The Application of the Differential Inequalities Method for Proving the Existence of Moving Front Solution of the Parabolic Equations System
https://doi.org/10.18255/1818-1015-2016-3-317-325
Abstract
Investigations of initial boundary value problems for parabolic equations solutions are an important component of mathematical modeling. In this regard of special interest for mathematical modeling are the boundary value problem solutions that undergo sharp changes in any area of space. Such areas are called internal transitional layers. In case when the position of a transitional layer changes over time, the solution of a parabolic equation behaves as a moving front. For the purpose of proving the existence of such initial boundary value problem solutions, the method of differential inequalities is very effective. According to this method the so-called upper and lower solutions are to be constructed for the initial boundary value problem. The essence of an asymptotic method of differential inequalities is in receiving the upper and lower solutions as modifications of asymptotic submissions of the solutions of boundary value problems. The existence of the upper and lower solutions is a sufficient condition of existence of a solution of a boundary value problem. While proving the differential inequalities the so-called ”quasimonotony” condition is essential. In the present work it is considered how to construct the upper and lower solutions for the system of the parabolic equations under various conditions of quasimonotony.
About the Authors
N. T. Levashova
Lomonosov Moscow State University
Russian Federation
Russian Federation
PhD, Faculty of Physics, Leninskiye Gory, 1, bld. 2, Moscow, 119991, Russia
A. A. Melnikova
Lomonosov Moscow State University
Russian Federation
Russian Federation
PhD, Faculty of Physics, Leninskiye Gory, 1, bld. 2, Moscow, 119991, Russia
S. V. Bytsyura
Lomonosov Moscow State University
Russian Federation
Russian Federation
student, Faculty of Physics, Leninskiye Gory, 1, bld. 2, Moscow, 119991, Russia
References
1. FitzHugh R. A., “Impulses and physiological states in theoretical model of nerve membrane”, Biophys. J., 1 (1961), 445–466.
2. Murray J. D., Mathematical Biology II: Spatial Models and Biomedical Applications, Third Edition, Springer, 2003.
3. Сидорова А.Э., Левашова Н.Т., Мельникова А.А., Яковенко Л.В., “Популяционная модель урбоэкосистем в представлениях активных сред”, Биофизика, 60:3 (2015), 574– 582; English transl.: Sidorova A. E., Levashova N. T., Melnikova A. A., Yakovenko L. V, “A model of a human dominated urban ecosystem as an active medium”, Biophysics, 60:3 (2015), 466–473.
4. Бутузов В.Ф., Неделько И.В., “Контрастная структура типа ступеньки в системе двух сингулярно возмущенных параболических уравнений”, Матем. моделирование, 13:12 (2000), 23–42; English transl.: Butuzov V. F., Nedelko I. V., “Step-type contrast structure in a system of two singularly perturbed parabolic equations”, Matem. Mod., 60:3 (2001), 23–42.
5. Левашова Н.Т., Мельникова А.А., “Контрастная структура типа ступеньки в сингулярно возмущенной системе параболических уравнений”, Дифференциальные уравнения, 51:3 (2015), 339–358; English transl.: Levashova N. T., Melnikova A. A., “Step-like
6. contrast structure in a singularly perturbed system of parabolic equations”, Differential Equations, 51:3 (2015), 342–361.
7. Fife P. C., McLeod J. B., “The Approach of Solutions of Nonlinear Diffusion. Equations to Travelling Front Solutions”, Arch. ration. mech. anal., 65:4 (1977), 335–361.
8. Pao C. V., Nonlinear Parabolic and Elliptic Equations, Plenum Press. New York, 1992.
9. Волков В.Т., Нефeдов Н.Н., “Развитие асимптотического метода дифференциальных неравенств для исследования периодических контрастных структур в уравнениях реакция-диффузия”, Ж. вычисл. матем. и матем. физ., 46:4 (2006), 615–623; English transl.: Volkov V. T., Nefedov N. N., “Development of the asymptotic method of differential inequalities for investigation of periodic contrast structures in reaction-diffusion equations”, Zh. Vychisl. Mat. Mat. Fiz, 46:4 (2006), 585–593.
10. Божевольнов Ю.В.,Нефeдов Н.Н., “Движение фронта в параболической задаче реакция-диффузия”, Ж. вычисл. матем. и матем. физ., 50:2 (2010), 276–285; English transl.: Bozhevol’nov Yu. V., Nefedov N. N., “Front motion in a parabolic reaction-diffusion problem”, Zh. Vychisl. Mat. Mat. Fiz, 50:2 (2010), 264–273.
11. Антипов Е.А., Левашова Н.Т., Нефедов Н.Н., “Асимптотика движения фронта в задаче реакция-диффузия-адвекция”, Ж. вычисл. матем. и матем. физ., 54:10 (2014), 1594–1607; English transl.: Antipov E. A., Levashova N. T., Nefedov N.N., “Asymptotics of the front motion in the reaction-diffusion-advection problem”, Zh. Vychisl. Mat. Mat. Fiz., 54:10 (2014), 1536–1549.
12. Васильева А.Б., Бутузов В.Ф, Асимптотические методы в теории сингулярных возмущений, Высш. школа, М, 1990, 208 с.; [Vasil’eva A.B., Butuzov V.F, Asimptoticheskie metody v teorii singuljarnyh vozmushhenij, Vysshaja shkola, Moskva, 1990 (in Russian).] 208 pp.
13. Левашова Н.Т., Петровская Е.С., “Применение метода дифференциальных неравенств для обоснования асимптотики решения системы двух обыкновенных дифференциальных уравнений в виде контрастной структуры типа ступеньки”, Ученые записки физического факультета, 2014, № 1, 1–13; [Levashova N. T., Petrovskaya E. S., “Primenenie metoda differentsialnykh neravenstv dlya obosnovaniya asimptotiki resheniya sistemy dvukh obyknovennykh differentsialnykh uravneniy v vide kontrastnoy struktury tipa stupenki”, Uchenye zapiski fizicheskogo fakulteta, 2014, № 1, 1–13 (in Russian).]
Review
For citations:
Levashova N.T., Melnikova A.A., Bytsyura S.V. The Application of the Differential Inequalities Method for Proving the Existence of Moving Front Solution of the Parabolic Equations System. Modeling and Analysis of Information Systems. 2016;23(3):317-325. (In Russ.) https://doi.org/10.18255/1818-1015-2016-3-317-325
Views:
929
Email this article 