Asymptotic-Numerical Method for Moving Fronts in Two-Dimensional R-D-A Problems

  • Vladimir Volkov
  • Nikolay NefedovEmail author
  • Eugene Antipov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9045)


A singularly perturbed initial-boundary value problem for a parabolic equation known in applications as the reaction-diffusion equation is considered. An asymptotic expansion of the solution with moving front is constructed. Using the asymptotic method of differential inequalities we prove the existence and estimate the asymptotic expansion for such solutions. The method is based on well-known comparison theorems and formal asymptotics for the construction of upper and lower solutions in singularly perturbed problems with internal and boundary layers.


Singularly perturbed parabolic problems Reaction-diffusion equation Internal layers Fronts Asymptotic methods Differential inequalities 



This work is supported by RFBR, pr. N 13-01-00200.


  1. 1.
    Vasilieva, A.B., Butuzov, V.F., Nefedov, N.N.: Contrast structures in singularly perturbed problems. J. Fund. Prikl. Math. 4(3), 799–851 (1998)Google Scholar
  2. 2.
    Volkov, V.T., Grachev, N.E., Nefedov, N.N., Nikolaev, A.N.: On the formation of sharp transition layers in two-dimensional reaction-diffusion models. J. Comp. Math. Math. Phys. 47(8), 1301–1309 (2007)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Butuzov, V.F., Nefedov, N.N., Schneider, K.R.: On generation and propagation of sharp transition layers in parabolic problems. Vestnik MGU 3(1), 9–13 (2005)MathSciNetGoogle Scholar
  4. 4.
    Bozhevolnov, Y.V., Nefedov, N.N.: Front motion in parabolic reaction-diffusion problem. J. Comp. Math. Math. Phys. 50(2), 264–273 (2010)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Nefedov, N.N.: The method of diff. inequalities for some classes of nonlinear singul. perturbed problems. J. Diff. Uravn. 31(7), 1142–1149 (1995)MathSciNetGoogle Scholar
  6. 6.
    Fife, P.C., Hsiao, L.: The generation and propagation of internal layers. Nonlinear Anal. Theory Meth. Appl. 12(1), 19–41 (1998)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Volkov, V., Nefedov, N.: Asymptotic-numerical investigation of generation and motion of fronts in phase transition models. In: Dimov, I., Faragó, I., Vulkov, L. (eds.) NAA 2012. LNCS, vol. 8236, pp. 524–531. Springer, Heidelberg (2013) CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Vladimir Volkov
    • 1
  • Nikolay Nefedov
    • 1
    Email author
  • Eugene Antipov
    • 1
  1. 1.Department of Mathematics, Faculty of PhysicsLomonosov Moscow State UniversityMoscowRussia

Personalised recommendations