{"id":"https://openalex.org/W4412552372","doi":"https://doi.org/10.3390/axioms14080547","title":"A Cubic Spline Numerical Method for a Singularly Perturbed Two-Parameter Ordinary Differential Equation","display_name":"A Cubic Spline Numerical Method for a Singularly Perturbed Two-Parameter Ordinary Differential Equation","publication_year":2025,"publication_date":"2025-07-22","ids":{"openalex":"https://openalex.org/W4412552372","doi":"https://doi.org/10.3390/axioms14080547"},"language":"en","primary_location":{"id":"doi:10.3390/axioms14080547","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms14080547","pdf_url":"https://www.mdpi.com/2075-1680/14/8/547/pdf?version=1753167281","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/2075-1680/14/8/547/pdf?version=1753167281","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5089971351","display_name":"Hassan J. Al Salman","orcid":"https://orcid.org/0000-0001-8169-5294"},"institutions":[{"id":"https://openalex.org/I4626487","display_name":"King Faisal University","ror":"https://ror.org/00dn43547","country_code":"SA","type":"education","lineage":["https://openalex.org/I4626487"]}],"countries":["SA"],"is_corresponding":true,"raw_author_name":"Hassan J. Al Salman","raw_affiliation_strings":["Department of Mathematics and Statistics, College of Science, King Faisal University, Hofuf 319832, Al Ahsa, Saudi Arabia"],"raw_orcid":"https://orcid.org/0000-0001-8169-5294","affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, College of Science, King Faisal University, Hofuf 319832, Al Ahsa, Saudi Arabia","institution_ids":["https://openalex.org/I4626487"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5074193703","display_name":"Fasika Wondimu Gelu","orcid":"https://orcid.org/0000-0002-4693-9051"},"institutions":[{"id":"https://openalex.org/I4210136909","display_name":"Dilla University","ror":"https://ror.org/04ahz4692","country_code":"ET","type":"education","lineage":["https://openalex.org/I4210136909"]}],"countries":["ET"],"is_corresponding":true,"raw_author_name":"Fasika Wondimu Gelu","raw_affiliation_strings":["Department of Mathematics, College of Natural and Computational Sciences, Dilla University, Dilla 419, Ethiopia"],"raw_orcid":"https://orcid.org/0000-0002-4693-9051","affiliations":[{"raw_affiliation_string":"Department of Mathematics, College of Natural and Computational Sciences, Dilla University, Dilla 419, Ethiopia","institution_ids":["https://openalex.org/I4210136909"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5029149642","display_name":"Ahmed A. Al Ghafli","orcid":"https://orcid.org/0000-0003-1111-3054"},"institutions":[{"id":"https://openalex.org/I4626487","display_name":"King Faisal University","ror":"https://ror.org/00dn43547","country_code":"SA","type":"education","lineage":["https://openalex.org/I4626487"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Ahmed A. Al Ghafli","raw_affiliation_strings":["Department of Mathematics and Statistics, College of Science, King Faisal University, Hofuf 319832, Al Ahsa, Saudi Arabia"],"raw_orcid":"https://orcid.org/0000-0003-1111-3054","affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, College of Science, King Faisal University, Hofuf 319832, Al Ahsa, Saudi Arabia","institution_ids":["https://openalex.org/I4626487"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5074193703","https://openalex.org/A5089971351"],"corresponding_institution_ids":["https://openalex.org/I4210136909","https://openalex.org/I4626487"],"apc_list":{"value":1400,"currency":"CHF","value_usd":1515},"apc_paid":{"value":1400,"currency":"CHF","value_usd":1515},"fwci":1.9805,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.8494969,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":95,"max":98},"biblio":{"volume":"14","issue":"8","first_page":"547","last_page":"547"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12727","display_name":"Differential Equations and Numerical Methods","score":1.0,"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":1.0,"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":0.9506999850273132,"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"}},{"id":"https://openalex.org/T13584","display_name":"Coal Combustion and Slurry Processing","score":0.9377999901771545,"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/ordinary-differential-equation","display_name":"Ordinary differential equation","score":0.7465181350708008},{"id":"https://openalex.org/keywords/monotone-cubic-interpolation","display_name":"Monotone cubic interpolation","score":0.6957809329032898},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6205903887748718},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.5090824961662292},{"id":"https://openalex.org/keywords/cubic-function","display_name":"Cubic function","score":0.44561976194381714},{"id":"https://openalex.org/keywords/exact-differential-equation","display_name":"Exact differential