{"id":"https://openalex.org/W1970182106","doi":"https://doi.org/10.1080/00207160008804976","title":"Legendre spectral method for solving integral and integro-differential equations","display_name":"Legendre spectral method for solving integral and integro-differential equations","publication_year":2000,"publication_date":"2000-01-01","ids":{"openalex":"https://openalex.org/W1970182106","doi":"https://doi.org/10.1080/00207160008804976","mag":"1970182106"},"language":"en","primary_location":{"id":"doi:10.1080/00207160008804976","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00207160008804976","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"],"host_organization_lineage_names":["Taylor & Francis"],"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":null,"display_name":"H.M. El-Hawary","orcid":null},"institutions":[{"id":"https://openalex.org/I91041137","display_name":"Assiut University","ror":"https://ror.org/01jaj8n65","country_code":"EG","type":"education","lineage":["https://openalex.org/I91041137"]}],"countries":["EG"],"is_corresponding":true,"raw_author_name":"H.M. El-Hawary","raw_affiliation_strings":["Dept. of Math., Faculty of Science , Assuit University , Assuit, Egypt"],"affiliations":[{"raw_affiliation_string":"Dept. of Math., Faculty of Science , Assuit University , Assuit, Egypt","institution_ids":["https://openalex.org/I91041137"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5021641892","display_name":"M.S. Salim","orcid":"https://orcid.org/0000-0001-5856-3525"},"institutions":[{"id":"https://openalex.org/I184834183","display_name":"Al-Azhar University","ror":"https://ror.org/05fnp1145","country_code":"EG","type":"education","lineage":["https://openalex.org/I184834183"]}],"countries":["EG"],"is_corresponding":false,"raw_author_name":"M.S. Salim","raw_affiliation_strings":["Dept. of Math., Faculty of Science , El-Azhar University , Assuit, Egypt"],"affiliations":[{"raw_affiliation_string":"Dept. of Math., Faculty of Science , El-Azhar University , Assuit, Egypt","institution_ids":["https://openalex.org/I184834183"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5085037139","display_name":"H.S. Hussein","orcid":null},"institutions":[{"id":"https://openalex.org/I189575948","display_name":"South Valley University","ror":"https://ror.org/00jxshx33","country_code":"EG","type":"education","lineage":["https://openalex.org/I189575948"]}],"countries":["EG"],"is_corresponding":false,"raw_author_name":"H.S. Hussein","raw_affiliation_strings":["Dept. of Math., Faculty of Science , South valley University , Qena, Egypt","Dept. of Math., Faculty of Science, South Valley University, Qena, Egypt"],"affiliations":[{"raw_affiliation_string":"Dept. of Math., Faculty of Science , South valley University , Qena, Egypt","institution_ids":["https://openalex.org/I189575948"]},{"raw_affiliation_string":"Dept. of Math., Faculty of Science, South Valley University, Qena, Egypt","institution_ids":["https://openalex.org/I189575948"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I91041137"],"apc_list":null,"apc_paid":null,"fwci":1.8693,"has_fulltext":false,"cited_by_count":6,"citation_normalized_percentile":{"value":0.87352445,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"75","issue":"2","first_page":"187","last_page":"203"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9991999864578247,"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":0.9991999864578247,"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/T11416","display_name":"Numerical methods for differential equations","score":0.9991000294685364,"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/T12727","display_name":"Differential Equations and Numerical Methods","score":0.9991000294685364,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/legendre-polynomials","display_name":"Legendre polynomials","score":0.9343158006668091},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8480062484741211},{"id":"https://openalex.org/keywords/associated-legendre-polynomials","display_name":"Associated Legendre polynomials","score":0.694200873374939},{"id":"https://openalex.org/keywords/legendre-wavelet","display_name":"Legendre wavelet","score":0.6938632726669312},{"id":"https://openalex.org/keywords/legendre-function","display_name":"Legendre function","score":0.6678569316864014},{"id":"https://openalex.org/keywords/residual","display_name":"Residual","score":0.6084390878677368},{"id":"https://openalex.org/keywords/method-of-mean-weighted-residuals","display_name":"Method of mean weighted residuals","score":0.5729477405548096},{"id":"https://openalex.org/keywords/legendres-equation","display_name":"Legendre's equation","score":0.5453313589096069},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.5221548080444336},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.5096266269683838},{"id":"https://openalex.org/keywords/integral-equation","display_name":"Integral equation","score":0.4982116222381592},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.47279995679855347},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.459468811750412},{"id":"https://openalex.org/keywords/orthogonal-polynomials","display_name":"Orthogonal polynomials","score":0.18737944960594177},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.14760592579841614},{"id":"https://openalex.org/keywords/classical-orthogonal-polynomials","display_name":"Classical