{"id":"https://openalex.org/W2064785338","doi":"https://doi.org/10.1155/2014/168568","title":"Spectral Method for Solving the Nonlinear Thomas-Fermi Equation Based on Exponential Functions","display_name":"Spectral Method for Solving the Nonlinear Thomas-Fermi Equation Based on Exponential Functions","publication_year":2014,"publication_date":"2014-01-01","ids":{"openalex":"https://openalex.org/W2064785338","doi":"https://doi.org/10.1155/2014/168568","mag":"2064785338"},"language":"en","primary_location":{"id":"doi:10.1155/2014/168568","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2014/168568","pdf_url":"https://downloads.hindawi.com/journals/jam/2014/168568.pdf","source":{"id":"https://openalex.org/S190082376","display_name":"Journal of Applied Mathematics","issn_l":"1110-757X","issn":["1110-757X","1687-0042"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869","https://openalex.org/P4310320595"],"host_organization_lineage_names":["Hindawi Publishing Corporation","Wiley"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Applied Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://downloads.hindawi.com/journals/jam/2014/168568.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5008685555","display_name":"Raka Jovanovi\u0107","orcid":"https://orcid.org/0000-0001-8167-1516"},"institutions":[{"id":"https://openalex.org/I92528248","display_name":"Qatar Foundation","ror":"https://ror.org/01cawbq05","country_code":"QA","type":"funder","lineage":["https://openalex.org/I92528248"]}],"countries":["QA"],"is_corresponding":false,"raw_author_name":"Raka Jovanovic","raw_affiliation_strings":["Qatar Environment and Energy Research Institute (QEERI), Qatar Foundation, P.O. Box 5825, Doha, Qatar","Qatar Environment and Energy Research Institute (QEERI), Qatar Foundation, P.O. Box 5825, Doha, Qatar,"],"raw_orcid":"https://orcid.org/0000-0001-8167-1516","affiliations":[{"raw_affiliation_string":"Qatar Environment and Energy Research Institute (QEERI), Qatar Foundation, P.O. Box 5825, Doha, Qatar","institution_ids":["https://openalex.org/I92528248"]},{"raw_affiliation_string":"Qatar Environment and Energy Research Institute (QEERI), Qatar Foundation, P.O. Box 5825, Doha, Qatar,","institution_ids":["https://openalex.org/I92528248"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5017775824","display_name":"Sabre Kais","orcid":"https://orcid.org/0000-0003-0574-5346"},"institutions":[{"id":"https://openalex.org/I92528248","display_name":"Qatar Foundation","ror":"https://ror.org/01cawbq05","country_code":"QA","type":"funder","lineage":["https://openalex.org/I92528248"]}],"countries":["QA"],"is_corresponding":false,"raw_author_name":"Sabre Kais","raw_affiliation_strings":["Qatar Environment and Energy Research Institute (QEERI), Qatar Foundation, P.O. Box 5825, Doha, Qatar","Qatar Environment and Energy Research Institute (QEERI), Qatar Foundation, P.O. Box 5825, Doha, Qatar,"],"raw_orcid":"https://orcid.org/0000-0003-0574-5346","affiliations":[{"raw_affiliation_string":"Qatar Environment and Energy Research Institute (QEERI), Qatar Foundation, P.O. Box 5825, Doha, Qatar","institution_ids":["https://openalex.org/I92528248"]},{"raw_affiliation_string":"Qatar Environment and Energy Research Institute (QEERI), Qatar Foundation, P.O. Box 5825, Doha, Qatar,","institution_ids":["https://openalex.org/I92528248"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5078672526","display_name":"Fahhad H. Alharbi","orcid":"https://orcid.org/0000-0002-4131-8491"},"institutions":[{"id":"https://openalex.org/I92528248","display_name":"Qatar Foundation","ror":"https://ror.org/01cawbq05","country_code":"QA","type":"funder","lineage":["https://openalex.org/I92528248"]}],"countries":["QA"],"is_corresponding":false,"raw_author_name":"Fahhad H. Alharbi","raw_affiliation_strings":["Qatar Environment and Energy Research Institute (QEERI), Qatar Foundation, P.O. Box 5825, Doha, Qatar","Qatar Environment and Energy Research Institute (QEERI), Qatar Foundation, P.O. Box 5825, Doha, Qatar,"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Qatar Environment and Energy Research Institute (QEERI), Qatar Foundation, P.O. Box 5825, Doha, Qatar","institution_ids":["https://openalex.org/I92528248"]},{"raw_affiliation_string":"Qatar Environment and Energy Research Institute (QEERI), Qatar Foundation, P.O. Box 5825, Doha, Qatar,","institution_ids":["https://openalex.org/I92528248"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I92528248"],"apc_list":{"value":1050,"currency":"USD","value_usd":1050},"apc_paid":{"value":1050,"currency":"USD","value_usd":1050},"fwci":1.3985,"has_fulltext":true,"cited_by_count":16,"citation_normalized_percentile":{"value":0.80834526,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":97},"biblio":{"volume":"2014","issue":null,"first_page":"1","last_page":"8"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.995199978351593,"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":0.995199978351593,"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/T11177","display_name":"Spectroscopy and Quantum Chemical Studies","score":0.9908999800682068,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9898999929428101,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/basis-function","display_name":"Basis function","score":0.6058582663536072},{"id":"https://openalex.org/keywords/exponential-function","display_name":"Exponential function","score":0.6044828295707703},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6009106040000916},{"id":"https://openalex.org/keywords/solver","display_name":"Solver","score":0.5975005626678467},{"id":"https://openalex.org/keywords/spectral-method","display_name":"Spectral method","score":0.5776810646057129},{"id":"https://openalex.org/keywords/algebraic-equation","display_name":"Algebraic