{"id":"https://openalex.org/W4417312510","doi":"https://doi.org/10.1007/s40314-025-03539-7","title":"An investigation of discontinuities in time-dependent 2D and 3D parabolic partial differential equations utilizing collocation methods: a comparative analysis of complex interface problems","display_name":"An investigation of discontinuities in time-dependent 2D and 3D parabolic partial differential equations utilizing collocation methods: a comparative analysis of complex interface problems","publication_year":2025,"publication_date":"2025-12-15","ids":{"openalex":"https://openalex.org/W4417312510","doi":"https://doi.org/10.1007/s40314-025-03539-7"},"language":"en","primary_location":{"id":"doi:10.1007/s40314-025-03539-7","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s40314-025-03539-7","pdf_url":null,"source":{"id":"https://openalex.org/S4210195884","display_name":"Computational and Applied Mathematics","issn_l":"1807-0302","issn":["1807-0302","2238-3603"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computational and Applied Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5113665999","display_name":"Muhammad Faheem","orcid":"https://orcid.org/0009-0000-2274-6821"},"institutions":[{"id":"https://openalex.org/I2802999120","display_name":"Government of Khyber Pakhtunkhwa","ror":"https://ror.org/04zyfmb02","country_code":"PK","type":"government","lineage":["https://openalex.org/I2802999120"]},{"id":"https://openalex.org/I36001604","display_name":"University of Peshawar","ror":"https://ror.org/02t2qwf81","country_code":"PK","type":"education","lineage":["https://openalex.org/I36001604"]}],"countries":["PK"],"is_corresponding":true,"raw_author_name":"Muhammad Faheem","raw_affiliation_strings":["Department of Mathematics, University of Peshawar, Peshawar, Khyber Pakhtunkhwa, Pakistan","Govt Degree College Badaber, Govt Degree College Badaber, Higher Education Department, Peshawar, Khyber Pakhtunkhwa, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Peshawar, Peshawar, Khyber Pakhtunkhwa, Pakistan","institution_ids":["https://openalex.org/I36001604"]},{"raw_affiliation_string":"Govt Degree College Badaber, Govt Degree College Badaber, Higher Education Department, Peshawar, Khyber Pakhtunkhwa, Pakistan","institution_ids":["https://openalex.org/I2802999120"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5063170670","display_name":"M. Salman Asif","orcid":"https://orcid.org/0000-0001-5993-3903"},"institutions":[{"id":"https://openalex.org/I36001604","display_name":"University of Peshawar","ror":"https://ror.org/02t2qwf81","country_code":"PK","type":"education","lineage":["https://openalex.org/I36001604"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Muhammad Asif","raw_affiliation_strings":["Department of Mathematics, University of Peshawar, Peshawar, Khyber Pakhtunkhwa, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Peshawar, Peshawar, Khyber Pakhtunkhwa, Pakistan","institution_ids":["https://openalex.org/I36001604"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5075592215","display_name":"Rohul Amin","orcid":"https://orcid.org/0000-0002-7000-3958"},"institutions":[{"id":"https://openalex.org/I36001604","display_name":"University of Peshawar","ror":"https://ror.org/02t2qwf81","country_code":"PK","type":"education","lineage":["https://openalex.org/I36001604"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Rohul Amin","raw_affiliation_strings":["Department of Mathematics, University of Peshawar, Peshawar, Khyber Pakhtunkhwa, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Peshawar, Peshawar, Khyber Pakhtunkhwa, Pakistan","institution_ids":["https://openalex.org/I36001604"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5025190663","display_name":"Nadeem Haider","orcid":null},"institutions":[{"id":"https://openalex.org/I2802999120","display_name":"Government of Khyber Pakhtunkhwa","ror":"https://ror.org/04zyfmb02","country_code":"PK","type":"government","lineage":["https://openalex.org/I2802999120"]},{"id":"https://openalex.org/I36001604","display_name":"University of Peshawar","ror":"https://ror.org/02t2qwf81","country_code":"PK","type":"education","lineage":["https://openalex.org/I36001604"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Nadeem Haider","raw_affiliation_strings":["Department of Mathematics, University of Peshawar, Peshawar, Khyber Pakhtunkhwa, Pakistan","Govt Degree College Badaber, Govt Degree College Badaber, Higher Education