{"id":"https://openalex.org/W4226028217","doi":"https://doi.org/10.3390/sym14040740","title":"Impact of Brownian Motion on the Analytical Solutions of the Space-Fractional Stochastic Approximate Long Water Wave Equation","display_name":"Impact of Brownian Motion on the Analytical Solutions of the Space-Fractional Stochastic Approximate Long Water Wave Equation","publication_year":2022,"publication_date":"2022-04-04","ids":{"openalex":"https://openalex.org/W4226028217","doi":"https://doi.org/10.3390/sym14040740"},"language":"en","primary_location":{"id":"doi:10.3390/sym14040740","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14040740","pdf_url":"https://www.mdpi.com/2073-8994/14/4/740/pdf?version=1649063330","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/14/4/740/pdf?version=1649063330","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5072340361","display_name":"Farah M. Al\u2010Askar","orcid":null},"institutions":[{"id":"https://openalex.org/I106778892","display_name":"Princess Nourah bint Abdulrahman University","ror":"https://ror.org/05b0cyh02","country_code":"SA","type":"education","lineage":["https://openalex.org/I106778892"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Farah M. Al-Askar","raw_affiliation_strings":["Department of Mathematical Science, Collage of Science, Princess Nourah Bint Abdulrahman University, Riyadh 11671, Saudi Arabia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematical Science, Collage of Science, Princess Nourah Bint Abdulrahman University, Riyadh 11671, Saudi Arabia","institution_ids":["https://openalex.org/I106778892"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5061698030","display_name":"Wael W. Mohammed","orcid":"https://orcid.org/0000-0002-1402-7584"},"institutions":[{"id":"https://openalex.org/I159247623","display_name":"Mansoura University","ror":"https://ror.org/01k8vtd75","country_code":"EG","type":"education","lineage":["https://openalex.org/I159247623"]},{"id":"https://openalex.org/I4210088963","display_name":"University of Ha'il","ror":"https://ror.org/013w98a82","country_code":"SA","type":"education","lineage":["https://openalex.org/I4210088963"]}],"countries":["EG","SA"],"is_corresponding":true,"raw_author_name":"Wael W. Mohammed","raw_affiliation_strings":["Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt","Department of Mathematics, Faculty of Science, University of Ha\u2019il, Ha\u2019il 81411, Saudi Arabia","Department of Mathematics, Faculty of Science, University of Ha'il, Ha'il 81411, Saudi Arabia"],"raw_orcid":"https://orcid.org/0000-0002-1402-7584","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt","institution_ids":["https://openalex.org/I159247623"]},{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, University of Ha\u2019il, Ha\u2019il 81411, Saudi Arabia","institution_ids":["https://openalex.org/I4210088963"]},{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, University of Ha'il, Ha'il 81411, Saudi Arabia","institution_ids":["https://openalex.org/I4210088963"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5056162561","display_name":"Mohammad Alshammari","orcid":"https://orcid.org/0000-0002-4030-9083"},"institutions":[{"id":"https://openalex.org/I4210088963","display_name":"University of Ha'il","ror":"https://ror.org/013w98a82","country_code":"SA","type":"education","lineage":["https://openalex.org/I4210088963"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Mohammad Alshammari","raw_affiliation_strings":["Department of Mathematics, Faculty of Science, University of Ha\u2019il, Ha\u2019il 81411, Saudi Arabia","Department of Mathematics, Faculty of Science, University of Ha'il, Ha'il 81411, Saudi Arabia"],"raw_orcid":"https://orcid.org/0000-0002-4030-9083","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, University of Ha\u2019il, Ha\u2019il 81411, Saudi Arabia","institution_ids":["https://openalex.org/I4210088963"]},{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, University of Ha'il, Ha'il 81411, Saudi Arabia","institution_ids":["https://openalex.org/I4210088963"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5061698030"],"corresponding_institution_ids":["https://openalex.org/I159247623","https://openalex.org/I4210088963"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":2.8893,"has_fulltext":false,"cited_by_count":22,"citation_normalized_percentile":{"value":0.91319834,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":99},"biblio":{"volume":"14","issue":"4","first_page":"740","last_page":"740"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9994000196456909,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"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/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9994000196456909,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"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/T10248","display_name":"Nonlinear Waves and Solitons","score":0.9991999864578247,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12727","display_name":"Differential Equations and Numerical Methods","score":0.9968000054359436,"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/fractional-brownian-motion","display_name":"Fractional Brownian motion","score":0.7122390270233154},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.620858907699585},{"id":"https://openalex.org/keywords/geometric-brownian-motion","display_name":"Geometric Brownian motion","score":0.5710297226905823},{"id":"https://openalex.org/keywords/brownian-motion","display_name":"Brownian motion","score":0.515475332736969},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.5134381055831909},{"id":"https://openalex.org/keywords/wave-equation","display_name":"Wave equation","score":0.47265613079071045},{"id":"https://openalex.org/keywords/riccati-equation","display_name":"Riccati equation","score":0.44823992252349854},{"id":"https://openalex.org/keywords/work","display_name":"Work (physics)","score":0.4281766712665558},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.39558491110801697},{"id":"https://openalex.org/keywords/diffusion-process","display_name":"Diffusion