Analysis and Simulations of a Nonlocal Gray-Scott Model

The Gray-Scott model is a set of reaction-diffusion equations that describes chemical systems far from equilibrium. Interest in this model stems from its ability to generate spatio-temporal structures, including pulses, spots, stripes, and self-replicating patterns. We consider an extension of this...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2024-03
Main Authors: Cappanera, Loic, Jaramillo, Gabriela, Ward, Cory
Format: Article
Language:English
Subjects:
Convolution
Diffusion effects
Dirichlet problem
Mathematical models
Reaction-diffusion equations
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
cited_by
cites
container_end_page
container_issue
container_start_page
container_title arXiv.org
container_volume
creator Cappanera, Loic
Jaramillo, Gabriela
Ward, Cory
description The Gray-Scott model is a set of reaction-diffusion equations that describes chemical systems far from equilibrium. Interest in this model stems from its ability to generate spatio-temporal structures, including pulses, spots, stripes, and self-replicating patterns. We consider an extension of this model in which the spread of the different chemicals is assumed to be nonlocal, and can thus be represented by an integral operator. In particular, we focus on the case of strictly positive, symmetric, \(L^1\) convolution kernels that have a finite second moment. Modeling the equations on a finite interval, we prove the existence of small-time weak solutions in the case of nonlocal Dirichlet and Neumann boundary constraints. We then use this result to develop a finite element numerical scheme that helps us explore the effects of nonlocal diffusion on the formation of pulse solutions.
format article
fullrecord <record><control><sourceid>proquest</sourceid><recordid>TN_cdi_proquest_journals_2756876707</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2756876707</sourcerecordid><originalsourceid>FETCH-proquest_journals_27568767073</originalsourceid><addsrcrecordid>eNqNzL0KwjAUQOEgCBbtO1xwDsTENl1F_Fl0qXu5tCmkxFztTYe-vQ4-gNNZPs5CZNqYnaz2Wq9EzjwopXRpdVGYTNhDxDCzZ8DYQe2fU8DkKTJQDwh3ioFaDHAZcZZ1SynBjToXNmLZY2CX_7oW2_PpcbzK10jvyXFqBprG75sbbYuysqVV1vynPtBINUM</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2756876707</pqid></control><display><type>article</type><title>Analysis and Simulations of a Nonlocal Gray-Scott Model</title><source>Publicly Available Content (ProQuest)</source><creator>Cappanera, Loic ; Jaramillo, Gabriela ; Ward, Cory</creator><creatorcontrib>Cappanera, Loic ; Jaramillo, Gabriela ; Ward, Cory</creatorcontrib><description>The Gray-Scott model is a set of reaction-diffusion equations that describes chemical systems far from equilibrium. Interest in this model stems from its ability to generate spatio-temporal structures, including pulses, spots, stripes, and self-replicating patterns. We consider an extension of this model in which the spread of the different chemicals is assumed to be nonlocal, and can thus be represented by an integral operator. In particular, we focus on the case of strictly positive, symmetric, \(L^1\) convolution kernels that have a finite second moment. Modeling the equations on a finite interval, we prove the existence of small-time weak solutions in the case of nonlocal Dirichlet and Neumann boundary constraints. We then use this result to develop a finite element numerical scheme that helps us explore the effects of nonlocal diffusion on the formation of pulse solutions.</description><identifier>EISSN: 2331-8422</identifier><language>eng</language><publisher>Ithaca: Cornell University Library, arXiv.org</publisher><subject>Convolution ; Diffusion effects ; Dirichlet problem ; Mathematical models ; Reaction-diffusion equations</subject><ispartof>arXiv.org, 2024-03</ispartof><rights>2024. This work is published under http://arxiv.org/licenses/nonexclusive-distrib/1.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://www.proquest.com/docview/2756876707?pq-origsite=primo$$EHTML$$P50$$Gproquest$$Hfree_for_read</linktohtml><link.rule.ids>777,781,25734,36993,44571</link.rule.ids></links><search><creatorcontrib>Cappanera, Loic</creatorcontrib><creatorcontrib>Jaramillo, Gabriela</creatorcontrib><creatorcontrib>Ward, Cory</creatorcontrib><title>Analysis and Simulations of a Nonlocal Gray-Scott Model</title><title>arXiv.org</title><description>The Gray-Scott model is a set of reaction-diffusion equations that describes chemical systems far from equilibrium. Interest in this model stems from its ability to generate spatio-temporal structures, including pulses, spots, stripes, and self-replicating patterns. We consider an extension of this model in which the spread of the different chemicals is assumed to be nonlocal, and can thus be represented by an integral operator. In particular, we focus on the case of strictly positive, symmetric, \(L^1\) convolution kernels that have a finite second moment. Modeling the equations on a finite interval, we prove the existence of small-time weak solutions in the case of nonlocal Dirichlet and Neumann boundary constraints. We then use this result to develop a finite element numerical scheme that helps us explore the effects of nonlocal diffusion on the formation of pulse solutions.