{"id":"https://openalex.org/W4389300417","doi":"https://doi.org/10.3390/sym15122152","title":"A Modified Residual Power Series Method for the Approximate Solution of Two-Dimensional Fractional Helmholtz Equations","display_name":"A Modified Residual Power Series Method for the Approximate Solution of Two-Dimensional Fractional Helmholtz Equations","publication_year":2023,"publication_date":"2023-12-04","ids":{"openalex":"https://openalex.org/W4389300417","doi":"https://doi.org/10.3390/sym15122152"},"language":"en","primary_location":{"id":"doi:10.3390/sym15122152","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15122152","pdf_url":"https://www.mdpi.com/2073-8994/15/12/2152/pdf?version=1701671585","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/2073-8994/15/12/2152/pdf?version=1701671585","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5043251411","display_name":"Jinxing Liu","orcid":"https://orcid.org/0000-0003-1071-2231"},"institutions":[{"id":"https://openalex.org/I4210131342","display_name":"Yibin University","ror":"https://ror.org/03w8m2977","country_code":"CN","type":"education","lineage":["https://openalex.org/I4210131342"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Jinxing Liu","raw_affiliation_strings":["Faculty of Science, Yibin University, Yibin 644000, China"],"raw_orcid":"https://orcid.org/0000-0003-1071-2231","affiliations":[{"raw_affiliation_string":"Faculty of Science, Yibin University, Yibin 644000, China","institution_ids":["https://openalex.org/I4210131342"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5069957230","display_name":"Muhammad Nadeem","orcid":"https://orcid.org/0000-0002-9349-4729"},"institutions":[{"id":"https://openalex.org/I4210105365","display_name":"Qujing Normal University","ror":"https://ror.org/02ad7ap24","country_code":"CN","type":"education","lineage":["https://openalex.org/I4210105365"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Muhammad Nadeem","raw_affiliation_strings":["School of Mathematics and Statistics, Qujing Normal University, Qujing 655011, China"],"raw_orcid":"https://orcid.org/0000-0002-9349-4729","affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, Qujing Normal University, Qujing 655011, China","institution_ids":["https://openalex.org/I4210105365"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101963275","display_name":"Asad Islam","orcid":"https://orcid.org/0000-0002-4192-7211"},"institutions":[{"id":"https://openalex.org/I899713450","display_name":"Air University","ror":"https://ror.org/03yfe9v83","country_code":"PK","type":"education","lineage":["https://openalex.org/I899713450"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Asad Islam","raw_affiliation_strings":["Department of Mechanical and Aerospace Engineering, Air University, Islamabad 44000, Pakistan"],"raw_orcid":"https://orcid.org/0000-0002-4192-7211","affiliations":[{"raw_affiliation_string":"Department of Mechanical and Aerospace Engineering, Air University, Islamabad 44000, Pakistan","institution_ids":["https://openalex.org/I899713450"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5016978313","display_name":"Sorin Mure\u015fan","orcid":"https://orcid.org/0000-0002-1882-3748"},"institutions":[{"id":"https://openalex.org/I162871001","display_name":"University of Oradea","ror":"https://ror.org/00wzhv093","country_code":"RO","type":"education","lineage":["https://openalex.org/I162871001"]}],"countries":["RO"],"is_corresponding":false,"raw_author_name":"Sorin Mure\u015fan","raw_affiliation_strings":["Department of Mathematics and Computer Science, University of Oradea, 1 University Street, 410087 Oradea, Romania"],"raw_orcid":"https://orcid.org/0000-0002-1882-3748","affiliations":[{"raw_affiliation_string":"Department of Mathematics and Computer Science, University of Oradea, 1 University Street, 410087 Oradea, Romania","institution_ids":["https://openalex.org/I162871001"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5075452466","display_name":"Loredana Florentina Iambor","orcid":"https://orcid.org/0000-0001-8845-3095"},"institutions":[{"id":"https://openalex.org/I162871001","display_name":"University of