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

Article

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.

Keywords

Piecewise polynomial collocation Delay Volterra integro-differential equations Global convergence Optimal order of superconvergence 

Mathematics Subject Classification (2000)

45E99 45K05 92D25 

References

  1. 1.
    Volterra V (1927) Variazioni e fluttuazioni del numero d’individui in specie animali conviventi. Memorie del R. Comitato talassografico italiano, Mem. CXXXIGoogle Scholar
  2. 2.
    Cushing JM (1977) Integrodifferential equations and delay models in population dynamics. Lecture notes in biomathematics, vol 20. Springer, BerlinGoogle Scholar
  3. 3.
    Volterra V (1939) The general equations of biological strife in the case of historical actions. Proc Edinburgh Math Soc 2: 4–10CrossRefGoogle Scholar
  4. 4.
    Bocharov GA, Rihan FA (2000) Numerical modelling in biosciences using delay differential equations. J Comput Appl Math 125: 183–199MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Gopalsamy K (1992) Stability and oscillation in delay differential equations of population dynamics. Kluwer, BostonGoogle Scholar
  6. 6.
    Kuang Y (1993) Delay differential equations with applications in population dynamic. Academic Press, San Diego, CAGoogle Scholar
  7. 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: 5–10MATHMathSciNetGoogle Scholar
  8. 8.
    Enright WH, Hu M (1997) Continuous Runge–Kutta methods for neutral Volterra integro-differential equations with delay. Appl Numer Math 24: 175–190MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Brunner H (1988) The approximate solution of initial- value problems for general Volterra integro-differential equations. Computing 40: 125–137MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Makroglou A (1983) A bloc-by-block method for the numerical solution of Volterra delay integro-differential equation. Computing 30: 49–62MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Brunner H (1989) Collocation methods for nonlinear Volterra Integro-differential equations with infinite delay. Math Comp 53: 571–587MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Brunner H (1994) The numerical solution of neutral Volterra integro-differential equations with delay arguments. Ann Numer Math 1: 309–322MATHMathSciNetGoogle Scholar
  13. 13.
    Brunner H, Zhang W (1999) Primary discontinuities in solutions for delay integro-differential equations. Methods Appl Anal 6: 525–534MATHMathSciNetGoogle Scholar
  14. 14.
    Baker CTH, Willé D (2000) On the propagation of derivative discontinuities in Volterra retarded integro-differential equations. N Z J Math 29: 103–113MATHGoogle Scholar
  15. 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, RIGoogle Scholar
  16. 16.
    Brunner H (1984) Implicit Runge–Kutta methods of optimal order for volterra integro-differential equations. Math Comp 42: 95–109MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Atkinson KE (1989) Introduction to numerical analysis, 2nd edn. Wiley, New YorkMATHGoogle Scholar
  18. 18.
    McKee S (1982) Generalized discrete Gronwall lemmas. Z Angew Math Mech 62: 429–434MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Brunner H (2004) Collocation methods for Volterra integral and related functional equations. Cambridge University Press, CambridgeMATHGoogle Scholar
  20. 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 109–119Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer Science, Department of Applied MathematicsAmirkabir University of TechnologyTehranIran

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