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Abstract
Creating adequate mathematical models of processes in living nature is an important task of modern biophysics. Blood clotting, nerve impulse propagation, reduction of the heart muscle, the pattern-formation in nature are auto-wave processes. FitzHugh–Nagumo system of equations is used to describe the auto-wave processes in active media. Such math problems are usually solved by numerical methods. The use of resource-intensive algorithms is required in the case of auto-wave solutions with sharp gradients. Therefore, it is appropriate to use the analytical methods for this type of problems. In this paper, the asymptotic method of contrast structures theory is used to obtain an approximate solution of a singularly perturbed system of FitzHugh–Nagumo type. The method allows to reduce the non-linear system of equations to a number of problems that can be solved analytically or with a stable numerical algorithm. This study presents the asymptotic approximation of a stationary auto-wave solution of the considered system. Additionally, this paper provides a formula that specifies the location of internal transition layers. The results were compared with the numerical solution. The application of contrast structures theory to the study of active media models can be used for analytical studies of other such systems, improving existing models and increasing the efficiency of the numerical calculations.
Highlights
This study presents the asymptotic approximation of a stationary auto-wave solution of the considered system
This paper provides a formula that specifies the location of internal transition layers
A., Argun R.L., "Asymptotic Approximation of the Stationary Solution with Internal Layer for FitzHugh–Nagumo System", Modeling and Analysis of Information Systems, 23:5 (2016), 559–567
Summary
U0(x), v(x, 0) = v0(x), x ∈ [0; L], где u ∈ Iu ∈ R+, v ∈ Iv ∈ R+ – неизвестные функции, ε ∈ (0; 1) малый параметр, γ > 0 – параметр системы, функция α(x) ∈ (0; 1) при x ∈ [0; L]. В работе [1] доказана теорема существования решения с внутренним переходным слоем для задачи типа (2). Предполагается, что параметр γ и значения функции α(x) таковы, что первое уравнение (3) имеет три решения: φ1(v, x) = 0, φ2,3(v, x) = 0.5 α(x) + 1 ∓ (α(x) − 1)2 − 4v. Подставим устойчивые корни φ1,3(v, x) (для них выполнено условие fu(φ1,3(v, x), v, x) > 0 ) во второе уравнение (3) и разрешим полученные уравнения относительно переменной v: v1(x) = 0, v3(x) = 0.5γ−1 α(x) + 1 − γ−1 + (α(x) + 1 − γ−1)2 − 4α(x).
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