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High-order collocation methods for singular Volterra functional equations of neutral type. (English) Zbl 1119.65121

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

d dt[y(t)-(𝒯 θ y)(t)]=F(t,y(t),y(θ(t)),(𝒱y)(t)),tI:=[t 0 ,T],y(t)=ϕ(t),tt 0 ·(1)

Here, θ(t):=t-τ(t) (τ(t)>0) is a delay function, and the (nonlinear) operator 𝒯 θ is either the Nemytskij operator (or: substitution operator) 𝒩 θ with delay, 𝒩 θ f(t):=G(t,f(θ(0)), tI, or the weakly singular delay Volterra integral operator 𝒱 θ,α ,

(𝒱 θ,α f)(t):= t 0 t k α (t-s)G(s,f(θ(s)))ds,

with kernel k α given by

k α (s-t)=k 0 (t-s),α=0,λ·(t-s) -α ,0<α<1,λ·log(t-s),α=1·

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 𝒱 of the form (𝒱y)(t):= 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.

MSC:
65R20Integral equations (numerical methods)
65-06Proceedings of conferences (numerical analysis)
65L20Stability and convergence of numerical methods for ODE
65L80Numerical methods for differential-algebraic equations
34K28Numerical approximation of solutions of functional-differential equations
47H30Particular nonlinear operators
45G10Nonsingular nonlinear integral equations
45J05Integro-ordinary differential equations
34K40Neutral functional-differential equations