Skip to main content
SpringerLink
Log in

On the numerical solution of nonlinear systems of Volterra integro-differential equations with delay arguments

  • Published:
Computing Aims and scope Submit manuscript

Abstract

Particular cases of nonlinear systems of delay Volterra integro-differential equations (denoted by DVIDEs) with constant delay τ > 0, arise in mathematical modelling of ‘predator–prey’ dynamics in Ecology. In this paper, we give an analysis of the global convergence and local superconvergence properties of piecewise polynomial collocation for systems of this type. Then, from the perspective of applied mathematics, we consider the Volterra’s integro-differential system of ‘predator–prey’ dynamics arising in Ecology. We analyze the numerical issues of the introduced collocation method applied to the ‘predator–prey’ system and confirm that we can achieve the expected theoretical orders of convergence.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Volterra V (1927) Variazioni e fluttuazioni del numero d’individui in specie animali conviventi. Memorie del R. Comitato talassografico italiano, Mem. CXXXI

  2. Cushing JM (1977) Integrodifferential equations and delay models in population dynamics. Lecture notes in biomathematics, vol 20. Springer, Berlin

    Google Scholar 

  3. Volterra V (1939) The general equations of biological strife in the case of historical actions. Proc Edinburgh Math Soc 2: 4–10

    Article  Google Scholar 

  4. Bocharov GA, Rihan FA (2000) Numerical modelling in biosciences using delay differential equations. J Comput Appl Math 125: 183–199

    Article  MATH  MathSciNet  Google Scholar 

  5. Gopalsamy K (1992) Stability and oscillation in delay differential equations of population dynamics. Kluwer, Boston

    Google Scholar 

  6. Kuang Y (1993) Delay differential equations with applications in population dynamic. Academic Press, San Diego, CA

    Google Scholar 

  7. Baker CTH, Ford NJ (1990) Asymptotic error expansions for linear multistep methods for a class of delay integro-differential equations. Bull Greek Math Soc 31: 5–10

    MATH  MathSciNet  Google Scholar 

  8. Enright WH, Hu M (1997) Continuous Runge–Kutta methods for neutral Volterra integro-differential equations with delay. Appl Numer Math 24: 175–190

    Article  MATH  MathSciNet  Google Scholar 

  9. Brunner H (1988) The approximate solution of initial- value problems for general Volterra integro-differential equations. Computing 40: 125–137

    Article  MATH  MathSciNet  Google Scholar 

  10. Makroglou A (1983) A bloc-by-block method for the numerical solution of Volterra delay integro-differential equation. Computing 30: 49–62

    Article  MATH  MathSciNet  Google Scholar 

  11. Brunner H (1989) Collocation methods for nonlinear Volterra Integro-differential equations with infinite delay. Math Comp 53: 571–587

    Article  MATH  MathSciNet  Google Scholar 

  12. Brunner H (1994) The numerical solution of neutral Volterra integro-differential equations with delay arguments. Ann Numer Math 1: 309–322

    MATH  MathSciNet  Google Scholar 

  13. Brunner H, Zhang W (1999) Primary discontinuities in solutions for delay integro-differential equations. Methods Appl Anal 6: 525–534

    MATH  MathSciNet  Google Scholar 

  14. Baker CTH, Willé D (2000) On the propagation of derivative discontinuities in Volterra retarded integro-differential equations. N Z J Math 29: 103–113

    MATH  Google Scholar 

  15. Brunner H (1998) The use of splines in the numerical solution of Volterra integral and integro-differential equations. In: Dubuc S (ed) Splines and the theory of wavelets I, II, CRM Proceedings and Lecture Notes. American Mathematical Society, Providence, RI

  16. Brunner H (1984) Implicit Runge–Kutta methods of optimal order for volterra integro-differential equations. Math Comp 42: 95–109

    Article  MATH  MathSciNet  Google Scholar 

  17. Atkinson KE (1989) Introduction to numerical analysis, 2nd edn. Wiley, New York

    MATH  Google Scholar 

  18. McKee S (1982) Generalized discrete Gronwall lemmas. Z Angew Math Mech 62: 429–434

    Article  MATH  MathSciNet  Google Scholar 

  19. Brunner H (2004) Collocation methods for Volterra integral and related functional equations. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  20. Enright WH (2005) Tools for verification of approximate solutions to differential equations. In: Einarsson B(eds) Handbook for scientific computing. SIAM Press, Philadelphia, pp 109–119

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Department of Applied Mathematics, Amirkabir University of Technology, No. 424, Hafez Avenue, 15914, Tehran, Iran

    M. Shakourifar & M. Dehghan

Authors
  1. M. Shakourifar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Dehghan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Dehghan.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shakourifar, M., Dehghan, M. On the numerical solution of nonlinear systems of Volterra integro-differential equations with delay arguments. Computing 82, 241–260 (2008). https://doi.org/10.1007/s00607-008-0009-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00607-008-0009-4

Keywords

Mathematics Subject Classification (2000)

Search

Navigation