LIOUVILLE-GREEN APPROXIMATION FOR LINEARLY COUPLED SYSTEMS: ASYMPTOTIC ANALYSIS WITH APPLICATIONS TO REACTION-DIFFUSION SYSTEMS

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21340%2F22%3A00363806" target="_blank" >RIV/68407700:21340/22:00363806 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.3934/dcdss.2022133" target="_blank" >https://doi.org/10.3934/dcdss.2022133</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.3934/dcdss.2022133" target="_blank" >10.3934/dcdss.2022133</a>

Jazyk výsledku

angličtina

Název v původním jazyce

LIOUVILLE-GREEN APPROXIMATION FOR LINEARLY COUPLED SYSTEMS: ASYMPTOTIC ANALYSIS WITH APPLICATIONS TO REACTION-DIFFUSION SYSTEMS

Popis výsledku v původním jazyce

Asymptotic analysis has become a common approach in investigations of reaction-diffusion equations and pattern formation, especially when considering generalizations of the original model, such as spatial heterogeneity, where finding an analytic solution even to the linearized equations is generally not possible. The Liouville-Green approximation (also known as WKBJ method), one of the more robust asymptotic approaches for investigating dissipative phenomena captured by linear equations, has recently been applied to the Turing model in a heterogeneous environment. It demonstrated the anticipated modifications to the results obtained in a homogeneous setting, such as localized patterns and local Turing conditions. In this context, we attempt a generalization of the scalar Liouville-Green approximation to multicomponent systems. Our broader mathematical approach results in general approximation theorems for systems of ODEs without turning points. We discuss the cases of exponential and oscillatory behaviour first before treating the general case. Subsequently, we demonstrate the spectral properties utilized in the approximation theorems for a typical Turing system, hence showing that Liouville-Green approximation is plausible for an arbitrary number of coupled species outside of turning points and generally valid for fast growing modes as long as the diffusivities are distinct. Note that our line of approach is via showing that the solution is close (using suitable weight functions for measuring the error) to a linear combination of Airy-like functions.

Název v anglickém jazyce

LIOUVILLE-GREEN APPROXIMATION FOR LINEARLY COUPLED SYSTEMS: ASYMPTOTIC ANALYSIS WITH APPLICATIONS TO REACTION-DIFFUSION SYSTEMS

Popis výsledku anglicky

Asymptotic analysis has become a common approach in investigations of reaction-diffusion equations and pattern formation, especially when considering generalizations of the original model, such as spatial heterogeneity, where finding an analytic solution even to the linearized equations is generally not possible. The Liouville-Green approximation (also known as WKBJ method), one of the more robust asymptotic approaches for investigating dissipative phenomena captured by linear equations, has recently been applied to the Turing model in a heterogeneous environment. It demonstrated the anticipated modifications to the results obtained in a homogeneous setting, such as localized patterns and local Turing conditions. In this context, we attempt a generalization of the scalar Liouville-Green approximation to multicomponent systems. Our broader mathematical approach results in general approximation theorems for systems of ODEs without turning points. We discuss the cases of exponential and oscillatory behaviour first before treating the general case. Subsequently, we demonstrate the spectral properties utilized in the approximation theorems for a typical Turing system, hence showing that Liouville-Green approximation is plausible for an arbitrary number of coupled species outside of turning points and generally valid for fast growing modes as long as the diffusivities are distinct. Note that our line of approach is via showing that the solution is close (using suitable weight functions for measuring the error) to a linear combination of Airy-like functions.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10102 - Applied mathematics

Projekt

<a href="/cs/project/GA20-22092S" target="_blank" >GA20-22092S: Víceškálová termodynamika: okrajové podmínky, integrace a aplikace</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2022

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Discrete and Continuous Dynamical Systems. Series S

ISSN

1937-1632

e-ISSN

1937-1179

Svazek periodika

15

Číslo periodika v rámci svazku

9

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

27

Strana od-do

2553-2579

Kód UT WoS článku

000824221100001

EID výsledku v databázi Scopus

2-s2.0-85135446156