High-Order Methods for Weakly Singular Volterra Integro-Differential Equations

Abstract

Let $\mathcal{R} = ( - \infty ,\infty ),\,\,b > 0,\,\Delta _b = \{ (t,\,s)\,\, \in \,\mathcal{Z}^2 \,:\,0\, \leq \,t\, \leq \,b,\,0\, \leq \,s\, < \,t\} ,\,\overline \Delta _b = \{ (t,s) \in \mathcal{R}^2 :0 \leq s\, \leq \,t \leq \,b\} .$ We consider a linear integro-differential equation of the form

$$y^\prime(t) = p(t)y(t) + q(t) + \int\limits^t_0 K_0(t, s)y(s)ds + \int\limits^t_0 K_1(t, s)y^\prime(s)ds, \quad 0 \leq t \leq b,$$

(14.1)

with given initial condition

$$y(0) = y_0, \quad y_0 \in \mathcal{R}$$

(14.2)

We assume that $K_0, K_1 \in W^{m,\nu} (\Delta_b), p, \ q, \ \in C^{m, \nu}(0, b], m\ \in \mathbb{N} = \{1, 2, \ldots\}, \ \nu \in \mathcal{R}, \ \nu < 1.$

For given $$$$ and ?∞ < ν < 1 we define W ^m,ν(Δ_b) as the set of all m-times continuously differentiable functions $K : \Delta_b \to \mathcal{R}$ satisfying

$$\left|\left(\frac{\partial}{\partial t}\right)^i \left(\frac{\partial}{\partial t} + \frac{\partial}{\partial s}\right)^j K(t, s)\right|\leq c\begin{cases}1 \qquad\qquad\qquad \quad{\rm if} \ \ \nu + i < 0,\\ 1 + |\log(t - s)| \quad {\rm if} \ \ \nu + i = 0,\\ (t - s)^{-\nu-i} \ \ \quad\quad {\rm if} \ \ \nu + i > 0,\end{cases}$$

(14.3)

with a constant c = c(K) for all (t, s) ∈ Δ_b and all nonnegative integers i and j such that i + j ? m.