Abstract and Applied Analysis

Analytical Techniques for a Numerical Solution of the Linear Volterra Integral Equation of the Second Kind

M. I. Berenguer, D. Gámez, A. I. Garralda-Guillem, M. Ruiz Galán, and M. C. Serrano Pérez

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Abstract

In this work we use analytical tools—Schauder bases and Geometric Series theorem—in order to develop a new method for the numerical resolution of the linear Volterra integral equation of the second kind.

Article information

Source
Abstr. Appl. Anal., Volume 2009 (2009), Article ID 149367, 12 pages.

Dates
First available in Project Euclid: 16 March 2010

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1268745615

Digital Object Identifier
doi:10.1155/2009/149367

Mathematical Reviews number (MathSciNet)
MR2564002

Zentralblatt MATH identifier
1187.65140

Citation

Berenguer, M. I.; Gámez, D.; Garralda-Guillem, A. I.; Ruiz Galán, M.; Serrano Pérez, M. C. Analytical Techniques for a Numerical Solution of the Linear Volterra Integral Equation of the Second Kind. Abstr. Appl. Anal. 2009 (2009), Article ID 149367, 12 pages. doi:10.1155/2009/149367. https://projecteuclid.org/euclid.aaa/1268745615


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References

  • C. T. H. Baker, The Numerical Treatment of Integral Equations, Monographs on Numerical Analysis, Clarendon Press, Oxford, UK, 1977.
  • C. T. H. Baker, ``A perspective on the numerical treatment of Volterra equations,'' Journal of Computational and Applied Mathematics, vol. 125, no. 1-2, pp. 217--249, 2000.
  • L. M. Delves and J. L. Mohamed, Computational Methods for Integral Equations, Cambridge University Press, Cambridge, UK, 1985.
  • Z. Wan, Y. Chen, and Y. Huang, ``Legendre spectral Galerkin method for second-kind Volterra integral equations,'' Frontiers of Mathematics in China, vol. 4, no. 1, pp. 181--193, 2009.
  • M. I. Berenguer, M. A. Fortes, A. I. Garralda Guillem, and M. Ruiz Galán, ``Linear Volterra integro-differential equation and Schauder bases,'' Applied Mathematics and Computation, vol. 159, no. 2, pp. 495--507, 2004.
  • E. Castro, D. Gámez, A. I. Garralda Guillem, and M. Ruiz Galán, ``High order linear initial-value problems and Schauder bases,'' Applied Mathematical Modelling, vol. 31, no. 12, pp. 2629--2638, 2007.
  • D. Gámez, A. I. Garralda Guillem, and M. Ruiz Galán, ``High-order nonlinear initial-value problems countably determined,'' Journal of Computational and Applied Mathematics, vol. 228, no. 1, pp. 77--82, 2009.
  • D. Gámez, A. I. Garralda Guillem, and M. Ruiz Galán, ``Nonlinear initial-value problems and Schauder bases,'' Nonlinear Analysis: Theory, Methods & Applications, vol. 63, no. 1, pp. 97--105, 2005.
  • A. Palomares, M. Pasadas, V. Ramírez, and M. Ruiz Galán, ``A convergence result for a least-squares method using Schauder bases,'' Mathematics and Computers in Simulation, vol. 77, no. 2-3, pp. 274--281, 2008.
  • A. Palomares and M. Ruiz Galán, ``Isomorphisms, Schauder bases in Banach spaces, and numerical solution of integral and differential equations,'' Numerical Functional Analysis and Optimization, vol. 26, no. 1, pp. 129--137, 2005.
  • H. Brunner and P. J. van der Houwen, The Numerical Solution of Volterra Equations, vol. 3 of CWI Monographs, North-Holland, Amsterdam, The Netherlands, 1986.
  • H. Brunner, L. Qun, and Y. Ningning, ``The iterative correction method for Volterra integral equations,'' BIT. Numerical Mathematics, vol. 36, no. 2, pp. 221--228, 1996.
  • K. Maleknejad and N. Aghazadeh, ``Numerical solution of Volterra integral equations of the second kind with convolution kernel by using Taylor-series expansion method,'' Applied Mathematics and Computation, vol. 161, no. 3, pp. 915--922, 2005.
  • K. Atkinson and W. Han, Theoretical Numerical Analysis: A Functional Analysis Framework, vol. 39 of Texts in Applied Mathematics, Springer, Dordrecht, The Netherlands, 3rd edition, 2009.
  • Z. Semadeni, ``Product Schauder bases and approximation with nodes in spaces of continuous functions,'' Bulletin de l'Académie Polonaise des Sciences, vol. 11, pp. 387--391, 1963.
  • B. R. Gelbaum and J. Gil de Lamadrid, ``Bases of tensor products of Banach spaces,'' Pacific Journal of Mathematics, vol. 11, pp. 1281--1286, 1961.