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On the Number of Spikes of Solutions for a Singularly Perturbed Boundary-Value Problem

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Numerical Analysis and Its Applications (NAA 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5434))

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Abstract

We study the stationary solutions of the famous Fisher - Kolmogorov - Petrovsky - Piscounov (FKPP) equation

$$ u_t = D u_{x x} + \gamma u (1 - u), $$

where the diffusion parameter is small - D = ε 2 and γ is normalized to be 1. Next, we state a singularly perturbed boundary-value problem (BVP) on the interval [0, 1].

$$ \left| \begin{array}{l} \epsilon^2 \ddot{u} + u (1 - u) = 0, \\ u( 0 ) = a, \quad u( 1 ) = a, \quad a \in (0, 1). \end{array} \right. $$

Asymptotic formulas, as ε→0 +  are obtained for the solutions of the above (BVP). The solutions of the (BVP) can have ”spikes”. Estimates for the number of spikes and number of solutions to the (BVP) are given.

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References

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Author information

Authors and Affiliations

  1. Faculty of Mathematics and Informatics, Sofia University, 5 J.Bouchier blvd., 1164, Sofia, Bulgaria

    Ognyan Christov

Authors
  1. Ognyan Christov
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Editors and Affiliations

  1. Institute for Parallel Processing, Bulgarian Academy of Sciences, Acad. G. Bonchev Str. Bl. 25-A, 1113, Sofia, Bulgaria

    Svetozar Margenov

  2. FNSE, Department of Numerical Methods and Statistics, University of Rousse, 8 Studentska str., 7017, Rousse, Bulgaria

    Lubin G. Vulkov

  3. Department of Informatics and Mathematical Modelling, Technical University of Denmark, 2800, Kongens, Lyngby, Denmark

    Jerzy Waśniewski

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Christov, O. (2009). On the Number of Spikes of Solutions for a Singularly Perturbed Boundary-Value Problem. In: Margenov, S., Vulkov, L.G., Waśniewski, J. (eds) Numerical Analysis and Its Applications. NAA 2008. Lecture Notes in Computer Science, vol 5434. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00464-3_24

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  • DOI: https://doi.org/10.1007/978-3-642-00464-3_24

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-00463-6

  • Online ISBN: 978-3-642-00464-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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