ISRN Computational Mathematics
Volume 2012 (2012), Article ID 139514, 5 pages
http://dx.doi.org/10.5402/2012/139514
Research Article
On Some Volterra and Fredholm Problems via the Unified Integrodifferential Quadrature Method
Laboratory of Applied Mathematics, Mohamed Khider University of Biskra, BP145, 07000 Biskra, Algeria
Received 6 September 2011; Accepted 20 October 2011
Academic Editors: V. Rai and K. Wang
Copyright © 2012 Abdelwahab Zerarka et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Linked References
- R. K. Miller and A. Feldstein, “Smoothness of solutions of Volterra integral equations with weakly singular kernels,” SIAM Journal on Mathematical Analysis, vol. 2, pp. 242–258, 1971. View at Google Scholar
- J. Norbury and A. M. Stuart, “Volterra integral equations and a new Gronwall inequality. I. The linear case,” Proceedings of the Royal Society of Edinburgh Section A, vol. 106, no. 3-4, pp. 361–373, 1987. View at Google Scholar
- J. Norbury and A. M. Stuart, “Volterra integral equations and a new Gronwall inequality. II. The nonlinear case,” Proceedings of the Royal Society of Edinburgh Section A, vol. 106, no. 3-4, pp. 375–384, 1987. View at Google Scholar
- A. Zerarka, S. Hassouni, H. Saidi, and Y. Boumedjane, “Energy spectra of the Schrödinger equation and the differential quadrature method,” Communications in Nonlinear Science and Numerical Simulation, vol. 10, no. 7, pp. 737–745, 2005. View at Publisher · View at Google Scholar
- H. Brunner and P. J. van der Houwen, The Numerical Solution of Volterra Equations, vol. 3 of CWI Monographs, North-Holland Publishing, Amsterdam, The Netherlands, 1986.
- C. T. H. Baker, The Numerical Treatment of Integral Equations, Clarendon Press, Oxford, UK, 1977.
- L. M. Delves and J. Walsh, Numerical Solution of Integral Equations, Clarendon Press, Oxford, UK, 1974.
- J. A. Cochran, The Analysis of Linear Integral Equations, McGraw-Hill, New York, NY, USA, 1972.
- K. E. Atkinson, A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 1976.
- P. H. M. Wolkenfelt, “The construction of reducible quadrature rules for Volterra integral and integro-differential equations,” IMA Journal of Numerical Analysis, vol. 2, no. 2, pp. 131–152, 1982. View at Publisher · View at Google Scholar
- H.-J. Dobner, “Bounds of high quality for first kind Volterra integral equations,” Reliable Computing, vol. 2, no. 1, pp. 35–45, 1996. View at Google Scholar
- D. Bugajewski, “On the Volterra integral equation in locally convex spaces,” Demonstratio Mathematica, vol. 25, no. 4, pp. 747–754, 1992. View at Google Scholar
- L. M. Delves and J. L. Mohamed, Computational Methods for Integral Equations, Cambridge University Press, Cambridge, UK, 1985. View at Publisher · View at Google Scholar
- H. Brunner, A. Pedas, and G. Vainikko, “The piecewise polynomial collocation method for nonlinear weakly singular Volterra equations,” Mathematics of Computation, vol. 68, no. 227, pp. 1079–1095, 1999. View at Publisher · View at Google Scholar
- J. P. Kauthen and H. Brunner, “Continuous collocation approximations to solutions of first kind Volterra equations,” Mathematics of Computation, vol. 66, no. 220, pp. 1441–1459, 1997. View at Publisher · View at Google Scholar
- Q. Hu, “Multilevel correction for discrete collocation solutions of Volterra integral equations with delay arguments,” Applied Numerical Mathematics, vol. 31, no. 2, pp. 159–171, 1999. View at Publisher · View at Google Scholar
- H. Brunner, Q. Hu, and Q. Lin, “Geometric meshes in collocation methods for Volterra integral equations with proportional delays,” IMA Journal of Numerical Analysis, vol. 21, no. 4, pp. 783–798, 2001. View at Publisher · View at Google Scholar
- V. Karlin, V. G. Maz'ya, A. B. Movchan, J. R. Willis, and R. Bullough, “Numerical solution of nonlinear hypersingular integral equations of the Peierls type in dislocation theory,” SIAM Journal on Applied Mathematics, vol. 60, no. 2, pp. 664–678, 2000. View at Publisher · View at Google Scholar
- M. H. Fahmy, M. A. Abdou, and M. A. Darwish, “Integral equations and potential-theoretic type integrals of orthogonal polynomials,” Journal of Computational and Applied Mathematics, vol. 106, no. 2, pp. 245–254, 1999. View at Publisher · View at Google Scholar
