{"id":"https://openalex.org/W4392456566","doi":"https://doi.org/10.3390/computation12030051","title":"Extension of Cubic B-Spline for Solving the Time-Fractional Allen\u2013Cahn Equation in the Context of Mathematical Physics","display_name":"Extension of Cubic B-Spline for Solving the Time-Fractional Allen\u2013Cahn Equation in the Context of Mathematical Physics","publication_year":2024,"publication_date":"2024-03-05","ids":{"openalex":"https://openalex.org/W4392456566","doi":"https://doi.org/10.3390/computation12030051"},"language":"en","primary_location":{"id":"doi:10.3390/computation12030051","is_oa":true,"landing_page_url":"https://doi.org/10.3390/computation12030051","pdf_url":"https://www.mdpi.com/2079-3197/12/3/51/pdf?version=1709652161","source":{"id":"https://openalex.org/S2738402919","display_name":"Computation","issn_l":"2079-3197","issn":["2079-3197"],"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":"Computation","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2079-3197/12/3/51/pdf?version=1709652161","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5113133954","display_name":"Mubeen Fatima","orcid":null},"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":"Mubeen Fatima","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/A5055431798","display_name":"Ravi P. Agarwal","orcid":"https://orcid.org/0000-0003-0634-2370"},"institutions":[{"id":"https://openalex.org/I181414168","display_name":"Texas A&M University \u2013 Kingsville","ror":"https://ror.org/05abs3w97","country_code":"US","type":"education","lineage":["https://openalex.org/I181414168"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Ravi P. Agarwal","raw_affiliation_strings":["Department of Mathematics, Texas A & M University-Kingsville, Kingsville, TX 78363, USA"],"raw_orcid":"https://orcid.org/0000-0003-0634-2370","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Texas A & M University-Kingsville, Kingsville, TX 78363, USA","institution_ids":["https://openalex.org/I181414168"]}]},{"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/A5059025015","display_name":"Pshtiwan Othman Mohammed","orcid":null},"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, Sulaymaniyah 46001, Iraq","Research and Development Center, University of Sulaimani, Sulaymaniyah 46001, Iraq"],"raw_orcid":"https://orcid.org/0000-0001-6837-8075","affiliations":[{"raw_affiliation_string":"Department of Mathematics, College of Education, University of Sulaimani, Sulaymaniyah 46001, Iraq","institution_ids":["https://openalex.org/I64952554"]},{"raw_affiliation_string":"Research and Development Center, University of Sulaimani, Sulaymaniyah 46001, Iraq","institution_ids":["https://openalex.org/I64952554"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5018429906","display_name":"Madiha Shafiq","orcid":null},"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":"Madiha Shafiq","raw_affiliation_strings":["Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan"],"raw_orcid":"https://orcid.org/0009-0002-4084-442X","affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan","institution_ids":["https://openalex.org/I160968435"]}]},{"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":"https://orcid.org/0000-0001-8833-6585","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":4,"institutions_distinct_count":6,"corresponding_author_ids":["https://openalex.org/A5059025015"],"corresponding_institution_ids":["https://openalex.org/I64952554"],"apc_list":{"value":1400,"currency":"CHF","value_usd":1515},"apc_paid":{"value":1400,"currency":"CHF","value_usd":1515},"fwci":2.6692,"has_fulltext":true,"cited_by_count":7,"citation_normalized_percentile":{"value":0.88651244,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":95,"max":98},"biblio":{"volume":"12","issue":"3","first_page":"51","last_page":"51"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9997000098228455,"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":0.9997000098228455,"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/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/T10248","display_name":"Nonlinear Waves and Solitons","score":0.9927999973297119,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/extension","display_name":"Extension (predicate logic)","score":0.7484925389289856},{"id":"https://openalex.org/keywords/allen\u2013cahn-equation","display_name":"Allen\u2013Cahn equation","score":0.714094877243042},{"id":"https://openalex.org/keywords/context","display_name":"Context (archaeology)","score":0.5292783379554749},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.42075207829475403},{"id":"https://openalex.org/keywords/mathematical-physics","display_name":"Mathematical physics","score":0.4110277593135834},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.36922913789749146},{"id":"https://openalex.org/keywords/calculus","display_name":"Calculus (dental)","score":0.3510677218437195},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.1870644986629486},{"id":"https://openalex.org/keywords/geology","display_name":"Geology","score":0.06540203094482422}],"concepts":[{"id":"https://openalex.org/C2778029271","wikidata":"https://www.wikidata.org/wiki/Q5421931","display_name":"Extension (predicate logic)","level":2,"score":0.7484925389289856},{"id":"https://openalex.org/C2777166216","wikidata":"https://www.wikidata.org/wiki/Q4732089","display_name":"Allen\u2013Cahn equation","level":2,"score":0.714094877243042},{"id":"https://openalex.org/C2779343474","wikidata":"https://www.wikidata.org/wiki/Q3109175","display_name":"Context (archaeology)","level":2,"score":0.5292783379554749},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.42075207829475403},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.4110277593135834},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.36922913789749146},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.3510677218437195},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.1870644986629486},{"id":"https://openalex.org/C127313418","wikidata":"https://www.wikidata.org/wiki/Q1069","display_name":"Geology","level":0,"score":0.06540203094482422},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"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/C199343813","wikidata":"https://www.wikidata.org/wiki/Q12128","display_name":"Dentistry","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/computation12030051","is_oa":true,"landing_page_url":"https://doi.org/10.3390/computation12030051","pdf_url":"https://www.mdpi.com/2079-3197/12/3/51/pdf?version=1709652161","source":{"id":"https://openalex.org/S2738402919","display_name":"Computation","issn_l":"2079-3197","issn":["2079-3197"],"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":"Computation","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:c13d096910a24b23972422118b4787c8","is_oa":true,"landing_page_url":"https://doaj.org/article/c13d096910a24b23972422118b4787c8","pdf_url":null,"source":{"id":"https://openalex.org/S112646816","display_name":"SHILAP Revista de lepidopterolog\u00eda","issn_l":"0300-5267","issn":["0300-5267","2340-4078"],"is_oa":true,"is_in_doaj":true,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Computation, Vol 12, Iss 3, p 51 (2024)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/computation12030051","is_oa":true,"landing_page_url":"https://doi.org/10.3390/computation12030051","pdf_url":"https://www.mdpi.com/2079-3197/12/3/51/pdf?version=1709652161","source":{"id":"https://openalex.org/S2738402919","display_name":"Computation","issn_l":"2079-3197","issn":["2079-3197"],"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":"Computation","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":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4392456566.pdf","grobid_xml":"https://content.openalex.org/works/W4392456566.grobid-xml"},"referenced_works_count":53,"referenced_works":["https://openalex.org/W852831235","https://openalex.org/W941945513","https://openalex.org/W1523640391","https://openalex.org/W1787698886","https://openalex.org/W1972262069","https://openalex.org/W1984647578","https://openalex.org/W1990282067","https://openalex.org/W1998413407","https://openalex.org/W2021095734","https://openalex.org/W2041001120","https://openalex.org/W2045849932","https://openalex.org/W2079184891","https://openalex.org/W2084797093","https://openalex.org/W2086737536","https://openalex.org/W2089666393","https://openalex.org/W2140447289","https://openalex.org/W2141112962","https://openalex.org/W2142631307","https://openalex.org/W2146423481","https://openalex.org/W2170339701","https://openalex.org/W2172191622","https://openalex.org/W2265245553","https://openalex.org/W2315857262","https://openalex.org/W2339657375","https://openalex.org/W2551065309","https://openalex.org/W2561735245","https://openalex.org/W2565752446","https://openalex.org/W2568377229","https://openalex.org/W2589656637","https://openalex.org/W2642280594","https://openalex.org/W2731531705","https://openalex.org/W2747592978","https://openalex.org/W2767618852","https://openalex.org/W2806868147","https://openalex.org/W2811177600","https://openalex.org/W2886181830","https://openalex.org/W2914754930","https://openalex.org/W2952055402","https://openalex.org/W2963929757","https://openalex.org/W2971430436","https://openalex.org/W3023384070","https://openalex.org/W3031710104","https://openalex.org/W3133855275","https://openalex.org/W3162085095","https://openalex.org/W3184975075","https://openalex.org/W3188810325","https://openalex.org/W4210702245","https://openalex.org/W4226307167","https://openalex.org/W4311978818","https://openalex.org/W4367310173","https://openalex.org/W4387730329","https://openalex.org/W4389829127","https://openalex.org/W4389885514"],"related_works":["https://openalex.org/W2787174914","https://openalex.org/W4302407027","https://openalex.org/W2075768550","https://openalex.org/W3022218857","https://openalex.org/W2369178846","https://openalex.org/W2964308048","https://openalex.org/W2370289839","https://openalex.org/W4388722486","https://openalex.org/W2082716031","https://openalex.org/W1727769347"],"abstract_inverted_index":{"A":[0],"B-spline":[1,30],"is":[2,13,58,69,78,88],"defined":[3],"by":[4],"the":[5,50,62,72,82,85],"degree":[6],"and":[7,11,24,39,81,94],"quantity":[8],"of":[9,20,84],"knots,":[10],"it":[12],"observed":[14],"to":[15,48,60],"provide":[16],"a":[17,66],"higher":[18],"level":[19],"flexibility":[21],"in":[22,45],"curve":[23],"surface":[25],"layout.":[26],"The":[27,91],"extended":[28],"cubic":[29],"(ExCBS)":[31],"functions":[32],"with":[33],"new":[34,67],"approximation":[35],"for":[36,71],"second":[37],"derivative":[38],"finite":[40],"difference":[41],"technique":[42],"are":[43,96],"incorporated":[44],"this":[46],"study":[47],"solve":[49],"time-fractional":[51,63],"Allen\u2013Cahn":[52],"equation":[53],"(TFACE).":[54],"Initially,":[55],"Caputo\u2019s":[56],"formula":[57],"used":[59,70],"discretize":[61],"derivative,":[64],"while":[65],"ExCBS":[68],"spatial":[73],"derivative\u2019s":[74],"discretization.":[75],"Convergence":[76],"analysis":[77],"carried":[79],"out":[80],"stability":[83],"proposed":[86],"method":[87],"also":[89],"analyzed.":[90],"scheme\u2019s":[92],"applicability":[93],"feasibility":[95],"demonstrated":[97],"through":[98],"numerical":[99],"analysis.":[100]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":4}],"updated_date":"2026-03-27T05:58:40.876381","created_date":"2025-10-10T00:00:00"}