Palomares, A.; Pasadas, M.; Ramírez, V.; Ruiz Galán, M. A convergence result for a least-squares method using Schauder bases. (English) Zbl 1135.65326 Math. Comput. Simul. 77, No. 2-3, 274-281 (2008). Summary: We introduce a method, by using the least-squares method and a Schauder basis, which provides a numerical solution for a wide class of linear differential or integral equations. In addition, we give a convergence result and an application. Cited in 1 Document MSC: 65J10 Numerical solutions to equations with linear operators 47A50 Equations and inequalities involving linear operators, with vector unknowns 65R20 Numerical methods for integral equations 45D05 Volterra integral equations Keywords:least-squares; Schauder basis; Banach space; numerical example; linear operator equation; Volterra integral equation; convergence PDFBibTeX XMLCite \textit{A. Palomares} et al., Math. Comput. Simul. 77, No. 2--3, 274--281 (2008; Zbl 1135.65326) Full Text: DOI References: [1] Bedivan, D. M.; Fix, G. J., Analysis of finite element approximation and quadrature of Volterra integral equations, Numer. Methods Partial Differential Eq., 13, 663-672 (1997) · Zbl 0889.65138 [2] Bochev, P. B.; Gunzburger, M. D., Finite element methods of least-squares type, SIAM Rev., 40, 789-837 (1998) · Zbl 0914.65108 [3] Ciesielski, Z., A construction of basis in \(C^{(1)}(I^2)\), Stud. Math., 33, 243-247 (1969) · Zbl 0185.37601 [4] Ciesielski, Z.; Domsta, J., Construction of an orthonormal basis in \(C^m(I^d)\) and \(W_p^m(I^d)\), Stud. Math., 41, 211-224 (1972) · Zbl 0235.46047 [5] Lindenstrauss, J.; Tzafiri, L., Classical Banach Spaces I (1977), Springer-Verlag: Springer-Verlag Berlin [6] Megginson, R. E., An Introduction to Banach Space Theory (1998), Springer-Verlag: Springer-Verlag New York · Zbl 0910.46008 [7] Onumanyi, P.; Ortiz, E. L., Numerical solution of stiff and singularly perturbed boundary value problems with a segmented-adaptive formulation of the tau method, Math. Comp., 43, 189-203 (1984) · Zbl 0574.65091 [8] Ortiz, E. L., The tau method, SIAM J. Numer. Anal., 6, 480-492 (1969) · Zbl 0195.45701 [9] Palomares, A.; Pasadas, M.; Ramírez, V.; Ruiz Galán, M., Schauder bases in Banach spaces: application to numerical solutions of differential equations, Comput. Math. Appl., 44, 619-622 (2002) · Zbl 1035.65082 [10] Palomares, A.; Ruiz Galán, M., Isomorphisms, Schauder bases in Banach spaces, and numerical solution of integral and differential equations, Numer. Funct. Anal. Optimiz., 26, 129-137 (2005) · Zbl 1079.65067 [11] Schonefeld, S., Schauder bases in spaces of differentiable functions, Bull. Am. Math. Soc., 75, 586-590 (1969) · Zbl 0201.16101 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.