Linear Volterra integro-differential equation and Schauder bases. (English) Zbl 1068.65143

Berenguer, M. I.; Fortes, M. A.; Garralda Guillem, A. I.; Ruiz Galán, M.

Appl. Math. Comput. 159, No. 2, 495-507 (2004).

Upon rewriting the initial-value problem for the linear Volterra integro-differential equation \[ y'(t)= q(t)+ p(t) y(t)+ q(t)+ \int^t_0 K(t, s)y(s)\,ds,\quad t\in [0,1], \] as a second-kind Volterra integral equation, the authors show that instead of resorting to the Banach fixed-point theorem for generating its solution \(y\), the use of Schauder bases for the spaces \(C([0, 1])\) and \(C([0, 1]^2)\) yields computable approximations for \(y\). A simple example illustrates the convergence analysis. No comparison with other global numerical methods (e.g. collocation) is given.

Reviewer: Hermann Brunner (St. John’s)

MSC:

65R20	Numerical methods for integral equations
45J05	Integro-ordinary differential equations

Keywords:

Schauder bases; fixed point theorem; initial-value problem; linear Volterra integro-differential equation; convergence

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