Berenguer, M. I.; Fortes, M. A.; Garralda Guillem, A. I.; Ruiz Galán, M. Linear Volterra integro-differential equation and Schauder bases. (English) Zbl 1068.65143 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) Cited in 19 Documents 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 PDF BibTeX XML Cite \textit{M. I. Berenguer} et al., Appl. Math. Comput. 159, No. 2, 495--507 (2004; Zbl 1068.65143) Full Text: DOI References: [1] Jamenson, G.J.O., Topology and normed spaces, (1974), Chapman & Hall London [2] Brunner, H., High-order methods for the numerical solution of Volterra integro-differential equations, J. comput. appl. math, 15, 301-309, (1986) · Zbl 0634.65143 [3] Brunner, H., A survey of recent advances in the numerical treatment of Volterra integral and integro-differential equations, J. comput. appl. math, 8, 3, 213-229, (1982) · Zbl 0485.65087 [4] Brunner, H.; van der Houwen, P.J., The numerical solution of Volterra equations, (1986), North-Holland Amsterdam · Zbl 0611.65092 [5] Brunner, H., The numerical treatment of Volterra integro-differential equations with unbounded delay, J. comput. appl. math, 28, 5-23, (1989) · Zbl 0687.65131 [6] Crisci, M.R.; Russo, E.; Vecchio, A., Time point relaxation methods for Volterra integro-differential equations, Comput. math. appl, 36, 9, 59-70, (1998) · Zbl 0933.65150 [7] Lin, T.; Lin, Y.; Rao, M.; Zhang, S., Petrov – galerkin methods for linear Volterra integro-differential equations, SIAM J. numer. anal, 38, 3, 937-963, (2000) · Zbl 0983.65138 [8] Semadeni, Z., Product Schauder bases and approximation with nodes in spaces of continuous functions, Bull. acad. polon. sci, 11, 387-391, (1963) · Zbl 0124.31703 [9] Gelbaum, B.; Gil de Lamadrid, J., Bases on tensor products of Banach spaces, Pacific J. math, 11, 1281-1286, (1961) · Zbl 0106.08604 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.