</description><subject>Convolution</subject><subject>Diffusion effects</subject><subject>Dirichlet problem</subject><subject>Mathematical models</subject><subject>Reaction-diffusion equations</subject><issn>2331-8422</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><sourceid>PIMPY</sourceid><recordid>eNqNzL0KwjAUQOEgCBbtO1xwDsTENl1F_Fl0qXu5tCmkxFztTYe-vQ4-gNNZPs5CZNqYnaz2Wq9EzjwopXRpdVGYTNhDxDCzZ8DYQe2fU8DkKTJQDwh3ioFaDHAZcZZ1SynBjToXNmLZY2CX_7oW2_PpcbzK10jvyXFqBprG75sbbYuysqVV1vynPtBINUM</recordid><startdate>20240322</startdate><enddate>20240322</enddate><creator>Cappanera, Loic</creator><creator>Jaramillo, Gabriela</creator><creator>Ward, Cory</creator><general>Cornell University Library, arXiv.org</general><scope>8FE</scope><scope>8FG</scope><scope>ABJCF</scope><scope>ABUWG</scope><scope>AFKRA</scope><scope>AZQEC</scope><scope>BENPR</scope><scope>BGLVJ</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>HCIFZ</scope><scope>L6V</scope><scope>M7S</scope><scope>PIMPY</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PRINS</scope><scope>PTHSS</scope></search><sort><creationdate>20240322</creationdate><title>Analysis and Simulations of a Nonlocal Gray-Scott Model</title><author>Cappanera, Loic ; Jaramillo, Gabriela ; Ward, Cory</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-proquest_journals_27568767073</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Convolution</topic><topic>Diffusion effects</topic><topic>Dirichlet problem</topic><topic>Mathematical models</topic><topic>Reaction-diffusion equations</topic><toplevel>online_resources</toplevel><creatorcontrib>Cappanera, Loic</creatorcontrib><creatorcontrib>Jaramillo, Gabriela</creatorcontrib><creatorcontrib>Ward, Cory</creatorcontrib><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>Materials Science &amp; Engineering Collection</collection><collection>ProQuest Central (Alumni)</collection><collection>ProQuest Central</collection><collection>ProQuest Central Essentials</collection><collection>AUTh Library subscriptions: ProQuest Central</collection><collection>Technology Collection</collection><collection>ProQuest One Community College</collection><collection>ProQuest Central</collection><collection>SciTech Premium Collection</collection><collection>ProQuest Engineering Collection</collection><collection>Engineering Database</collection><collection>Publicly Available Content (ProQuest)</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>ProQuest Central China</collection><collection>Engineering collection</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Cappanera, Loic</au><au>Jaramillo, Gabriela</au><au>Ward, Cory</au><format>book</format><genre>document</genre><ristype>GEN</ristype><atitle>Analysis and Simulations of a Nonlocal Gray-Scott Model</atitle><jtitle>arXiv.org</jtitle><date>2024-03-22</date><risdate>2024</risdate><eissn>2331-8422</eissn><abstract>The Gray-Scott model is a set of reaction-diffusion equations that describes chemical systems far from equilibrium. Interest in this model stems from its ability to generate spatio-temporal structures, including pulses, spots, stripes, and self-replicating patterns. We consider an extension of this model in which the spread of the different chemicals is assumed to be nonlocal, and can thus be represented by an integral operator. In particular, we focus on the case of strictly positive, symmetric, \(L^1\) convolution kernels that have a finite second moment. Modeling the equations on a finite interval, we prove the existence of small-time weak solutions in the case of nonlocal Dirichlet and Neumann boundary constraints. We then use this result to develop a finite element numerical scheme that helps us explore the effects of nonlocal diffusion on the formation of pulse solutions.</abstract><cop>Ithaca</cop><pub>Cornell University Library, arXiv.org</pub><oa>free_for_read</oa></addata></record>
fulltext fulltext
identifier EISSN: 2331-8422
ispartof arXiv.org, 2024-03
issn 2331-8422
language eng
recordid cdi_proquest_journals_2756876707
source Publicly Available Content (ProQuest)
subjects Convolution
Diffusion effects
Dirichlet problem
Mathematical models
Reaction-diffusion equations
title Analysis and Simulations of a Nonlocal Gray-Scott Model
url http://sfxeu10.hosted.exlibrisgroup.com/loughborough?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-01-21T08%3A03%3A52IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-proquest&rft_val_fmt=info:ofi/fmt:kev:mtx:book&rft.genre=document&rft.atitle=Analysis%20and%20Simulations%20of%20a%20Nonlocal%20Gray-Scott%20Model&rft.jtitle=arXiv.org&rft.au=Cappanera,%20Loic&rft.date=2024-03-22&rft.eissn=2331-8422&rft_id=info:doi/&rft_dat=%3Cproquest%3E2756876707%3C/proquest%3E%3Cgrp_id%3Ecdi_FETCH-proquest_journals_27568767073%3C/grp_id%3E%3Coa%3E%3C/oa%3E%3Curl%3E%3C/url%3E&rft_id=info:oai/&rft_pqid=2756876707&rft_id=info:pmid/&rfr_iscdi=true