collocation methods for singular Volterra functional equations of neutral type. (English)
[J] Appl. Numer. Math. 57, No. 5-7, 533-548 (2007). ISSN 0168-9274

The author presents a survey of recent results on the attainable order of (super-)convergence of collocation solutions for systems of Volterra functional integro-differential equations of the form $$ \aligned &\frac{d}{dt}[y(t) - (\Cal T_\theta y)(t)] = F(t, y(t), y(\theta (t)),(\Cal Vy)(t)),\quad t\in I:= [t_0, T],\\ &y(t) = \phi(t),\quad t\leqslant t_0. \endaligned\tag1 $$ Here, $\theta(t) :=t -\tau(t)$ $(\tau(t) > 0)$ is a delay function, and the (nonlinear) operator $\Cal T_\theta$ is either the Nemytskij operator (or: substitution operator) $\Cal N_\theta$ with delay, $\Cal N_\theta f(t):=G(t,f(\theta(0))$, $t\in I$, or the weakly singular delay Volterra integral operator $\Cal V_{\theta, \alpha}$, $$(\Cal V_{\theta, \alpha}f)(t):=\int_{t_0}^tk_\alpha(t-s)G(s,f(\theta(s)))\, ds,$$ with kernel $k_\alpha$ given by $$ k_\alpha(s-t)=\left\{ \aligned &k_0(t-s),\quad \alpha=0,\\ &\lambda\cdot(t-s)^{-\alpha},\quad0<\alpha<1,\\ &\lambda\cdot\log(t-s),\quad\alpha=1. \endaligned \right. $$ While the right-hand side $F$ in (1) could also depend on more general (non-Hammerstein) operators, including delay operators, the author restricts the analysis to Volterra integral operators $\Cal V$ of the form $(\Cal V y)(t):=\int_{t_0}^t K(t,s)Q(s,y(s))\,ds$. The functions $F$, $G$, $k_0$, $K$ and $Q$ are assumed to be smooth on their respective domains. Related functional equations and theoretical and computational aspects of collocation methods for their solution are described.
[Nikolay Yakovlevich Tikhonenko (Odessa)]

MSC 2000:: *65R20 Integral equations (numerical methods)
65-06 Proceedings of conferences (numerical analysis)
65L20 Stability of numerical methods for ODE
65L80 Methods for differential-algebraic equations
34K28 Numerical approximation of solutions of FDE
47H30 Particular nonlinear operators
45G10 Nonsingular nonlinear integral equations
45J05 Integro-ordinary differential equations
34K40 Neutral equations

Keywords: Volterra functional integro-differential equations; neutral equations; weakly singular kernels; variable delays; collocation; piecewise polynomial; vanishing proportional delays; state-dependent delays; convergence

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Zentralblatt MATH Berlin [Germany]