{"id":"https://openalex.org/W4387936466","doi":"https://doi.org/10.3390/axioms12111004","title":"On the Spatiotemporal Pattern Formation in Nonlinear Coupled Reaction\u2013Diffusion Systems","display_name":"On the Spatiotemporal Pattern Formation in Nonlinear Coupled Reaction\u2013Diffusion Systems","publication_year":2023,"publication_date":"2023-10-25","ids":{"openalex":"https://openalex.org/W4387936466","doi":"https://doi.org/10.3390/axioms12111004"},"language":"en","primary_location":{"id":"doi:10.3390/axioms12111004","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms12111004","pdf_url":"https://www.mdpi.com/2075-1680/12/11/1004/pdf?version=1698237608","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2075-1680/12/11/1004/pdf?version=1698237608","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5083491222","display_name":"Satyvir Singh","orcid":"https://orcid.org/0000-0001-6669-5296"},"institutions":[{"id":"https://openalex.org/I60054993","display_name":"Graphic Era University","ror":"https://ror.org/03wqgqd89","country_code":"IN","type":"education","lineage":["https://openalex.org/I60054993"]},{"id":"https://openalex.org/I887968799","display_name":"RWTH Aachen University","ror":"https://ror.org/04xfq0f34","country_code":"DE","type":"education","lineage":["https://openalex.org/I887968799"]}],"countries":["DE","IN"],"is_corresponding":true,"raw_author_name":"Satyvir Singh","raw_affiliation_strings":["Applied and Computational Mathematics, RWTH Aachen University, 52062 Aachen, Germany","Department of Mathematics, Graphic Era Deemed to be University, Dehradun 248002, India"],"raw_orcid":"https://orcid.org/0000-0001-6669-5296","affiliations":[{"raw_affiliation_string":"Applied and Computational Mathematics, RWTH Aachen University, 52062 Aachen, Germany","institution_ids":["https://openalex.org/I887968799"]},{"raw_affiliation_string":"Department of Mathematics, Graphic Era Deemed to be University, Dehradun 248002, India","institution_ids":["https://openalex.org/I60054993"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5009015518","display_name":"Ahmed Hussein Msmali","orcid":"https://orcid.org/0000-0002-0186-0526"},"institutions":[{"id":"https://openalex.org/I204824540","display_name":"University of Wollongong","ror":"https://ror.org/00jtmb277","country_code":"AU","type":"education","lineage":["https://openalex.org/I204824540"]},{"id":"https://openalex.org/I91450128","display_name":"Jazan University","ror":"https://ror.org/02bjnq803","country_code":"SA","type":"education","lineage":["https://openalex.org/I91450128"]}],"countries":["AU","SA"],"is_corresponding":false,"raw_author_name":"Ahmed Hussein Msmali","raw_affiliation_strings":["Department of Mathematics, College of Science, Jazan University, Jazan 45142, Saudi Arabia","School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2500, Australia"],"raw_orcid":"https://orcid.org/0000-0002-0186-0526","affiliations":[{"raw_affiliation_string":"Department of Mathematics, College of Science, Jazan University, Jazan 45142, Saudi Arabia","institution_ids":["https://openalex.org/I91450128"]},{"raw_affiliation_string":"School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2500, Australia","institution_ids":["https://openalex.org/I204824540"]}]}],"institutions":[],"countries_distinct_count":4,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5083491222"],"corresponding_institution_ids":["https://openalex.org/I60054993","https://openalex.org/I887968799"],"apc_list":{"value":1400,"currency":"CHF","value_usd":1515},"apc_paid":{"value":1400,"currency":"CHF","value_usd":1515},"fwci":1.9706,"has_fulltext":false,"cited_by_count":5,"citation_normalized_percentile":{"value":0.84380476,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":95,"max":97},"biblio":{"volume":"12","issue":"11","first_page":"1004","last_page":"1004"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12727","display_name":"Differential Equations and Numerical Methods","score":0.9993000030517578,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12727","display_name":"Differential Equations and Numerical Methods","score":0.9993000030517578,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11187","display_name":"Nonlinear Dynamics and Pattern Formation","score":0.9979000091552734,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11416","display_name":"Numerical methods for differential equations","score":0.9929999709129333,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/brusselator","display_name":"Brusselator","score":0.7975940704345703},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.6922367811203003},{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.5993024110794067},{"id":"https://openalex.org/keywords/reaction\u2013diffusion-system","display_name":"Reaction\u2013diffusion