Department, Peshawar, Khyber Pakhtunkhwa, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Peshawar, Peshawar, Khyber Pakhtunkhwa, Pakistan","institution_ids":["https://openalex.org/I36001604"]},{"raw_affiliation_string":"Govt Degree College Badaber, Govt Degree College Badaber, Higher Education Department, Peshawar, Khyber Pakhtunkhwa, Pakistan","institution_ids":["https://openalex.org/I2802999120"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5085047470","display_name":"Fahd Jarad","orcid":"https://orcid.org/0000-0002-3303-0623"},"institutions":[{"id":"https://openalex.org/I106245021","display_name":"\u00c7ankaya University","ror":"https://ror.org/056wqre19","country_code":"TR","type":"education","lineage":["https://openalex.org/I106245021"]},{"id":"https://openalex.org/I26889942","display_name":"Azerbaijan Technical University","ror":"https://ror.org/01qe1a488","country_code":"AZ","type":"education","lineage":["https://openalex.org/I26889942"]}],"countries":["AZ","TR"],"is_corresponding":false,"raw_author_name":"Fahd Jarad","raw_affiliation_strings":["Department of Engineering Mathematics and Artificial Intelligence, Azerbaijan Technical University, Baku, Azerbaijan","Department of Mathematics, \u00c7ankaya University, Etimesgut, 06790, Ankara, Turkey"],"raw_orcid":"https://orcid.org/0000-0002-3303-0623","affiliations":[{"raw_affiliation_string":"Department of Engineering Mathematics and Artificial Intelligence, Azerbaijan Technical University, Baku, Azerbaijan","institution_ids":["https://openalex.org/I26889942"]},{"raw_affiliation_string":"Department of Mathematics, \u00c7ankaya University, Etimesgut, 06790, Ankara, Turkey","institution_ids":["https://openalex.org/I106245021"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5113665999"],"corresponding_institution_ids":["https://openalex.org/I2802999120","https://openalex.org/I36001604"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.41693575,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"45","issue":"3","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10514","display_name":"Numerical methods in engineering","score":0.5932999849319458,"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"}},"topics":[{"id":"https://openalex.org/T10514","display_name":"Numerical methods in engineering","score":0.5932999849319458,"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"}},{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.27709999680519104,"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/T11565","display_name":"Thermoelastic and Magnetoelastic Phenomena","score":0.017899999395012856,"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/radial-basis-function","display_name":"Radial basis function","score":0.6729000210762024},{"id":"https://openalex.org/keywords/classification-of-discontinuities","display_name":"Classification of discontinuities","score":0.670799970626831},{"id":"https://openalex.org/keywords/collocation-method","display_name":"Collocation method","score":0.6378999948501587},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.5727999806404114},{"id":"https://openalex.org/keywords/collocation","display_name":"Collocation (remote sensing)","score":0.5030999779701233},{"id":"https://openalex.org/keywords/orthogonal-collocation","display_name":"Orthogonal collocation","score":0.46230000257492065},{"id":"https://openalex.org/keywords/norm","display_name":"Norm (philosophy)","score":0.4311999976634979},{"id":"https://openalex.org/keywords/haar-wavelet","display_name":"Haar wavelet","score":0.3991999924182892}],"concepts":[{"id":"https://openalex.org/C98856871","wikidata":"https://www.wikidata.org/wiki/Q1588488","display_name":"Radial basis function","level":3,"score":0.6729000210762024},{"id":"https://openalex.org/C15627037","wikidata":"https://www.wikidata.org/wiki/Q541961","display_name":"Classification of discontinuities","level":2,"score":0.670799970626831},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.652400016784668},{"id":"https://openalex.org/C2126413","wikidata":"https://www.wikidata.org/wiki/Q1779479","display_name":"Collocation method","level":4,"score":0.6378999948501587},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.5727999806404114},{"id":"https://openalex.org/C80023036","wikidata":"https://www.wikidata.org/wiki/Q5147531","display_name":"Collocation (remote sensing)","level":2,"score":0.5030999779701233},{"id":"https://openalex.org/C114609681","wikidata":"https://www.wikidata.org/wiki/Q24839227","display_name":"Orthogonal