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1. 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
2. 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
3. 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
4. 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
5. 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.
6. C. T. H. Baker, The Numerical Treatment of Integral Equations, Clarendon Press, Oxford, UK, 1977.
7. L. M. Delves and J. Walsh, Numerical Solution of Integral Equations, Clarendon Press, Oxford, UK, 1974.
8. J. A. Cochran, The Analysis of Linear Integral Equations, McGraw-Hill, New York, NY, USA, 1972.
9. 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.
10. 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
11. 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
12. 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
13. 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
14. 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
15. 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
16. 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
17. 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
18. 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
19. 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
20. 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
21. 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
22. 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
23. 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
24. 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
25. 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
26. P. Linz, Analytical and Numerical Methods for Volterra Equations, vol. 7, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 1985.
27. U. Kulisch and W. L. Miranker, A New Approach to Scientific Computation, Academic Press, New York, NY, USA, 1983.
28. R. Hammer, M. Hocks, U. Kulisch, and D. Ratz, Numerical Toolbox for Verified Computing I, Springer, Berlin, Germany, 1995.
29. 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
30. 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
31. E. T. Copson, Partial Differential Equations, Cambridge University Press, Cambridge, UK, 1975.
32. D. Bugajewska, Topological properties of solution sets of some problems for differential equations, Ph.D. thesis, Poznan, 1999.
33. 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
34. 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
35. M. Maron and R. Lopez, Numerical Analysis, Wadsworth Publishing Company, Belmont, Calif, USA, 1991.
36. 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
37. 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
38. 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
39. 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
40. 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
41. 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
42. 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