Applied Numerical Mathematics 62 (2012) 297–304 Contents lists available at ScienceDirect Applied Numerical Mathematics www.elsevier.com/locate/apnum A sequential approach for solving the Fredholm integro-differential equation M.I. Berenguer, M.V. Fernández Muñoz ∗ , A.I. Garralda-Guillem, M. Ruiz Galán Universidad de Granada, Dpto. Matemática Aplicada, E.T.S. Ingeniería de Edificación, c/ Severo Ochoa s/n, 18071 Granada, Spain article info abstract Article history: Received 24 June 2010 Received in revised form 6 December 2010 Accepted 22 March 2011 Available online 29 March 2011 Keywords: Fredholm integro-differential equation Numerical methods Schauder bases A numerical approximation method for the solution of Fredholm integro-differential equations is presented. The method provides a sequential solution and makes use of appropriate Schauder bases in adequate Banach spaces of continuous functions as well as of classical fixed-point results. The method is computationally attractive and some numerical examples are provided to illustrate its high accuracy.  2011 IMACS. Published by Elsevier B.V. All rights reserved. 1. Introduction Several numerical methods for approximating integro-differential equations are known, since such equations model a wide class of applied problems (see for instance [6,10,11]). These methods often transform an integro-differential equation into a linear or nonlinear system of algebraic equations that can be solved by direct or iterative methods. See [3,9,14] and the references therein for a more general view of the classical methods and some recent advances. This paper considers the specific case of the nonlinear Fredholm integro-differential equation: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ y ′ (x) = g (x) + b  a G ( x, t , y(t ) ) dt ( x ∈[a, b] ) , y(a) = y 0 (1) where y 0 ∈ R and g :[a, b]→ R and G :[a, b]×[a, b]× R → R are continuous functions. The linear case (G(x, t , u) = h(x, t )u for certain continuous function h :[a, b]×[a, b]→ R) is an obvious particular case of the equation above. Our starting point is the formulation of the Fredholm integro-differential equation (1) in terms of an operator T . By defining within the Banach space C([a, b]) of those continuous and real valued functions defined on [a, b] (usual sup-norm) the integral operator T : C([a, b]) → C([a, b]) for each y ∈ C([a, b]) as T ( y)(x) = y 0 + x  a g (s) ds + x  a b  a G ( s, t , y(t ) ) ds dt ( x ∈[a, b] ) , (2) * Corresponding author. E-mail addresses: maribel@ugr.es (M.I. Berenguer), mvfm@ugr.es (M.V. Fernández Muñoz), agarral@ugr.es (A.I. Garralda-Guillem), mruizg@ugr.es (M. Ruiz Galán). 0168-9274/$30.00  2011 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.apnum.2011.03.009