Oradea","ror":"https://ror.org/00wzhv093","country_code":"RO","type":"education","lineage":["https://openalex.org/I162871001"]}],"countries":["RO"],"is_corresponding":true,"raw_author_name":"Loredana Florentina Iambor","raw_affiliation_strings":["Department of Mathematics and Computer Science, University of Oradea, 1 University Street, 410087 Oradea, Romania"],"raw_orcid":"https://orcid.org/0000-0001-8845-3095","affiliations":[{"raw_affiliation_string":"Department of Mathematics and Computer Science, University of Oradea, 1 University Street, 410087 Oradea, Romania","institution_ids":["https://openalex.org/I162871001"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5069957230","https://openalex.org/A5075452466"],"corresponding_institution_ids":["https://openalex.org/I162871001","https://openalex.org/I4210105365"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":1.0653,"has_fulltext":true,"cited_by_count":5,"citation_normalized_percentile":{"value":0.74890928,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":97},"biblio":{"volume":"15","issue":"12","first_page":"2152","last_page":"2152"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9998000264167786,"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":0.9998000264167786,"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":0.9839000105857849,"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/T10514","display_name":"Numerical methods in engineering","score":0.9593999981880188,"subfield":{"id":"https://openalex.org/subfields/2211","display_name":"Mechanics of Materials"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/residual","display_name":"Residual","score":0.7750077247619629},{"id":"https://openalex.org/keywords/series","display_name":"Series (stratigraphy)","score":0.7439395785331726},{"id":"https://openalex.org/keywords/power-series","display_name":"Power series","score":0.6934463977813721},{"id":"https://openalex.org/keywords/helmholtz-equation","display_name":"Helmholtz equation","score":0.6476208567619324},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6386328339576721},{"id":"https://openalex.org/keywords/helmholtz-free-energy","display_name":"Helmholtz free energy","score":0.6231545209884644},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5590930581092834},{"id":"https://openalex.org/keywords/fractional-calculus","display_name":"Fractional calculus","score":0.46936801075935364},{"id":"https://openalex.org/keywords/power","display_name":"Power (physics)","score":0.44465214014053345},{"id":"https://openalex.org/keywords/work","display_name":"Work (physics)","score":0.43986648321151733},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3147696554660797},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.2859131693840027},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.07445046305656433}],"concepts":[{"id":"https://openalex.org/C155512373","wikidata":"https://www.wikidata.org/wiki/Q287450","display_name":"Residual","level":2,"score":0.7750077247619629},{"id":"https://openalex.org/C143724316","wikidata":"https://www.wikidata.org/wiki/Q312468","display_name":"Series (stratigraphy)","level":2,"score":0.7439395785331726},{"id":"https://openalex.org/C73905626","wikidata":"https://www.wikidata.org/wiki/Q206925","display_name":"Power series","level":2,"score":0.6934463977813721},{"id":"https://openalex.org/C18591234","wikidata":"https://www.wikidata.org/wiki/Q860615","display_name":"Helmholtz equation","level":3,"score":0.6476208567619324},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6386328339576721},{"id":"https://openalex.org/C27592594","wikidata":"https://www.wikidata.org/wiki/Q865821","display_name":"Helmholtz free energy","level":2,"score":0.6231545209884644},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5590930581092834},{"id":"https://openalex.org/C154249771","wikidata":"https://www.wikidata.org/wiki/Q1339058","display_name":"Fractional calculus","level":2,"score":0.46936801075935364},{"id":"https://openalex.org/C163258240","wikidata":"https://www.wikidata.org/wiki/Q25342","display_name":"Power (physics)","level":2,"score":0.44465214014053345},{"id":"https://openalex.org/C18762648","wikidata":"https://www.wikidata.org/wiki/Q42213","display_name":"Work (physics)","level":2,"score":0.43986648321151733},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3147696554660797},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.2859131693840027},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.07445046305656433},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","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/C151730666","wikidata":"https://www.wikidata.org/wiki/Q7205","display_name":"Paleontology","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/sym15122152","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15122152","pdf_url":"https://www.mdpi.com/2073-8994/15/12/2152/pdf?version=1701671585","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:7c59119527c84d1babc3d1130da06d29","is_oa":false,"landing_page_url":"https://doaj.org/article/7c59119527c84d1babc3d1130da06d29","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 15, Iss 12, p 2152 (2023)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/sym15122152","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15122152","pdf_url":"https://www.mdpi.com/2073-8994/15/12/2152/pdf?version=1701671585","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":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":false},"content_urls":{"pdf":"https://content.openalex.org/works/W4389300417.pdf"},"referenced_works_count":32,"referenced_works":["https://openalex.org/W2034832920","https://openalex.org/W2040987249","https://openalex.org/W2052302718","https://openalex.org/W2057323358","https://openalex.org/W2073806305","https://openalex.org/W2093651704","https://openalex.org/W2101961371","https://openalex.org/W2301349605","https://openalex.org/W2320122857","https://openalex.org/W2794210887","https://openalex.org/W2897394303","https://openalex.org/W2921117343","https://openalex.org/W2922153620","https://openalex.org/W2995503801","https://openalex.org/W2997074511","https://openalex.org/W3019954140","https://openalex.org/W3028167646","https://openalex.org/W3085151210","https://openalex.org/W3109390860","https://openalex.org/W3172025975","https://openalex.org/W3190242550","https://openalex.org/W3207442474","https://openalex.org/W4200272077","https://openalex.org/W4224292209","https://openalex.org/W4281259876","https://openalex.org/W4309916329","https://openalex.org/W4311119663","https://openalex.org/W4323356190","https://openalex.org/W4362571199","https://openalex.org/W6698191729","https://openalex.org/W6786532235","https://openalex.org/W6810356550"],"related_works":["https://openalex.org/W2560215812","https://openalex.org/W2092968426","https://openalex.org/W2949601986","https://openalex.org/W2540416959","https://openalex.org/W2788972299","https://openalex.org/W2521347458","https://openalex.org/W4365420316","https://openalex.org/W3193896743","https://openalex.org/W3176862315","https://openalex.org/W3120022531"],"abstract_inverted_index":{"In":[0],"this":[1,61],"paper,":[2],"we":[3],"suggest":[4],"a":[5,34],"modification":[6],"for":[7,76,98],"the":[8,24,37,42,69,74],"residual":[9,26,43],"power":[10,27,44],"series":[11,28,45,87],"method":[12,29,46],"that":[13,64,88],"is":[14,22,63],"used":[15],"to":[16,55,90,106],"solve":[17],"fractional-order":[18],"Helmholtz":[19],"equations,":[20],"which":[21],"called":[23],"Shehu-transform":[25],"(ST-RPSM).":[30],"This":[31],"scheme":[32],"uses":[33],"combination":[35],"of":[36,60,101],"Shehu":[38],"transform":[39],"(ST)":[40],"and":[41,72],"(RPSM).":[47],"The":[48,58,80,94],"fractional":[49,70,102],"derivatives":[50],"are":[51,104],"taken":[52],"with":[53,110],"respect":[54],"Caputo":[56],"order.":[57],"novelty":[59],"approach":[62],"it":[65],"does":[66],"not":[67],"restrict":[68],"order":[71],"reduces":[73],"need":[75],"heavy":[77],"computational":[78],"work.":[79],"results":[81,109],"were":[82],"obtained":[83],"using":[84],"an":[85,91],"iterative":[86],"led":[89],"exact":[92,111],"solution.":[93],"3D":[95],"graphical":[96],"plots":[97],"different":[99],"values":[100],"orders":[103],"shown":[105],"compare":[107],"ST-RPSM":[108],"solutions.":[112]},"counts_by_year":[{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":2}],"updated_date":"2026-05-21T09:19:25.381259","created_date":"2025-10-10T00:00:00"}