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

  • Ognyan Christov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5434)


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.


Phase Portrait Asymptotic Formula Shooting Method Internal Boundary Layer Travel Wave Front 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Ognyan Christov
    • 1
  1. 1.Faculty of Mathematics and InformaticsSofia UniversitySofiaBulgaria

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