Abstract
In the present work the model boundary value problem for a stationary singularly perturbed reaction-diffusion-advection equation arising at the description of gas impurity transfer processes in an ecosystem ”forest – swamp” is considered. Application of a boundary functions method and an asymptotic method of differential inequalities allow to construct an asymptotics of the boundary layer type solution, to prove the existence of the solution with such an asymptotics and its asymptotic stability by Lyapunov as the stationary solution of the corresponding parabolic problem with the definition of local area of boundary layer type solution formation. The latter has a certain importance for applications, since it allows to reveal the solution describing one of the most probable conditions of the ecosystem. In the final part of the work sufficient conditions for existence of solutions with interior transitional layers (contrast structures) are discussed.
Highlights
singularly perturbed reaction-diffusion-advection equation arising at the description of gas impurity transfer processes
an asymptotic method of differential inequalities allow to construct an asymptotics of the boundary layer type solution
Davydova М.А. , Levashova N.T., Zakharova S.А., "The Asymptotical Analysis for the Problem of Modeling the Gas Admixture in the Surface Layer of the Atmosphere", Modeling and Analysis of Information Systems, 23:3 (2016), 283–290
Summary
Стационарное распределение концентрации парниковых газов в экосистеме лес– болото в предположении изотропности пространства по одной из горизонтальных координат имеет вид контрастной структуры с локализацией внутреннего переходного слоя в окрестности границы между лесополосой и болотом. Это явилось основанием для применения асимптотической теории контрастных структур к исследованию одномерной модельной краевой задачи для усредненного уравнения переноса газовой примеси ε2u − εA (x) u = B (u, x, ε) , x ∈ (−1, 1) , ε ∈ (0, 1) ,. Здесь u – безразмерная концентрация газовой примеси, A (x) – горизонтальная компонента безразмерной скорости ветра. B (u, x, ε) = (u − φ1 (x)) (u − φ2 (x)) (u − φ3 (x)) , где функции u = φi (x) , i = 1, 3 интерпретируются как безразмерные концентрации парниковых газов над лесом и болотом
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