Smoothness of Solutions of Volterra Integral Equations with Weakly Singular Kernels

Runge-Kutta theory for Volterra and Abel integral equations of the second kind

“…Now the value for a42 follows from (viii), and (v) gives a43. Finally, the coefficients a2x,... ,a5] are obtained from (9). By Theorem 10 the method has the asserted local order.…”

“…It is a straightforward extension of a result in Miller and Feldstein [9], and we state it without proof. Theorem 1.…”

“…The structure of these singularities is well understood for the special case a = -{-(see Miller and Feldstein [9], de Hoog and Weiss [7]). The following result characterizes the behavior of the solution near 0 for arbitrary a > -1.…”

Almost-periodic behavior of solutions of a nonlinear Volterra system.

“…The conditions F t C2[0, L] and (2.1) are sufficient to insure that / e C[0, °°) C\0, 00) and that f is locally of class L1 on 0 ^ t < °o. Since gi and g2 e C1, then it follows from results in [11] that xx(t) and x2(t) have these same smoothness properties, that is x(t) e C[0, °o) C\ C^O, «=) and x'it) t L1 near t = 0.…”

The Numerical Solution of Weakly Singular Volterra Integral Equations by Collocation on Graded Meshes

“…It is well-known that, were the exact solution y of (1.1) (or (1.2)) in Cm{I), then we would obtain, for a uniform mesh (where h" = h = TN-1), (1.12) \\y-u\\oo = 0(N-m). Unfortunately, smooth g and k (or K) in (1.1) (or in (1.2)) lead, for 0 < a < 1, to an exact solution y which behaves like y{t) = &{tl~a) near t = 0; it has thus unbounded derivatives at t = 0 (compare [16], [12], [14], [3]). As a consequence, the collocation approximation u e S^~}\{ZN) given by (1.8), with the underlying mesh being the uniform one, satisfies only (1.13) \\y-u\\x = cV{N^-"y), and this order is best possible for any m > 1.…”