collocation","level":5,"score":0.46230000257492065},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4433000087738037},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.4311999976634979},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.42489999532699585},{"id":"https://openalex.org/C2780423554","wikidata":"https://www.wikidata.org/wiki/Q766198","display_name":"Haar wavelet","level":5,"score":0.3991999924182892},{"id":"https://openalex.org/C51544822","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Ordinary differential equation","level":3,"score":0.3869999945163727},{"id":"https://openalex.org/C47432892","wikidata":"https://www.wikidata.org/wiki/Q831390","display_name":"Wavelet","level":2,"score":0.36730000376701355},{"id":"https://openalex.org/C113843644","wikidata":"https://www.wikidata.org/wiki/Q901882","display_name":"Interface (matter)","level":4,"score":0.3560999929904938},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.3246999979019165},{"id":"https://openalex.org/C185798385","wikidata":"https://www.wikidata.org/wiki/Q1161707","display_name":"Benchmark (surveying)","level":2,"score":0.3239000141620636},{"id":"https://openalex.org/C139945424","wikidata":"https://www.wikidata.org/wiki/Q1940696","display_name":"Mean squared error","level":2,"score":0.30820000171661377},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.2953000068664551},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.2818000018596649},{"id":"https://openalex.org/C136798861","wikidata":"https://www.wikidata.org/wiki/Q1826712","display_name":"Method of lines","level":5,"score":0.2689000070095062},{"id":"https://openalex.org/C3018824978","wikidata":"https://www.wikidata.org/wiki/Q2894891","display_name":"Error analysis","level":2,"score":0.26440000534057617},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.258899986743927},{"id":"https://openalex.org/C134466208","wikidata":"https://www.wikidata.org/wiki/Q1520657","display_name":"Weight function","level":2,"score":0.25619998574256897},{"id":"https://openalex.org/C121927907","wikidata":"https://www.wikidata.org/wiki/Q1952516","display_name":"Multiresolution analysis","level":5,"score":0.251800000667572}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s40314-025-03539-7","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s40314-025-03539-7","pdf_url":null,"source":{"id":"https://openalex.org/S4210195884","display_name":"Computational and Applied Mathematics","issn_l":"1807-0302","issn":["1807-0302","2238-3603"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computational and Applied Mathematics","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":45,"referenced_works":["https://openalex.org/W652409038","https://openalex.org/W1558361595","https://openalex.org/W1565025637","https://openalex.org/W1968086898","https://openalex.org/W1975804616","https://openalex.org/W2003261837","https://openalex.org/W2009271335","https://openalex.org/W2014475794","https://openalex.org/W2016896593","https://openalex.org/W2035176821","https://openalex.org/W2045618004","https://openalex.org/W2048221940","https://openalex.org/W2053169068","https://openalex.org/W2057120927","https://openalex.org/W2057265674","https://openalex.org/W2061586673","https://openalex.org/W2079089564","https://openalex.org/W2090315767","https://openalex.org/W2092265656","https://openalex.org/W2093012115","https://openalex.org/W2113251641","https://openalex.org/W2141613219","https://openalex.org/W2151125499","https://openalex.org/W2165665236","https://openalex.org/W2168806188","https://openalex.org/W2215837845","https://openalex.org/W2265564325","https://openalex.org/W2481312347","https://openalex.org/W2580700410","https://openalex.org/W2737528248","https://openalex.org/W2756329465","https://openalex.org/W2773605021","https://openalex.org/W2804983734","https://openalex.org/W2883429435","https://openalex.org/W2896226064","https://openalex.org/W2963364860","https://openalex.org/W3021871092","https://openalex.org/W3047196423","https://openalex.org/W3216731935","https://openalex.org/W4232769230","https://openalex.org/W4290793773","https://openalex.org/W4365152863","https://openalex.org/W4390504964","https://openalex.org/W4399798688","https://openalex.org/W4414705861"],"related_works":[],"abstract_inverted_index":null,"counts_by_year":[],"updated_date":"2025-12-15T23:14:13.489394","created_date":"2025-12-15T00:00:00"}
