Stability and freezing of waves in non-linear hyperbolic-parabolic systems

被引：4

作者：

Rottmann-Matthes, Jens ^{[1
]}

机构：

[1] Univ Bielefeld, Dept Math, D-33501 Bielefeld, Germany

来源：

IMA JOURNAL OF APPLIED MATHEMATICS | 2012年 / 77卷 / 03期

关键词：

travelling waves; stability; hyperbolic-parabolic PDEs; freezing method; EQUATIONS;

D O I：

10.1093/imamat/hxs031

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

In this note, we consider the application of the freezing method to the approximation of travelling waves, in general hyperbolic-parabolic systems that include the Hodgkin-Huxley model and the FitzHugh-Nagumo equation. The tuple consisting of the profile and the speed of a travelling wave is a stationary solution for the method and we prove its asymptotic stability with optimal rates. Therefore, the method is suitable for the approximation of travelling waves by time integration. Numerical experiments for the FitzHugh-Nagumo equations confirm our results.

引用

页码：420 / 429

页数：10

共 12 条

[1]

Bates PW., 1989, DYNAMICS REPORTED, P1, DOI DOI 10.1007/978-3-322-96657-5_

[2] Freezing solutions of equivariant evolution equations [J].

Beyn, WJ ;

Thümmler, V .

SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS, 2004, 3 (02) :85-116

[3] NERVE AXON EQUATIONS .3. STABILITY OF NERVE IMPULSE [J].

EVANS, JW .

INDIANA UNIVERSITY MATHEMATICS JOURNAL, 1972, 22 (06) :577-593

[4] STABILITY OF TRAVELING WAVES FOR A CLASS OF REACTION-DIFFUSION SYSTEMS THAT ARISE IN CHEMICAL REACTION MODELS [J].

Ghazaryan, Anna ;

Latushkin, Yuri ;

Schecter, Stephen .

SIAM JOURNAL ON MATHEMATICAL ANALYSIS, 2010, 42 (06) :2434-2472

[5] A QUANTITATIVE DESCRIPTION OF MEMBRANE CURRENT AND ITS APPLICATION TO CONDUCTION AND EXCITATION IN NERVE [J].

HODGKIN, AL ;

HUXLEY, AF .

JOURNAL OF PHYSIOLOGY-LONDON, 1952, 117 (04) :500-544

[6] ON THE CONVERGENCE TO STEADY-STATE OF SOLUTIONS OF NONLINEAR HYPERBOLIC-PARABOLIC SYSTEMS [J].

KREISS, G ;

KREISS, HO ;

PETERSSON, NA .

SIAM JOURNAL ON NUMERICAL ANALYSIS, 1994, 31 (06) :1577-1604

[7] A differentiation index for partial differential-algebraic equations [J].

Martinson, WS ;

Barton, PI .

SIAM JOURNAL ON SCIENTIFIC COMPUTING, 2000, 21 (06) :2295-2315

[8]

ROTTMANN-MATTHES J., 2010, Computation and Stability of Patterns in Hyperbolic-Parabolic Systems

[9]

Rottmann-Matthes J., 2012, 12005 BIEL U

[10]

Rottmann-Matthes J., 2011, 11016 BIEL U

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