{"id":"https://openalex.org/W4387730329","doi":"https://doi.org/10.3390/sym15101929","title":"A Novel Quintic B-Spline Technique for Numerical Solutions of the Fourth-Order Singular Singularly-Perturbed Problems","display_name":"A Novel Quintic B-Spline Technique for Numerical Solutions of the Fourth-Order Singular Singularly-Perturbed Problems","publication_year":2023,"publication_date":"2023-10-18","ids":{"openalex":"https://openalex.org/W4387730329","doi":"https://doi.org/10.3390/sym15101929"},"language":"en","primary_location":{"id":"doi:10.3390/sym15101929","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15101929","pdf_url":"https://www.mdpi.com/2073-8994/15/10/1929/pdf?version=1697604855","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/2073-8994/15/10/1929/pdf?version=1697604855","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5009646445","display_name":"Muhammad Zain Yousaf","orcid":"https://orcid.org/0000-0002-6664-8235"},"institutions":[{"id":"https://openalex.org/I160968435","display_name":"University of Sargodha","ror":"https://ror.org/0086rpr26","country_code":"PK","type":"education","lineage":["https://openalex.org/I160968435"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Muhammad Zain Yousaf","raw_affiliation_strings":["Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan","institution_ids":["https://openalex.org/I160968435"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5079512627","display_name":"H. M. Srivastava","orcid":"https://orcid.org/0000-0002-9277-8092"},"institutions":[{"id":"https://openalex.org/I212119943","display_name":"University of Victoria","ror":"https://ror.org/04s5mat29","country_code":"CA","type":"education","lineage":["https://openalex.org/I212119943"]},{"id":"https://openalex.org/I35928602","display_name":"Kyung Hee University","ror":"https://ror.org/01zqcg218","country_code":"KR","type":"education","lineage":["https://openalex.org/I35928602"]},{"id":"https://openalex.org/I4210155399","display_name":"UniNettuno University","ror":"https://ror.org/04q0nep37","country_code":"IT","type":"education","lineage":["https://openalex.org/I4210155399"]}],"countries":["CA","IT","KR"],"is_corresponding":true,"raw_author_name":"Hari Mohan Srivastava","raw_affiliation_strings":["Center for Converging Humanities, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul 02447, Republic of Korea","Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada","Section of Mathematics, International Telematic University Uninettuno, I-00186 Rome, Italy"],"raw_orcid":"https://orcid.org/0000-0002-9277-8092","affiliations":[{"raw_affiliation_string":"Center for Converging Humanities, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul 02447, Republic of Korea","institution_ids":["https://openalex.org/I35928602"]},{"raw_affiliation_string":"Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada","institution_ids":["https://openalex.org/I212119943"]},{"raw_affiliation_string":"Section of Mathematics, International Telematic University Uninettuno, I-00186 Rome, Italy","institution_ids":["https://openalex.org/I4210155399"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5036734666","display_name":"Muhammad Abbas","orcid":"https://orcid.org/0000-0002-0491-1528"},"institutions":[{"id":"https://openalex.org/I160968435","display_name":"University of Sargodha","ror":"https://ror.org/0086rpr26","country_code":"PK","type":"education","lineage":["https://openalex.org/I160968435"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Muhammad Abbas","raw_affiliation_strings":["Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan"],"raw_orcid":"https://orcid.org/0000-0002-0491-1528","affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan","institution_ids":["https://openalex.org/I160968435"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5080291734","display_name":"Tahir Nazir","orcid":"https://orcid.org/0000-0002-2529-0302"},"institutions":[{"id":"https://openalex.org/I160968435","display_name":"University of Sargodha","ror":"https://ror.org/0086rpr26","country_code":"PK","type":"education","lineage":["https://openalex.org/I160968435"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Tahir Nazir","raw_affiliation_strings":["Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan","institution_ids":["https://openalex.org/I160968435"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5059025015","display_name":"Pshtiwan Othman Mohammed","orcid":"https://orcid.org/0000-0001-6837-8075"},"institutions":[{"id":"https://openalex.org/I64952554","display_name":"University of Sulaimani","ror":"https://ror.org/00saanr69","country_code":"IQ","type":"education","lineage":["https://openalex.org/I64952554"]}],"countries":["IQ"],"is_corresponding":true,"raw_author_name":"Pshtiwan Othman Mohammed","raw_affiliation_strings":["Department of Mathematics, College of Education, University of Sulaimani, Sulaimani 46001, Iraq"],"raw_orcid":"https://orcid.org/0000-0001-6837-8075","affiliations":[{"raw_affiliation_string":"Department