Front Dynamics in an Activator-Inhibitor System of Equations

  • Alina MelnikovaEmail author
  • Natalia Levashova
  • Dmitry Lukyanenko
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10187)


We consider the construction of formal asymptotic approximation for solution of the singularly perturbed boundary value problem of an activator-inhibitor type with a solution in a form of moving front. Corresponding asymptotic analysis provides a priori information about the localization of the transition point for moving front that is further used for constructing of dynamic adapted mesh. This mesh significantly improves numerical stability of numerical calculations for the considered system.


Saddle Point Transition Layer Asymptotic Approximation Uniform Mesh Regular Part 
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This study was supported by grants of the Russian Foundation for Basic Research projects No. 16-01-00437, 15-01-04619 and 16-01-00755.

Supplementary material

Supplementary material 1 (mp4 282 KB)


  1. 1.
    Alshin, A.B., Alshina, E.A., Kalitkin, N.N., Koryagina, A.: Rosenbrock schemes with complex coefficients for stiff and differential algebraic systems. Comput. Math. Math. Phys. 46, 1320–1340 (2006)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Barkley, D.: A model for fast computer simulation of waves in excitable media. Phys. D: Nonlinear Phenom. 49, 445–466 (1991)CrossRefGoogle Scholar
  3. 3.
    Butuzov, V.F., Levashova, N.T., Mel’nikova, A.A.: Steplike contrast structure in a singularly perturbed system of equations with different powers of small parameter. Comput. Math. Math. Phys. 52, 1526–1546 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Fife, P., McLeod, J.: The approach of solutions of nonlinear diffusion equations to travelling front solutions. Arch. Ration. Mech. Anal. 65, 335–361 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    FitzHugh, R.: Impulses and physiological states in theoretical model of nerve membrane. Biophys. J. 1, 445–466 (1961)CrossRefGoogle Scholar
  6. 6.
    Levashova, N.T., Mel’nikova, A.A.: Step-like contrast sructure in a singularly perturbed system of parabolic equations. Differ. Equ. 51, 342–367 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Vasil’eva, A.B., Butuzov, V.F., Kalachev, L.V.: The Boundary Function Method for Singular Perturbation Problems. SIAM, Bangkok (1995)CrossRefzbMATHGoogle Scholar
  8. 8.
    Kalitkin, N.N., Alshin, A.B., Alshina, E.A., Rogov, B.V.: Computations on Quasi-Uniform Grids. Fizmatlit, Moscow (2005). (in Russian)zbMATHGoogle Scholar
  9. 9.
    Rosenbrock, H.H.: Some general implicit processes for the numerical solution of differential equations. Comput. J. 5(4), 329–330 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hairer, E., Wanner, G.: Solving of Ordinary Differential Equations. Stiff and Differential-Algebraic Problems. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  11. 11.
    Lukyanenko, D.V., Volkov, V.T., Nefedov, N.N., Recke, L., Schneider, K.: Analytic-numerical approach to solving singularly perturbed parabolic equations with the use of dynamic adapted meshes. Model. Anal. of Inf. Syst. 23(3), 334–341 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Alina Melnikova
    • 1
    Email author
  • Natalia Levashova
    • 1
  • Dmitry Lukyanenko
    • 1
  1. 1.Faculty of Physics, Department of MathematicsLomonosov Moscow State UniversityMoscowRussia

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