- T. Diogo and P. Lima, “Numerical solution of a nonuniquely solvable Volterra integral equation using extrapolation methods,” Journal of Computational and Applied Mathematics, vol. 140, no. 1-2, pp. 537–557, 2002. View at Publisher · View at Google Scholar
- F. Fagnani and L. Pandolfi, “A recursive algorithm for the approximate solution of Volterra integral equations of the first kind of convolution type,” Inverse Problems, vol. 19, no. 1, pp. 23–47, 2003. View at Publisher · View at Google Scholar
- P. Oja and D. Saveljeva, “Cubic spline collocation for Volterra integral equations,” Computing, vol. 69, no. 4, pp. 319–337, 2002. View at Publisher · View at Google Scholar
- I. Bock and J. Lovíšek, “On a reliable solution of a Volterra integral equation in a Hilbert space,” Applications of Mathematics, vol. 48, no. 6, pp. 469–486, 2003. View at Publisher · View at Google Scholar
- M. Federson, R. Bianconi, and L. Barbanti, “Linear Volterra integral equations,” Acta Mathematicae Applicatae Sinica, vol. 18, no. 4, pp. 553–560, 2002. View at Publisher · View at Google Scholar
- M. Federson, R. Bianconi, and L. Barbanti, “Linear Volterra integral equations as the limit of discrete systems,” Cadernos de Matemática, vol. 4, pp. 331–352, 2003. View at Google Scholar
- P. Linz, Analytical and Numerical Methods for Volterra Equations, vol. 7, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 1985.
- U. Kulisch and W. L. Miranker, A New Approach to Scientific Computation, Academic Press, New York, NY, USA, 1983.
- R. Hammer, M. Hocks, U. Kulisch, and D. Ratz, Numerical Toolbox for Verified Computing I, Springer, Berlin, Germany, 1995.
- D. Bugajewski, “On differential and integral equations in locally convex spaces,” Demonstratio Mathematica, vol. 28, no. 4, pp. 961–966, 1995. View at Google Scholar
- H.-J. Dobner, “Bounds for the solution of hyperbolic problems,” Computing, vol. 38, no. 3, pp. 209–218, 1987. View at Publisher · View at Google Scholar
- E. T. Copson, Partial Differential Equations, Cambridge University Press, Cambridge, UK, 1975.
- D. Bugajewska, Topological properties of solution sets of some problems for differential equations, Ph.D. thesis, Poznan, 1999.
- A. Constantin, “On the unicity of solutions for the differential equation x(n) = f(t, x),” Rendiconti del Circolo Matematico di Palermo Serie II, vol. 42, no. 1, pp. 59–64, 1993. View at Publisher · View at Google Scholar
- D. Bugajewski and S. Szufla, “Kneser's theorem for weak solutions of the Darboux problem in Banach spaces,” Nonlinear Analysis, Theory, Methods & Applications, vol. 20, no. 2, pp. 169–173, 1993. View at Publisher · View at Google Scholar
- M. Maron and R. Lopez, Numerical Analysis, Wadsworth Publishing Company, Belmont, Calif, USA, 1991.
- M. Hukuhara, “Théorèmes fondamentaux de la théorie des équations différentielles ordinaires dans l'espace vectoriel topologique,” Journal of the Faculty of Science, University of Tokyo, Section IA, vol. 8, pp. 111–138, 1959. View at Google Scholar
- R. K. Miller and A. Feldstein, “Smoothness of solutions of Volterra integral equations with weakly singular kernels,” SIAM Journal on Mathematical Analysis, vol. 2, pp. 242–258, 1971. View at Google Scholar
- G. Adomian, “Random Volterra integral equations,” Mathematical and Computer Modelling, vol. 22, no. 8, pp. 101–102, 1995. View at Publisher · View at Google Scholar
- H. Brunner, “Polynomial spline collocation methods for Volterra integrodifferential equations with weakly singular kernels,” IMA Journal of Numerical Analysis, vol. 6, no. 2, pp. 221–239, 1986. View at Publisher · View at Google Scholar
- E. Rawashdeh, D. McDowell, and L. Rakesh, “Polynomial spline collocation methods for second-order Volterra integrodifferential equations,” International Journal of Mathematics and Mathematical Sciences, vol. 2004, no. 56, pp. 3011–3022, 2004. View at Publisher · View at Google Scholar
- K. Al-Khaled and F. Allan, “Decomposition method for solving nonlinear integro-differential equations,” Journal of Applied Mathematics & Computing, vol. 19, no. 1-2, pp. 415–425, 2005. View at Publisher · View at Google Scholar
- A. Akyüz-Daşcıoğlu, “A Chebyshev polynomial approach for linear Fredholm-Volterra integro-differential equations in the most general form,” Applied Mathematics and Computation, vol. 181, no. 1, pp. 103–112, 2006. View at Publisher · View at Google Scholar