orthogonal polynomials","score":0.13070809841156006},{"id":"https://openalex.org/keywords/gegenbauer-polynomials","display_name":"Gegenbauer polynomials","score":0.10242125391960144},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.07547229528427124}],"concepts":[{"id":"https://openalex.org/C111458787","wikidata":"https://www.wikidata.org/wiki/Q215405","display_name":"Legendre polynomials","level":2,"score":0.9343158006668091},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8480062484741211},{"id":"https://openalex.org/C36956377","wikidata":"https://www.wikidata.org/wiki/Q228376","display_name":"Associated Legendre polynomials","level":5,"score":0.694200873374939},{"id":"https://openalex.org/C123769847","wikidata":"https://www.wikidata.org/wiki/Q6517888","display_name":"Legendre wavelet","level":5,"score":0.6938632726669312},{"id":"https://openalex.org/C153262748","wikidata":"https://www.wikidata.org/wiki/Q17098449","display_name":"Legendre function","level":3,"score":0.6678569316864014},{"id":"https://openalex.org/C155512373","wikidata":"https://www.wikidata.org/wiki/Q287450","display_name":"Residual","level":2,"score":0.6084390878677368},{"id":"https://openalex.org/C159674985","wikidata":"https://www.wikidata.org/wiki/Q6823718","display_name":"Method of mean weighted residuals","level":4,"score":0.5729477405548096},{"id":"https://openalex.org/C86287605","wikidata":"https://www.wikidata.org/wiki/Q4454970","display_name":"Legendre's equation","level":3,"score":0.5453313589096069},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5221548080444336},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.5096266269683838},{"id":"https://openalex.org/C27016315","wikidata":"https://www.wikidata.org/wiki/Q580101","display_name":"Integral equation","level":2,"score":0.4982116222381592},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.47279995679855347},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.459468811750412},{"id":"https://openalex.org/C10628310","wikidata":"https://www.wikidata.org/wiki/Q619458","display_name":"Orthogonal polynomials","level":2,"score":0.18737944960594177},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.14760592579841614},{"id":"https://openalex.org/C21736991","wikidata":"https://www.wikidata.org/wiki/Q17006917","display_name":"Classical orthogonal polynomials","level":3,"score":0.13070809841156006},{"id":"https://openalex.org/C78540521","wikidata":"https://www.wikidata.org/wiki/Q1498246","display_name":"Gegenbauer polynomials","level":4,"score":0.10242125391960144},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.07547229528427124},{"id":"https://openalex.org/C196216189","wikidata":"https://www.wikidata.org/wiki/Q2867","display_name":"Wavelet transform","level":3,"score":0.0},{"id":"https://openalex.org/C46286280","wikidata":"https://www.wikidata.org/wiki/Q2414958","display_name":"Discrete wavelet transform","level":4,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0},{"id":"https://openalex.org/C47432892","wikidata":"https://www.wikidata.org/wiki/Q831390","display_name":"Wavelet","level":2,"score":0.0},{"id":"https://openalex.org/C186899397","wikidata":"https://www.wikidata.org/wiki/Q1491980","display_name":"Galerkin method","level":3,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1080/00207160008804976","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00207160008804976","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"],"host_organization_lineage_names":["Taylor & Francis"],"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":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":11,"referenced_works":["https://openalex.org/W1971099220","https://openalex.org/W1977849344","https://openalex.org/W1982005385","https://openalex.org/W1984147483","https://openalex.org/W2014537528","https://openalex.org/W2017332736","https://openalex.org/W2033707211","https://openalex.org/W2066775291","https://openalex.org/W2087759033","https://openalex.org/W2120062331","https://openalex.org/W2318788438"],"related_works":["https://openalex.org/W1607269076","https://openalex.org/W2953361964","https://openalex.org/W3005861762","https://openalex.org/W3200178185","https://openalex.org/W3130427480","https://openalex.org/W1988418937","https://openalex.org/W4296546724","https://openalex.org/W2288349649","https://openalex.org/W2034418652","https://openalex.org/W2022538265"],"abstract_inverted_index":{"A":[0],"new":[1],"spectral":[2],"approximation":[3,10,52],"of":[4,14,18,42,48,68],"an":[5],"integral":[6,29],"based":[7],"on":[8],"Legendre":[9,50],"at":[11],"the":[12,15,19,38,43,49,54,64,69],"zeros":[13],"first":[16],"term":[17],"residual":[20],"is":[21,25],"presented.":[22],"The":[23,33],"method":[24,34],"used":[26],"to":[27,37,53,62],"solve":[28],"and":[30,66],"integro-differential":[31],"equations.":[32],"generates":[35],"approximations":[36],"lower":[39],"order":[40,56],"derivatives":[41],"function":[44],"through":[45],"successive":[46],"integrations":[47],"polynomial":[51],"highest":[55],"derivative.":[57],"Numerical":[58],"results":[59],"are":[60],"included":[61],"confirm":[63],"efficiency":[65],"accuracy":[67],"method.":[70]},"counts_by_year":[{"year":2021,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2013,"cited_by_count":1}],"updated_date":"2026-04-16T08:26:57.006410","created_date":"2025-10-10T00:00:00"}