of Mathematics, College of Education, University of Sulaimani, Sulaimani 46001, Iraq","institution_ids":["https://openalex.org/I64952554"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5066464647","display_name":"Miguel Jos\u00e9 Vivas Cortez","orcid":"https://orcid.org/0000-0002-1567-0264"},"institutions":[{"id":"https://openalex.org/I39398033","display_name":"Pontificia Universidad Cat\u00f3lica del Ecuador","ror":"https://ror.org/02qztda51","country_code":"EC","type":"education","lineage":["https://openalex.org/I39398033"]}],"countries":["EC"],"is_corresponding":true,"raw_author_name":"Miguel Vivas-Cortez","raw_affiliation_strings":["Faculty of Exact and Natural Sciences, School of Physical Sciences and Mathematics, Pontifical Catholic University of Ecuador, Av. 12 de Octubre 1076 y Roca, Apartado, Quito 17-01-2184, Ecuador"],"raw_orcid":"https://orcid.org/0000-0002-1567-0264","affiliations":[{"raw_affiliation_string":"Faculty of Exact and Natural Sciences, School of Physical Sciences and Mathematics, Pontifical Catholic University of Ecuador, Av. 12 de Octubre 1076 y Roca, Apartado, Quito 17-01-2184, Ecuador","institution_ids":["https://openalex.org/I39398033"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5088075563","display_name":"Nejmeddine Chorfi","orcid":"https://orcid.org/0000-0001-8833-6585"},"institutions":[{"id":"https://openalex.org/I28022161","display_name":"King Saud University","ror":"https://ror.org/02f81g417","country_code":"SA","type":"education","lineage":["https://openalex.org/I28022161"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Nejmeddine Chorfi","raw_affiliation_strings":["Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia","institution_ids":["https://openalex.org/I28022161"]}]}],"institutions":[],"countries_distinct_count":7,"institutions_distinct_count":7,"corresponding_author_ids":["https://openalex.org/A5059025015","https://openalex.org/A5066464647","https://openalex.org/A5079512627"],"corresponding_institution_ids":["https://openalex.org/I212119943","https://openalex.org/I35928602","https://openalex.org/I39398033","https://openalex.org/I4210155399","https://openalex.org/I64952554"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":2.6822,"has_fulltext":true,"cited_by_count":7,"citation_normalized_percentile":{"value":0.89547387,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":95,"max":98},"biblio":{"volume":"15","issue":"10","first_page":"1929","last_page":"1929"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12727","display_name":"Differential Equations and Numerical Methods","score":0.9998999834060669,"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":0.9998999834060669,"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/T13685","display_name":"Material Science and Thermodynamics","score":0.9656999707221985,"subfield":{"id":"https://openalex.org/subfields/2210","display_name":"Mechanical Engineering"},"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/boundary-value-problem","display_name":"Boundary value problem","score":0.5972113013267517},{"id":"https://openalex.org/keywords/linearization","display_name":"Linearization","score":0.5798746943473816},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5622251033782959},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5086591839790344},{"id":"https://openalex.org/keywords/singular-perturbation","display_name":"Singular perturbation","score":0.46564826369285583},{"id":"https://openalex.org/keywords/partition","display_name":"Partition (number theory)","score":0.453920841217041},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.3248017430305481},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.31605005264282227},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.1951335072517395},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.15360406041145325}],"concepts":[{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.5972113013267517},{"id":"https://openalex.org/C11210021","wikidata":"https://www.wikidata.org/wiki/Q1520713","display_name":"Linearization","level":3,"score":0.5798746943473816},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5622251033782959},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5086591839790344},{"id":"https://openalex.org/C165160513","wikidata":"https://www.wikidata.org/wiki/Q7524249","display_name":"Singular perturbation","level":2,"score":0.46564826369285583},{"id":"https://openalex.org/C42812","wikidata":"https://www.wikidata.org/wiki/Q1082910","display_name":"Partition (number theory)","level":2,"score":0.453920841217041},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3248017430305481},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.31605005264282227},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.1951335072517395},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.15360406041145325},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","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}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym15101929","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15101929","pdf_url":"https://www.mdpi.com/2073-8994/15/10/1929/pdf?version=1697604855","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:333358aefa2340b19675365121313a49","is_oa":false,"landing_page_url":"https://doaj.org/article/333358aefa2340b19675365121313a49","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 