Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-26T05:12:57.459Z Has data issue: false hasContentIssue false
  • English
  • Français

Collocation Methods for A Class of Volterra Integral Functional Equations with Multiple Proportional Delays

Published online by Cambridge University Press:  03 June 2015

Kai Zhang  and
Jie Li
Kai Zhang*
Affiliation:
Department of Mathematics, Jilin University, Changchun, Jilin 130023, China
Jie Li*
Affiliation:
Department of Mathematics, Jilin University, Changchun, Jilin 130023, China
*
Corresponding author. Email: [email protected]
Email: [email protected]
Get access

Abstract

In this paper, we apply the collocation methods to a class of Volterra integral functional equations with multiple proportional delays (VIFEMPDs). We shall present the existence, uniqueness and regularity properties of analytic solutions for this type of equations, and then analyze the convergence orders of the collocation solutions and give corresponding error estimates. The numerical results verify our theoretical analysis.

Keywords

65R2034K0634K28Volterra integral functional equationmultiple proportional delayscollocation method
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 4 , Issue 5 , October 2012 , pp. 575 - 602
Copyright
Copyright © Global-Science Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Ali, I., Brunner, H. and Tang, T., Spectral methods for pantograph-type differential and integral equations with multiple delays, Front. Math. China., 4 (2009), pp. 49–61.CrossRefGoogle Scholar
[2]Atkinson, E. K., The Numerical Solution of Integral Equations of the Second Kind, Cambridge: Cambridge University Press, 1997.CrossRefGoogle Scholar
[3]Bedivan, D. M. and Fix, G. J., Finite element approximation of Volterra integral equations, Stability. Control. Theory. Methods. Appl., 10 (2000), pp. 141–147.Google Scholar
[4]Bellen, A. and Zennaro, M., Numerical Methods for Delay Differential Equations, Cambridge: Oxford University Press, 2003.CrossRefGoogle Scholar
[5]Brunner, H., On the discretization of differential and Volterra integral equations with variable delay, BIT., 37 (1997), pp. 1–12.CrossRefGoogle Scholar
[6]Brunner, H., Collocation Methods for Volterra Integral and Related Functional Differential Equations, Cambridge: Cambridge University Press, 2004.CrossRefGoogle Scholar
[7]Brunner, H., Collocation methods for pantograph-type Volterra functional equations with multiple delays, Comput. Methods. Appl. Math., 8 (2008), pp. 207–222.CrossRefGoogle Scholar
[8]Brunner, H., Davies, P. J. and Duncan, D. B., Discontinuous Galerkin approximations for Volterra integral equations of the first kind, IMA J. Numer. Anal., 29 (2009), pp. 856–881.CrossRefGoogle Scholar
[9]Brunner, H., Hu, Q. Y. and Lin, Q., Geometric meshes in collocation methods for Volterra integral equations with proportional delays, IMA J. Numer. Anal., 21 (2001), pp. 783–798.CrossRefGoogle Scholar
[10]Brunner, H., Xie, H. H. and Zhang, R., Analysis of collocation solutions for a class of functional equations with vanishing delays, IMA J. Numer. Anal., 31 (2011), pp. 698–718.CrossRefGoogle Scholar
[11]Chen, Y. P. and Tang, T., Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel, Math. Comput., 79 (2010), pp. 147–167.CrossRefGoogle Scholar
[12]Enright, W. H. and Hu, M., Interpolating Runge-Kutta methods for vanishing delay differential equations, Computing., 55 (1995), pp. 223–236.CrossRefGoogle Scholar
[13]Hu, Q. Y., Multilevel correction for discrete collocation solutions of Volterra integral equations with delay arguments, Appl. Numer. Math., 31 (1999), pp. 159–170.CrossRefGoogle Scholar
[14]Iserles, A., On the generalized pantograph functional differential equation, Euro. J. Appl. Math., 4 (1993), pp. 1–38.CrossRefGoogle Scholar
[15]Koshy, T., Catalan Numbers with Applications, Oxford University Press, 2008.CrossRefGoogle Scholar
[16]Norio, T., Yoshiaki, M. and Emiko, I., On the attainable order of collocation methods for delay differential equations with proportional delay, BIT., 40 (2000), pp. 374–394.Google Scholar
[17]Zhang, C. J. and Liao, X. X., Stability of BDF methods for nonlinear Volterra integral equations with delay, Comput. Math. Appl., 43 (2002), pp. 95–102.CrossRefGoogle Scholar