equation","score":0.41401752829551697},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.41057834029197693},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.34500980377197266}],"concepts":[{"id":"https://openalex.org/C51544822","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Ordinary differential equation","level":3,"score":0.7465181350708008},{"id":"https://openalex.org/C64485984","wikidata":"https://www.wikidata.org/wiki/Q4044484","display_name":"Monotone cubic interpolation","level":5,"score":0.6957809329032898},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6205903887748718},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5090824961662292},{"id":"https://openalex.org/C65559279","wikidata":"https://www.wikidata.org/wiki/Q16667694","display_name":"Cubic function","level":2,"score":0.44561976194381714},{"id":"https://openalex.org/C185184677","wikidata":"https://www.wikidata.org/wiki/Q1366795","display_name":"Exact differential equation","level":4,"score":0.41401752829551697},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.41057834029197693},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.34500980377197266},{"id":"https://openalex.org/C188325010","wikidata":"https://www.wikidata.org/wiki/Q861176","display_name":"Trilinear interpolation","level":4,"score":0.0},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.0},{"id":"https://openalex.org/C171836373","wikidata":"https://www.wikidata.org/wiki/Q2266329","display_name":"Linear interpolation","level":3,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/axioms14080547","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms14080547","pdf_url":"https://www.mdpi.com/2075-1680/14/8/547/pdf?version=1753167281","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:doaj.org/article:de1e9eb400764f938313f893ddaec2ee","is_oa":true,"landing_page_url":"https://doaj.org/article/de1e9eb400764f938313f893ddaec2ee","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 14, Iss 8, p 547 (2025)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/axioms14080547","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms14080547","pdf_url":"https://www.mdpi.com/2075-1680/14/8/547/pdf?version=1753167281","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":[],"funders":[{"id":"https://openalex.org/F4320331156","display_name":"King Faisal University","ror":null}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4412552372.pdf","grobid_xml":"https://content.openalex.org/works/W4412552372.grobid-xml"},"referenced_works_count":34,"referenced_works":["https://openalex.org/W1216919098","https://openalex.org/W1521324844","https://openalex.org/W1986356787","https://openalex.org/W2000484384","https://openalex.org/W2003767127","https://openalex.org/W2010526126","https://openalex.org/W2039530548","https://openalex.org/W2041246659","https://openalex.org/W2043888020","https://openalex.org/W2055478405","https://openalex.org/W2059344753","https://openalex.org/W2085038319","https://openalex.org/W2132352808","https://openalex.org/W2284905204","https://openalex.org/W2291247981","https://openalex.org/W2560402931","https://openalex.org/W2588064327","https://openalex.org/W2594257090","https://openalex.org/W2886742345","https://openalex.org/W3043208634","https://openalex.org/W3087299284","https://openalex.org/W3130282808","https://openalex.org/W3133922783","https://openalex.org/W3159759591","https://openalex.org/W3212578535","https://openalex.org/W4300865581","https://openalex.org/W4386171383","https://openalex.org/W4389981909","https://openalex.org/W4390085473","https://openalex.org/W4390858815","https://openalex.org/W4406127279","https://openalex.org/W6650331248","https://openalex.org/W6695670010","https://openalex.org/W6780352528"],"related_works":["https://openalex.org/W2899709058","https://openalex.org/W4289304019","https://openalex.org/W2388707757","https://openalex.org/W2187606527","https://openalex.org/W2747983507","https://openalex.org/W2257098514","https://openalex.org/W2900445293","https://openalex.org/W2954165412","https://openalex.org/W2358596639","https://openalex.org/W2085569878"],"abstract_inverted_index":{"This":[0],"paper":[1],"presents":[2],"a":[3,11,32],"uniformly":[4],"convergent":[5],"cubic":[6,27],"spline":[7,28],"numerical":[8,29,47,66],"method":[9,30,53],"for":[10],"singularly":[12],"perturbed":[13],"two-perturbation":[14],"parameter":[15],"ordinary":[16],"differential":[17,21],"equation.":[18],"The":[19,35,46],"considered":[20,63],"equation":[22],"is":[23,42],"discretized":[24],"using":[25],"the":[26,39,51],"on":[31],"Bakhvalov-type":[33],"mesh.":[34],"uniform":[36,56],"convergence":[37],"via":[38],"error":[40],"analysis":[41],"established":[43],"very":[44],"well.":[45],"findings":[48],"indicate":[49],"that":[50],"proposed":[52],"achieves":[54],"second-order":[55],"convergence.":[57],"Four":[58],"test":[59],"examples":[60],"have":[61],"been":[62],"to":[64],"perform":[65],"experimentations.":[67]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-06-15T08:34:33.830935","created_date":"2025-10-10T00:00:00"}