10, p 1929 (2023)","raw_type":"article"},{"id":"pmh:oai:dspace.library.uvic.ca:1828/15787","is_oa":true,"landing_page_url":"http://hdl.handle.net/1828/15787","pdf_url":null,"source":{"id":"https://openalex.org/S4306401509","display_name":"UVic\u2019s Research and Learning Repository (University of Victoria)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I212119943","host_organization_name":"University of Victoria","host_organization_lineage":["https://openalex.org/I212119943"],"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":"","raw_type":"Article"}],"best_oa_location":{"id":"doi:10.3390/sym15101929","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15101929","pdf_url":"https://www.mdpi.com/2073-8994/15/10/1929/pdf?version=1697604855","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":[{"id":"https://openalex.org/F4320321145","display_name":"King Saud University","ror":"https://ror.org/02f81g417"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4387730329.pdf","grobid_xml":"https://content.openalex.org/works/W4387730329.grobid-xml"},"referenced_works_count":31,"referenced_works":["https://openalex.org/W1985467574","https://openalex.org/W1988973528","https://openalex.org/W2013663612","https://openalex.org/W2016562746","https://openalex.org/W2023151433","https://openalex.org/W2042561164","https://openalex.org/W2051397382","https://openalex.org/W2073103099","https://openalex.org/W2121118956","https://openalex.org/W2154392286","https://openalex.org/W2154542282","https://openalex.org/W2334773148","https://openalex.org/W2344420529","https://openalex.org/W2411694874","https://openalex.org/W2508415445","https://openalex.org/W2514460319","https://openalex.org/W2955129014","https://openalex.org/W3017070889","https://openalex.org/W3162085095","https://openalex.org/W3200651239","https://openalex.org/W4206606587","https://openalex.org/W4252247111","https://openalex.org/W4313377891","https://openalex.org/W4366493163","https://openalex.org/W4378229123","https://openalex.org/W4383498012","https://openalex.org/W4385155115","https://openalex.org/W4386310962","https://openalex.org/W6677972282","https://openalex.org/W6725451928","https://openalex.org/W6725802411"],"related_works":["https://openalex.org/W2553468687","https://openalex.org/W2055379029","https://openalex.org/W2953137247","https://openalex.org/W3161249280","https://openalex.org/W2011339542","https://openalex.org/W2267059662","https://openalex.org/W2362574016","https://openalex.org/W2666785071","https://openalex.org/W2001488143","https://openalex.org/W2392492385"],"abstract_inverted_index":{"Singular":[0],"singularly-perturbed":[1,34],"problems":[2,39,65,73,145],"(SSPPs)":[3],"are":[4,40,66,120,153],"a":[5,11,43,53,69,143],"powerful":[6],"mathematical":[7],"tool":[8],"for":[9],"modelling":[10],"variety":[12],"of":[13,71,93,100],"real":[14],"phenomena,":[15],"such":[16],"as":[17],"nuclear":[18],"reactions,":[19],"heat":[20],"explosions,":[21],"mechanics,":[22],"and":[23,36,98,155],"hydrodynamics.":[24],"In":[25],"this":[26],"paper,":[27],"the":[28,63,76,111,115,127,134,148,159],"numerical":[29,129],"solutions":[30],"to":[31,56,88,108,146],"fourth-order":[32],"singular":[33],"boundary":[35],"initial":[37],"value":[38,60],"presented":[41],"using":[42],"novel":[44],"quintic":[45],"B-spline":[46],"(QBS)":[47],"approximation":[48],"approach.":[49,78],"This":[50,103],"method":[51,104],"uses":[52],"quasi-linearization":[54,77],"approach":[55,130,139,161],"solve":[57,110],"SSPNL":[58],"initial/boundary":[59],"problems.":[61],"And":[62],"non-linear":[64],"transformed":[67],"into":[68],"sequence":[70],"linear":[72],"by":[74],"applying":[75],"The":[79,137,150],"QBS":[80],"functions":[81],"produce":[82],"more":[83],"accurate":[84],"results":[85,152],"when":[86],"compared":[87],"other":[89],"existing":[90],"approaches":[91],"because":[92],"their":[94],"local":[95],"support,":[96],"symmetry,":[97],"partition":[99],"unity":[101],"features.":[102],"can":[105],"be":[106],"applied":[107],"immediately":[109],"SSPPs":[112],"without":[113],"reducing":[114],"order":[116],"in":[117],"which":[118],"they":[119,156],"presented.":[121],"It":[122],"has":[123],"been":[124],"demonstrated":[125],"that":[126,158],"suggested":[128],"converges":[131],"uniformly":[132],"over":[133],"whole":[135],"domain.":[136],"proposed":[138,160],"is":[140],"implemented":[141],"on":[142],"few":[144],"validate":[147],"scheme.":[149],"computational":[151],"compared,":[154],"illustrate":[157],"performs":[162],"better.":[163]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":5}],"updated_date":"2026-06-09T15:46:55.921056","created_date":"2025-10-10T00:00:00"}