Skip to main content
SpringerLink
Log in

Passage to the limit in a singularly perturbed partial integro-differential system

  • Integral Equations
  • Published:
Differential Equations Aims and scope Submit manuscript

Abstract

We study an initial–boundary value problem for a singularly perturbed system of partial integro-differential equations. We prove a theorem on the passage to the limit. The result is used to decrease the dimension of a virus evolution model. We construct an asymptotic solution by the Tikhonov–Vasil’eva boundary function method. The analytic results obtained are compared with a numerical study of the system.

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. Mishchenko, E.F. andRozov, N.Kh., Differentsial’nye uravneniya s malym parametrom i relaksatsionnye kolebaniya (Differential Equations with Small Parameter and Relaxation Oscillations), Moscow, 1975.

    Google Scholar 

  2. Filatov, A.N. and Sharova, L.V., Integral’nye neravenstva i teoriya nelineinykh kolebanii (Integral Inequalities and Theory of Nonlinear Oscillations), Moscow, 1976.

    MATH  Google Scholar 

  3. Henry, D., Geometric Theory of Semilinear Parabolic Equations, Heidelberg: Springer-Verlag, 1981. Translated under the title Geometricheskaya teoriya polulineinykh parabolicheskikh uravnenii, Moscow, 1985.

    Book  MATH  Google Scholar 

  4. Korobeinikov, A. and Dempsey, C., A Continuous Phenotype Space Model of RNA Virus Evolution within a Host, Math. Biosci. Eng., 2014, vol. 11, no. 4, pp. 919–927.

    Article  MathSciNet  MATH  Google Scholar 

  5. Anderson, R.M. and May, R.M., The Population Dynamics of Microparasites and Their Invertebrate Hosts, Philos. Trans. R. Soc. Lond. Ser. B 291, 1981, pp. 451–524.

    Article  Google Scholar 

  6. Nowak, M.A. and May, R.M., Virus Dynamics: Mathematical Principles of Immunology and Virology, Oxford, 2000.

    MATH  Google Scholar 

  7. Huang, G., Takeuchi, Y., and Korobeinikov, A., HIV Evolution and Progression of the Infection to AIDS, J. Theoret. Biol., 2012, vol. 307, pp. 149–159.

    Article  MathSciNet  MATH  Google Scholar 

  8. Vasil’eva, A.B. and Butuzov, V.F., Asimptoticheskie metody v teorii singulyarnykh vozmushchenii (Asymptotic Methods in Singular Perturbation Theory), Moscow, 1990.

    Google Scholar 

  9. Vasil’eva, A.B. and Butuzov, V.F., Asimptoticheskie razlozheniya reshenii singulyarno vozmushchennykh uravnenii (Asymptotic Expansions of Solutions of Singularly Perturbed Equations), Moscow, 1973.

    Google Scholar 

  10. Vasil’eva, A.B. and Butuzov, V.F., Asymptotics of a Solution of Integro-Differential Equation with a Small Parameter Multiplying the Derivative, Zh. Vychisl. Mat. Mat. Fiz., 1964, vol. 4, no. 4, pp. 183–191.

    MathSciNet  MATH  Google Scholar 

  11. Voropaeva, N.V. and Sobolev, V.A., Geometricheskaya dekompozitsiya singulyarno vozmushchennykh sistem (Geometric Decomposition of Singularly Perturbed Systems), Moscow, 2009.

    Google Scholar 

  12. Nefedov, N.N. and Nikitin, A.G., Method of Differential Inequalities for Singularly Perturbed Integro-Differential Equations, Differ. Uravn., 2000, vol. 36, no. 10, pp. 1398–1404.

    MathSciNet  MATH  Google Scholar 

  13. Nefedov, N.N. and Nikitin, A.G., Method of Differential Inequalities for Step-Like Contrast Structures in Singularly Perturbed Integro-Differential Equations in the Spatially Two-Dimensional Case, Differ. Uravn., 2006, vol. 42, no. 5, pp. 690–700.

    MathSciNet  MATH  Google Scholar 

  14. Nefedov, N.N. and Nikitin, A.G., Initial–Boundary Value Problem for Nonlocal Singularly Perturbed Reaction–Diffusion Equation, Zh. Vychisl. Mat. Mat. Fiz., 2012, vol. 52, no. 6, pp. 1042–1047.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Samara State Aerospace University, Samara, Russia

    A. A. Archibasov & V. A. Sobolev

  2. Centre de Recerca Matematica, Barcelona, Spain

    A. Korobeinikov

Authors
  1. A. A. Archibasov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Korobeinikov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. V. A. Sobolev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. A. Archibasov.

Additional information

Original Russian Text © A.A. Archibasov, A. Korobeinikov, V.A. Sobolev, 2016, published in Differentsial’nye Uravneniya, 2016, Vol. 52, No. 9, pp. 1160–1167.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Archibasov, A.A., Korobeinikov, A. & Sobolev, V.A. Passage to the limit in a singularly perturbed partial integro-differential system. Diff Equat 52, 1115–1122 (2016). https://doi.org/10.1134/S0012266116090020

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0012266116090020

Search

Navigation