system","score":0.5817753672599792},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.49837207794189453},{"id":"https://openalex.org/keywords/basis-function","display_name":"Basis function","score":0.49142158031463623},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4678080081939697},{"id":"https://openalex.org/keywords/pattern-formation","display_name":"Pattern formation","score":0.455120325088501},{"id":"https://openalex.org/keywords/spatiotemporal-pattern","display_name":"Spatiotemporal pattern","score":0.42499104142189026},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.3910205066204071},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3684312105178833},{"id":"https://openalex.org/keywords/biological-system","display_name":"Biological system","score":0.36338168382644653},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.17667266726493835},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.1360202133655548}],"concepts":[{"id":"https://openalex.org/C14028793","wikidata":"https://www.wikidata.org/wiki/Q906505","display_name":"Brusselator","level":3,"score":0.7975940704345703},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.6922367811203003},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.5993024110794067},{"id":"https://openalex.org/C121231716","wikidata":"https://www.wikidata.org/wiki/Q901210","display_name":"Reaction\u2013diffusion system","level":2,"score":0.5817753672599792},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.49837207794189453},{"id":"https://openalex.org/C5917680","wikidata":"https://www.wikidata.org/wiki/Q2621825","display_name":"Basis function","level":2,"score":0.49142158031463623},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4678080081939697},{"id":"https://openalex.org/C207016750","wikidata":"https://www.wikidata.org/wiki/Q1955627","display_name":"Pattern formation","level":2,"score":0.455120325088501},{"id":"https://openalex.org/C2779108282","wikidata":"https://www.wikidata.org/wiki/Q22908968","display_name":"Spatiotemporal pattern","level":2,"score":0.42499104142189026},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3910205066204071},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3684312105178833},{"id":"https://openalex.org/C186060115","wikidata":"https://www.wikidata.org/wiki/Q30336093","display_name":"Biological system","level":1,"score":0.36338168382644653},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.17667266726493835},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.1360202133655548},{"id":"https://openalex.org/C54355233","wikidata":"https://www.wikidata.org/wiki/Q7162","display_name":"Genetics","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/axioms12111004","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms12111004","pdf_url":"https://www.mdpi.com/2075-1680/12/11/1004/pdf?version=1698237608","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},{"id":"pmh:oai:publications.rwth-aachen.de:983554","is_oa":true,"landing_page_url":"https://publications.rwth-aachen.de/record/983554","pdf_url":null,"source":{"id":"https://openalex.org/S4306401362","display_name":"RWTH Publications (RWTH Aachen)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I887968799","host_organization_name":"RWTH Aachen University","host_organization_lineage":["https://openalex.org/I887968799"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms : open access journal 12(11), 1004 (2023). doi:10.3390/axioms12111004","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:oai:doaj.org/article:5e3aff6ebf8b42488140fbb67afc364c","is_oa":true,"landing_page_url":"https://doaj.org/article/5e3aff6ebf8b42488140fbb67afc364c","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Axioms, Vol 12, Iss 11, p 1004 (2023)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/axioms12111004","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms12111004","pdf_url":"https://www.mdpi.com/2075-1680/12/11/1004/pdf?version=1698237608","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G5464405269","display_name":null,"funder_award_id":"DFG\u2013FOR5409","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"}],"funders":[{"id":"https://openalex.org/F4320320879","display_name":"Deutsche