Convergence of the sinc method applied to delay Volterra integral equations

Document Type : Research Paper

Authors

Department of Mathematics‎, ‎University of Mohagheghe Ardabili‎, ‎56199-11367‎, ‎Ardabil‎, ‎Iran.

Abstract

‎In this paper‎, ‎the numerical solutions of linear and nonlinear Volterra integral‎ ‎equations with nonvanishing delay are considered by two methods‎. ‎The methods are developed by means of‎ ‎the sinc approximation with the single exponential (SE) and double exponential (DE)‎ ‎transformations‎. ‎The existence and uniqueness of sinc-collocation solutions for these equations are provided‎. ‎These methods improve conventional results and achieve exponential convergence‎. ‎Numerical results are included to confirm the efficiency and accuracy of the methods.

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References

K.E. Atkinson and W. Han, Theoretical Numerical Analysis: A Functional Analysis Framework, Springer, New York, 2001.
K.E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge Univ. Press, 1997.
A. Bellour and M. Bousselsal, A Taylor collocation method for solving delay integral equations, Numer. Algorithms 65 (2014), no. 4, 843--857.
J.M. Bownds, J.M. Cushing and R. Schutte, Existence, uniqueness, and extendibility of solutions of Volterra integral systems with multiple, variable lags, Funkcial. Ekvac. 19 (1976), no. 2, 101--111.
H. Brunner, Collocation Methods for Volterra Integral and Related Functional Equations, Cambridge Univ. Press, 2004.
H. Brunner, The numerical analysis of functional integral and integro-differential equations of Volterra type, Acta Numer. 13 (2004) 55--145.
H. Brunner, Recent advances in the numerical analysis of Volterra functional differential equations with variable delays, J. Comput. Appl. Math. 228 (2009) 524--537.
H. Brunner and Q. Hu, Optimal superconvergence orders of iterated collocation solutions for Volterra integral equations with vanishing delays, SIAM J. Numer. Anal. 43 (2005), no. 5, 1934--1949.
H. Brunner, Q. Hu, and Q. Lin, Geometric meshes in collocation methods for Volterra integral equations with proportional delays, IMA J. Numer. Anal. 21 (2001), no. 4, 783--798.
H. Brunner, H. Xie, and R. Zhang, Analysis of collocation solutions for a class of functional equations with vanishing delays, IMA J. Numer. Anal. 31 (2011), no. 2, 698--718.
A.M. Denisov and A. Lorenzi, Existence results and regularisation techniques for severely ill-posed integrofunctional equations, Boll. Un. Mat. Ital. 11 (1997) 713--731.
S. Haber, Two formulas for numerical indefinite integration, Math. Comp. 60 (1993), no. 201, 279--296.
Q. Hu, Stieltjes derivatives and β-polynomial spline collocation for Volterra integro-differential equations with singularities, SIAM J. Numer. Anal. 33 (1996), no. 1, 208--220.
T. Okayama, T. Matsuo and M. Sugihara, Error estimates with explicit constants for Sinc approximation, Sinc quadrature and Sinc indefinite integration, Numer. Math. 124 (2013), no. 2, 361--394.
F. Stenger, Numerical Methods Based on Sinc and Analytic Functions, Springer-Verlag, New York, 1993.
F. Stenger, Handbook of Sinc Numerical Methods, CRC Press, Boca Raton, 2011.
M. Zarebnia and J. Rashidinia, Convergence of the Sinc method applied to Volterra integral equations, Appl. Appl. Math. 5 (2010), no. 1, 198--216.
W. Zhong-Qing and S. Chang-Tao, hp-spectral collocation method for nonlinear Volterra integral equations with vanishing variable delays, Math. Comp. 85 (2016), no. 298, 635--666.
Bulletin of the Iranian Mathematical Society
Volume 43, Issue 5
October 2017
Pages 1357-1375

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History

  • Receive Date: 16 June 2015
  • Revise Date: 01 June 2016
  • Accept Date: 10 June 2016

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