process","score":0.1824847161769867},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.1806158721446991},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.1541244387626648},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.12991037964820862}],"concepts":[{"id":"https://openalex.org/C108819105","wikidata":"https://www.wikidata.org/wiki/Q1143293","display_name":"Fractional Brownian motion","level":3,"score":0.7122390270233154},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.620858907699585},{"id":"https://openalex.org/C101615488","wikidata":"https://www.wikidata.org/wiki/Q1503307","display_name":"Geometric Brownian motion","level":4,"score":0.5710297226905823},{"id":"https://openalex.org/C112401455","wikidata":"https://www.wikidata.org/wiki/Q178036","display_name":"Brownian motion","level":2,"score":0.515475332736969},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5134381055831909},{"id":"https://openalex.org/C59696629","wikidata":"https://www.wikidata.org/wiki/Q193846","display_name":"Wave equation","level":2,"score":0.47265613079071045},{"id":"https://openalex.org/C45473103","wikidata":"https://www.wikidata.org/wiki/Q851503","display_name":"Riccati equation","level":3,"score":0.44823992252349854},{"id":"https://openalex.org/C18762648","wikidata":"https://www.wikidata.org/wiki/Q42213","display_name":"Work (physics)","level":2,"score":0.4281766712665558},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.39558491110801697},{"id":"https://openalex.org/C68710425","wikidata":"https://www.wikidata.org/wiki/Q5275442","display_name":"Diffusion process","level":3,"score":0.1824847161769867},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.1806158721446991},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.1541244387626648},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.12991037964820862},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0},{"id":"https://openalex.org/C56739046","wikidata":"https://www.wikidata.org/wiki/Q192060","display_name":"Knowledge management","level":1,"score":0.0},{"id":"https://openalex.org/C3017618536","wikidata":"https://www.wikidata.org/wiki/Q304994","display_name":"Innovation diffusion","level":2,"score":0.0},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym14040740","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14040740","pdf_url":"https://www.mdpi.com/2073-8994/14/4/740/pdf?version=1649063330","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:d43a3c6d531d4b32999ebf383837e7ca","is_oa":true,"landing_page_url":"https://doaj.org/article/d43a3c6d531d4b32999ebf383837e7ca","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":"Symmetry, Vol 14, Iss 4, p 740 (2022)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/14/4/740/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym14040740","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym14040740","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14040740","pdf_url":"https://www.mdpi.com/2073-8994/14/4/740/pdf?version=1649063330","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[{"score":0.6700000166893005,"id":"https://metadata.un.org/sdg/6","display_name":"Clean water and sanitation"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4226028217.pdf"},"referenced_works_count":36,"referenced_works":["https://openalex.org/W2002704263","https://openalex.org/W2010016067","https://openalex.org/W2011412688","https://openalex.org/W2013688013","https://openalex.org/W2036791637","https://openalex.org/W2043756190","https://openalex.org/W2045849932","https://openalex.org/W2045885666","https://openalex.org/W2046828453","https://openalex.org/W2053833421","https://openalex.org/W2063067239","https://openalex.org/W2067577527","https://openalex.org/W2074916122","https://openalex.org/W2080011055","https://openalex.org/W2086737536","https://openalex.org/W2089237451","https://openalex.org/W2547841742","https://openalex.org/W2748008518","https://openalex.org/W2795854746","https://openalex.org/W2898513326","https://openalex.org/W2963268379","https://openalex.org/W2994823043","https://openalex.org/W3091605082","https://openalex.org/W3153325369","https://openalex.org/W3204096462","https://openalex.org/W4200106792","https://openalex.org/W4200296747","https://openalex.org/W4205943247","https://openalex.org/W4210748381","https://openalex.org/W4220732021","https://openalex.org/W4221102633","https://openalex.org/W4225729877","https://openalex.org/W6750365412","https://openalex.org/W6805574696","https://openalex.org/W7048023969","https://openalex.org/W7074551014"],"related_works":["https://openalex.org/W3129602449","https://openalex.org/W2384879887","https://openalex.org/W2142979439","https://openalex.org/W2375736556","https://openalex.org/W3034439966","https://openalex.org/W2960646972","https://openalex.org/W2786290219","https://openalex.org/W2380610144","https://openalex.org/W2963836928","https://openalex.org/W4382939522"],"abstract_inverted_index":{"The":[0,14],"space-fractional":[1],"stochastic":[2,34,82],"approximate":[3],"long":[4],"water":[5],"wave":[6],"equation":[7,16,28],"(SFSALWWE)":[8],"is":[9,18],"considered":[10],"in":[11],"this":[12],"work.":[13],"Riccati":[15],"method":[17],"used":[19],"to":[20,72],"get":[21],"analytical":[22,80],"solutions":[23],"of":[24,55,61],"the":[25,40,45,50,53,59],"SFSALWWE.":[26],"This":[27],"has":[29],"never":[30],"been":[31],"examined":[32],"with":[33],"term":[35,47],"and":[36,75],"fractional":[37,81],"space":[38],"at":[39],"same":[41],"time.":[42],"In":[43],"general,":[44],"noise":[46],"that":[48],"preserves":[49],"symmetry":[51],"reduces":[52],"domain":[54],"instability.":[56],"To":[57],"check":[58],"impact":[60],"Brownian":[62],"motion":[63],"on":[64],"these":[65],"solutions,":[66],"we":[67],"use":[68],"a":[69],"MATLAB":[70],"package":[71],"plot":[73],"3D":[74],"2D":[76],"graphs":[77],"for":[78],"some":[79],"solutions.":[83]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":4},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":7},{"year":2022,"cited_by_count":8}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}
