Collocation methods with tension in spline for singularly perturbed Volterra integro-differential equations
by
Vilmos Horvat

The collocation methods with spline in tension are considered to singularly perturbed Volterra integro-differential equations (VIDE)

\epsilony'(t)=f(t, y(t))+ ó õ t 0  k(t, s, y(s))ds, t in I:=[0, T]

with y(0)=y₀, where \epsilon is a small parametar satisfying 0 < varepsilon << 1 and where f:I ×\R --> \R and k:S ×\R --> \R (with S={(t, s):0 <= s <= t <= T}) are given functions. Under appropriate conditions on f and k for every \epsilon > 0 the above equation have a unique continuous solution on [0, T]. The well known polynomial collocation methods are adequate only for \epsilon \approx 1. A new collocation method is developed to approximate the solution of singularly perturbed VIDE for 0 < \epsilon << 1. The convergence results of the new method are given and the efficiency of the new method is illustrated by a few examples.

Date received: March 18, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caex-37.