Forschungsgemeinschaft","ror":"https://ror.org/018mejw64"}],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4387936466.pdf"},"referenced_works_count":66,"referenced_works":["https://openalex.org/W122787279","https://openalex.org/W175949082","https://openalex.org/W1592415565","https://openalex.org/W1804065695","https://openalex.org/W1923323189","https://openalex.org/W1963942260","https://openalex.org/W1964456182","https://openalex.org/W1965312277","https://openalex.org/W1980661496","https://openalex.org/W1982465309","https://openalex.org/W1991812943","https://openalex.org/W1993923624","https://openalex.org/W1994732172","https://openalex.org/W1997736083","https://openalex.org/W1998666864","https://openalex.org/W1998939448","https://openalex.org/W2006640177","https://openalex.org/W2012938047","https://openalex.org/W2022997508","https://openalex.org/W2023954922","https://openalex.org/W2027244891","https://openalex.org/W2029870529","https://openalex.org/W2033596376","https://openalex.org/W2038220764","https://openalex.org/W2045212693","https://openalex.org/W2053096573","https://openalex.org/W2065908448","https://openalex.org/W2072952452","https://openalex.org/W2073519923","https://openalex.org/W2085961649","https://openalex.org/W2089840306","https://openalex.org/W2096050810","https://openalex.org/W2098974766","https://openalex.org/W2099407793","https://openalex.org/W2136242628","https://openalex.org/W2167829186","https://openalex.org/W2331886756","https://openalex.org/W2418911301","https://openalex.org/W2507000343","https://openalex.org/W2519246473","https://openalex.org/W2618246756","https://openalex.org/W2724513501","https://openalex.org/W2789802199","https://openalex.org/W2792027214","https://openalex.org/W2801393104","https://openalex.org/W2893557925","https://openalex.org/W2894210653","https://openalex.org/W2902941945","https://openalex.org/W3080877638","https://openalex.org/W3152521352","https://openalex.org/W3185298526","https://openalex.org/W3203183407","https://openalex.org/W4211138966","https://openalex.org/W4212953954","https://openalex.org/W4252060206","https://openalex.org/W4255777804","https://openalex.org/W4284960095","https://openalex.org/W4312289896","https://openalex.org/W4367598576","https://openalex.org/W4377141690","https://openalex.org/W6648818983","https://openalex.org/W6663767097","https://openalex.org/W6667081063","https://openalex.org/W6748830136","https://openalex.org/W6793391210","https://openalex.org/W6808261592"],"related_works":["https://openalex.org/W1972125010","https://openalex.org/W2373509866","https://openalex.org/W3168394368","https://openalex.org/W2168169421","https://openalex.org/W4225646802","https://openalex.org/W2387264491","https://openalex.org/W4327740401","https://openalex.org/W2210238746","https://openalex.org/W2089784332","https://openalex.org/W2311232675"],"abstract_inverted_index":{"Nonlinear":[0],"coupled":[1],"reaction\u2013diffusion":[2],"(NCRD)":[3],"systems":[4,24,155],"have":[5],"played":[6],"a":[7,54,90,119,157],"crucial":[8],"role":[9],"in":[10,26,62,100,163,188,207],"the":[11,42,46,95,127,135,153,180,189],"emergence":[12],"of":[13,44,57,97,134,159],"spatiotemporal":[14,47,59,176],"patterns":[15,48,60,177],"across":[16],"various":[17,30],"scientific":[18],"and":[19,39,70,83,139,210],"engineering":[20],"domains.":[21],"The":[22,149,175],"NCRD":[23,85,128,154],"considered":[25],"this":[27],"study":[28],"encompass":[29],"models,":[31],"such":[32],"as":[33,204],"linear,":[34],"Gray\u2013Scott,":[35],"Brusselator,":[36],"isothermal":[37],"chemical,":[38],"Schnakenberg,":[40],"with":[41,179],"aim":[43],"capturing":[45],"they":[49],"generate.":[50],"These":[51],"models":[52],"cover":[53],"diverse":[55],"range":[56],"intricate":[58],"found":[61,187],"nature,":[63],"including":[64],"spots,":[65],"spot":[66],"replication,":[67],"stripes,":[68],"hexagons,":[69],"more.":[71],"A":[72],"mixed-type":[73],"modal":[74,107],"discontinuous":[75],"Galerkin":[76],"approach":[77,88,151,182,192],"is":[78,124],"employed":[79],"for":[80,126],"solving":[81],"one-":[82],"two-dimensional":[84],"systems.":[86],"This":[87,191],"introduces":[89],"mathematical":[91],"formulation":[92],"to":[93,144,185,197],"handle":[94,198],"occurrence":[96],"second-order":[98],"derivatives":[99],"diffusion":[101],"terms.":[102,148],"For":[103],"spatial":[104],"discretization,":[105],"hierarchical":[106],"basis":[108],"functions":[109],"premised":[110],"on":[111],"orthogonal":[112],"scaled":[113],"Legendre":[114],"polynomials":[115],"are":[116,166,183],"used.":[117],"Moreover,":[118],"novel":[120],"reaction":[121,147],"term":[122],"treatment":[123],"proposed":[125,150],"systems,":[129],"demonstrating":[130],"an":[131,169],"intrinsic":[132],"feature":[133],"new":[136],"DG":[137],"scheme":[138],"preventing":[140],"erroneous":[141],"solutions":[142],"due":[143],"extremely":[145],"nonlinear":[146],"reduces":[152],"into":[156],"framework":[158],"ordinary":[160],"differential":[161],"equations":[162,206],"time,":[164],"which":[165],"addressed":[167],"by":[168],"explicit":[170],"third-order":[171],"TVD":[172],"Runge\u2013Kutta":[173],"algorithm.":[174],"generated":[178],"present":[181],"comparable":[184],"those":[186],"literature.":[190],"can":[193],"readily":[194],"be":[195],"expanded":[196],"large":[199],"multi-dimensional":[200],"problems":[201],"that":[202],"appear":[203],"model":[205],"developed":[208],"biological":[209],"chemical":[211],"applications.":[212]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":3}],"updated_date":"2026-05-06T08:25:59.206177","created_date":"2025-10-10T00:00:00"}