Theoretical analysis of Sinc-Nyström methods for Volterra integral equations

Okayama, Tomoaki; Matsuo, Takayasu; Sugihara, Masaaki

doi:10.1090/S0025-5718-2014-02929-3

Theoretical analysis of Sinc-Nyström methods for Volterra integral equations
HTML articles powered by AMS MathViewer

by Tomoaki Okayama, Takayasu Matsuo and Masaaki Sugihara;

Math. Comp. 84 (2015), 1189-1215

DOI: https://doi.org/10.1090/S0025-5718-2014-02929-3

Published electronically: December 30, 2014

PDF | Request permission

Abstract:

In this paper, we present three theoretical results on Sinc-Nyström methods for Volterra integral equations of the first and second kind, which were proposed by Muhammad et al. Their methods involve the following issues: 1) it is difficult to determine the tuning parameter unless the solution is given, and 2) convergence has not been proved in a precise sense. In a mathematically rigorous manner, we present an implementable way to estimate the tuning parameter and a rigorous proof of the convergence with its rate explicitly revealed. Furthermore, we show that the resulting system is well conditioned. Numerical examples that support the theoretical results are also presented.

References

Philip M. Anselone, Collectively compact operator approximation theory and applications to integral equations, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1971. With an appendix by Joel Davis. MR 443383
Kendall E. Atkinson, The numerical solution of integral equations of the second kind, Cambridge Monographs on Applied and Computational Mathematics, vol. 4, Cambridge University Press, Cambridge, 1997. MR 1464941, DOI 10.1017/CBO9780511626340
Richard Bellman and Kenneth L. Cooke, Differential-difference equations, Academic Press, New York-London, 1963. MR 147745
Fred Brauer and Carlos Castillo-Chávez, Mathematical models in population biology and epidemiology, Texts in Applied Mathematics, vol. 40, Springer-Verlag, New York, 2001. MR 1822695, DOI 10.1007/978-1-4757-3516-1
Hermann Brunner, Collocation methods for Volterra integral and related functional differential equations, Cambridge Monographs on Applied and Computational Mathematics, vol. 15, Cambridge University Press, Cambridge, 2004. MR 2128285, DOI 10.1017/CBO9780511543234
Seymour Haber, Two formulas for numerical indefinite integration, Math. Comp. 60 (1993), no. 201, 279–296. MR 1149292, DOI 10.1090/S0025-5718-1993-1149292-9
S. I. Kabanikhin and A. Lorenzi, Identification problems of wave phenomena, Inverse and Ill-posed Problems Series, VSP, Utrecht, 1999. Theory and numerics. MR 1741782, DOI 10.1515/9783110943290
Rainer Kress, Linear integral equations, 2nd ed., Applied Mathematical Sciences, vol. 82, Springer-Verlag, New York, 1999. MR 1723850, DOI 10.1007/978-1-4612-0559-3
Prem K. Kythe and Pratap Puri, Computational methods for linear integral equations, Birkhäuser Boston, Inc., Boston, MA, 2002. MR 1902732, DOI 10.1007/978-1-4612-0101-4
Patricia K. Lamm, A survey of regularization methods for first-kind Volterra equations, Surveys on solution methods for inverse problems, Springer, Vienna, 2000, pp. 53–82. MR 1766739
Peter Linz, Analytical and numerical methods for Volterra equations, SIAM Studies in Applied Mathematics, vol. 7, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1985. MR 796318, DOI 10.1137/1.9781611970852
Richard K. Miller and Andreas Unterreiter, Switching behavior of PN-diodes: Volterra integral equation models, J. Integral Equations Appl. 4 (1992), no. 2, 257–272. MR 1172892, DOI 10.1216/jiea/1181075684
Mayinur Muhammad and Masatake Mori, Double exponential formulas for numerical indefinite integration, J. Comput. Appl. Math. 161 (2003), no. 2, 431–448. MR 2017024, DOI 10.1016/j.cam.2003.05.002
Mayinur Muhammad, Ahniyaz Nurmuhammad, Masatake Mori, and Masaaki Sugihara, Numerical solution of integral equations by means of the Sinc collocation method based on the double exponential transformation, J. Comput. Appl. Math. 177 (2005), no. 2, 269–286. MR 2125318, DOI 10.1016/j.cam.2004.09.019
Tomoaki Okayama, Takayasu Matsuo, and Masaaki Sugihara, Sinc-collocation methods for weakly singular Fredholm integral equations of the second kind, J. Comput. Appl. Math. 234 (2010), no. 4, 1211–1227. MR 2609575, DOI 10.1016/j.cam.2009.07.049
Tomoaki Okayama, Takayasu Matsuo, and Masaaki Sugihara, Improvement of a sinc-collocation method for Fredholm integral equations of the second kind, BIT 51 (2011), no. 2, 339–366. MR 2806534, DOI 10.1007/s10543-010-0289-x
Tomoaki Okayama, Takayasu Matsuo, and Masaaki Sugihara, Error estimates with explicit constants for Sinc approximation, Sinc quadrature and Sinc indefinite integration, Numer. Math. 124 (2013), no. 2, 361–394. MR 3054356, DOI 10.1007/s00211-013-0515-y
Andrei D. Polyanin and Alexander V. Manzhirov, Handbook of integral equations, 2nd ed., Chapman & Hall/CRC, Boca Raton, FL, 2008. MR 2404728, DOI 10.1201/9781420010558
J. Rashidinia and M. Zarebnia, Solution of a Volterra integral equation by the sinc-collocation method, J. Comput. Appl. Math. 206 (2007), no. 2, 801–813. MR 2333714, DOI 10.1016/j.cam.2006.08.036
Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
Frank Stenger, Numerical methods based on sinc and analytic functions, Springer Series in Computational Mathematics, vol. 20, Springer-Verlag, New York, 1993. MR 1226236, DOI 10.1007/978-1-4612-2706-9
Frank Stenger, Summary of Sinc numerical methods, J. Comput. Appl. Math. 121 (2000), no. 1-2, 379–420. Numerical analysis in the 20th century, Vol. I, Approximation theory. MR 1780056, DOI 10.1016/S0377-0427(00)00348-4
Ken’ichiro Tanaka, Masaaki Sugihara, and Kazuo Murota, Function classes for successful DE-Sinc approximations, Math. Comp. 78 (2009), no. 267, 1553–1571. MR 2501063, DOI 10.1090/S0025-5718-08-02196-0