{"id":"https://openalex.org/W2953650892","doi":"https://doi.org/10.1080/00207160.2019.1634261","title":"Polynomial solution of singular differential equations using Weighted Sobolev gradients","display_name":"Polynomial solution of singular differential equations using Weighted Sobolev gradients","publication_year":2019,"publication_date":"2019-06-24","ids":{"openalex":"https://openalex.org/W2953650892","doi":"https://doi.org/10.1080/00207160.2019.1634261","mag":"2953650892"},"language":"en","primary_location":{"id":"doi:10.1080/00207160.2019.1634261","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00207160.2019.1634261","pdf_url":null,"source":{"id":"https://openalex.org/S124867444","display_name":"International Journal of Computer Mathematics","issn_l":"0020-7160","issn":["0020-7160","1026-7425","1029-0265"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547","https://openalex.org/P4310320449"],"host_organization_lineage_names":["Taylor & Francis","Informa"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Computer 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/A5071287576","display_name":"Nauman Raza","orcid":"https://orcid.org/0000-0003-0700-1033"},"institutions":[{"id":"https://openalex.org/I172780181","display_name":"University of the Punjab","ror":"https://ror.org/011maz450","country_code":"PK","type":"education","lineage":["https://openalex.org/I172780181"]}],"countries":["PK"],"is_corresponding":true,"raw_author_name":"Nauman Raza","raw_affiliation_strings":["Department of Mathematics, University of the Punjab, Lahore, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of the Punjab, Lahore, Pakistan","institution_ids":["https://openalex.org/I172780181"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5032456923","display_name":"Ahmad Javid","orcid":"https://orcid.org/0000-0001-7968-4377"},"institutions":[{"id":"https://openalex.org/I172780181","display_name":"University of the Punjab","ror":"https://ror.org/011maz450","country_code":"PK","type":"education","lineage":["https://openalex.org/I172780181"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Ahmad Javid","raw_affiliation_strings":["Department of Mathematics, University of the Punjab, Lahore, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of the Punjab, Lahore, Pakistan","institution_ids":["https://openalex.org/I172780181"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5051936432","display_name":"Sultan Sial","orcid":"https://orcid.org/0000-0002-3906-332X"},"institutions":[{"id":"https://openalex.org/I207789805","display_name":"Lahore University of Management Sciences","ror":"https://ror.org/05b5x4a35","country_code":"PK","type":"education","lineage":["https://openalex.org/I207789805"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Sultan Sial","raw_affiliation_strings":["Department of Mathematics, Lahore University of Management Sciences, Lahore, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Lahore University of Management Sciences, Lahore, Pakistan","institution_ids":["https://openalex.org/I207789805"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5027468784","display_name":"Muahammad Dur-e-Ahmad","orcid":null},"institutions":[{"id":"https://openalex.org/I151746483","display_name":"University of Waterloo","ror":"https://ror.org/01aff2v68","country_code":"CA","type":"education","lineage":["https://openalex.org/I151746483"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Muahammad Dur-e-Ahmad","raw_affiliation_strings":["Department of Applied Mathematics, University of Waterloo, Waterloo, Canada"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics, University of Waterloo, Waterloo, Canada","institution_ids":["https://openalex.org/I151746483"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5071287576"],"corresponding_institution_ids":["https://openalex.org/I172780181"],"apc_list":null,"apc_paid":null,"fwci":0.3001,"has_fulltext":false,"cited_by_count":2,"citation_normalized_percentile":{"value":0.48539674,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"97","issue":"7","first_page":"1545","last_page":"1561"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10514","display_name":"Numerical methods in engineering","score":0.9994000196456909,"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":0.9994000196456909,"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":0.9976999759674072,"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/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.9973000288009644,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"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/mathematics","display_name":"Mathematics","score":0.8633723258972168},{"id":"https://openalex.org/keywords/sobolev-space","display_name":"Sobolev space","score":0.7263389825820923},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.6034355759620667},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.5129525065422058},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.49448132514953613},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4500402510166168}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8633723258972168},{"id":"https://openalex.org/C99730327","wikidata":"https://www.wikidata.org/wiki/Q1501536","display_name":"Sobolev space","level":2,"score":0.7263389825820923},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.6034355759620667},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5129525065422058},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.49448132514953613},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4500402510166168}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1080/00207160.2019.1634261","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00207160.2019.1634261","pdf_url":null,"source":{"id":"https://openalex.org/S124867444","display_name":"International Journal of Computer Mathematics","issn_l":"0020-7160","issn":["0020-7160","1026-7425","1029-0265"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547","https://openalex.org/P4310320449"],"host_organization_lineage_names":["Taylor & Francis","Informa"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Computer Mathematics","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320311820","display_name":"Lahore University of Management Sciences","ror":"https://ror.org/05b5x4a35"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":20,"referenced_works":["https://openalex.org/W1752175949","https://openalex.org/W1978403133","https://openalex.org/W1978906501","https://openalex.org/W1994338432","https://openalex.org/W2002704696","https://openalex.org/W2018728392","https://openalex.org/W2023034619","https://openalex.org/W2025054824","https://openalex.org/W2035155528","https://openalex.org/W2071576131","https://openalex.org/W2074439361","https://openalex.org/W2081374615","https://openalex.org/W2082055055","https://openalex.org/W2087475975","https://openalex.org/W2101232040","https://openalex.org/W2134221034","https://openalex.org/W2168023894","https://openalex.org/W2572075230","https://openalex.org/W2989591803","https://openalex.org/W4299512035"],"related_works":["https://openalex.org/W4299978068","https://openalex.org/W2164189677","https://openalex.org/W2950745223","https://openalex.org/W4301407218","https://openalex.org/W4388311156","https://openalex.org/W4296771954","https://openalex.org/W4390620896","https://openalex.org/W2963735960","https://openalex.org/W1869870390","https://openalex.org/W91521151"],"abstract_inverted_index":{"In":[0],"this":[1],"work,":[2],"we":[3],"proposed":[4],"a":[5,49],"new":[6,87],"scheme":[7],"based":[8],"on":[9,100],"Sobolev":[10,73],"gradient":[11],"approach":[12],"for":[13,77],"finding":[14],"an":[15],"approximate":[16],"polynomial":[17,58],"solution":[18],"of":[19,31,45,65,71],"certain":[20],"first":[21,51],"and":[22],"second":[23],"order":[24,52],"ordinary":[25],"differential":[26,33],"equations.":[27],"Continuous":[28],"function":[29],"instead":[30],"discretized":[32],"operator":[34],"is":[35,54,75,89],"used":[36,76],"to":[37,61],"avoid":[38],"numerical":[39],"issues":[40],"posed":[41],"by":[42],"the":[43,63,66,69,78,86,93,98,101],"size":[44],"grid.":[46],"For":[47],"example,":[48],"simple":[50],"equation":[53],"solved":[55],"using":[56],"different":[57],"basis":[59],"functions":[60],"illustrate":[62],"effectiveness":[64],"algorithm.":[67],"Then":[68],"theory":[70],"weighted":[72],"gradients":[74],"singular":[79],"Legendre's":[80],"equation.":[81],"Numerical":[82],"experiments":[83],"indicate":[84],"that":[85],"algorithm":[88],"more":[90],"efficient":[91],"than":[92],"previous":[94],"algorithms":[95],"discussed":[96],"in":[97],"literature":[99],"subject.":[102]},"counts_by_year":[{"year":2022,"cited_by_count":1},{"year":2020,"cited_by_count":1}],"updated_date":"2026-08-15T07:11:24.734988","created_date":"2025-10-10T00:00:00"}