equation","score":0.5547477006912231},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.5307239294052124},{"id":"https://openalex.org/keywords/robustness","display_name":"Robustness (evolution)","score":0.5197479724884033},{"id":"https://openalex.org/keywords/basis","display_name":"Basis (linear algebra)","score":0.5089445114135742},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.49875926971435547},{"id":"https://openalex.org/keywords/exponential-growth","display_name":"Exponential growth","score":0.442781925201416},{"id":"https://openalex.org/keywords/ordinary-differential-equation","display_name":"Ordinary differential equation","score":0.43876758217811584},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.35548120737075806},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.2977749705314636},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2213035225868225}],"concepts":[{"id":"https://openalex.org/C5917680","wikidata":"https://www.wikidata.org/wiki/Q2621825","display_name":"Basis function","level":2,"score":0.6058582663536072},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.6044828295707703},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6009106040000916},{"id":"https://openalex.org/C2778770139","wikidata":"https://www.wikidata.org/wiki/Q1966904","display_name":"Solver","level":2,"score":0.5975005626678467},{"id":"https://openalex.org/C23463724","wikidata":"https://www.wikidata.org/wiki/Q2308831","display_name":"Spectral method","level":2,"score":0.5776810646057129},{"id":"https://openalex.org/C23917780","wikidata":"https://www.wikidata.org/wiki/Q50698","display_name":"Algebraic equation","level":3,"score":0.5547477006912231},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5307239294052124},{"id":"https://openalex.org/C63479239","wikidata":"https://www.wikidata.org/wiki/Q7353546","display_name":"Robustness (evolution)","level":3,"score":0.5197479724884033},{"id":"https://openalex.org/C12426560","wikidata":"https://www.wikidata.org/wiki/Q189569","display_name":"Basis (linear algebra)","level":2,"score":0.5089445114135742},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.49875926971435547},{"id":"https://openalex.org/C75235859","wikidata":"https://www.wikidata.org/wiki/Q582659","display_name":"Exponential growth","level":2,"score":0.442781925201416},{"id":"https://openalex.org/C51544822","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Ordinary differential equation","level":3,"score":0.43876758217811584},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.35548120737075806},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.2977749705314636},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2213035225868225},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"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/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1155/2014/168568","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2014/168568","pdf_url":"https://downloads.hindawi.com/journals/jam/2014/168568.pdf","source":{"id":"https://openalex.org/S190082376","display_name":"Journal of Applied Mathematics","issn_l":"1110-757X","issn":["1110-757X","1687-0042"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869","https://openalex.org/P4310320595"],"host_organization_lineage_names":["Hindawi Publishing Corporation","Wiley"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Applied Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:CULeuclid:euclid.jam/1425306060","is_oa":false,"landing_page_url":"https://projecteuclid.org/euclid.jam/1425306060","pdf_url":null,"source":{"id":"https://openalex.org/S4306400787","display_name":"Project Euclid (Cornell University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Text"},{"id":"pmh:oai:RePEc:hin:jnljam:168568","is_oa":false,"landing_page_url":"http://downloads.hindawi.com/journals/JAM/2014/168568.xml","pdf_url":null,"source":{"id":"https://openalex.org/S4306401271","display_name":"RePEc: Research Papers in Economics","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I77793887","host_organization_name":"Federal Reserve Bank of St. Louis","host_organization_lineage":["https://openalex.org/I77793887"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},{"id":"pmh:oai:doaj.org/article:d671096bd3314f3fbfe5bd7f24f23647","is_oa":true,"landing_page_url":"https://doaj.org/article/d671096bd3314f3fbfe5bd7f24f23647","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":"Journal of Applied Mathematics, Vol 2014 (2014)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.1155/2014/168568","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2014/168568","pdf_url":"https://downloads.hindawi.com/journals/jam/2014/168568.pdf","source":{"id":"https://openalex.org/S190082376","display_name":"Journal of Applied Mathematics","issn_l":"1110-757X","issn":["1110-757X","1687-0042"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319869","host_organization_name":"Hindawi Publishing Corporation","host_organization_lineage":["https://openalex.org/P4310319869","https://openalex.org/P4310320595"],"host_organization_lineage_names":["Hindawi Publishing Corporation","Wiley"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Applied Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2064785338.pdf","grobid_xml":"https://content.openalex.org/works/W2064785338.grobid-xml"},"referenced_works_count":44,"referenced_works":["https://openalex.org/W1549252824","https://openalex.org/W1550061415","https://openalex.org/W1632774950","https://openalex.org/W1677218873","https://openalex.org/W1701580692","https://openalex.org/W1965032003","https://openalex.org/W1974210344","https://openalex.org/W1975608189","https://openalex.org/W1991100480","https://openalex.org/W1994010302","https://openalex.org/W1994371245","https://openalex.org/W1995907640","https://openalex.org/W1999140638","https://openalex.org/W2032839612","https://openalex.org/W2038394077","https://openalex.org/W2039869174","https://openalex.org/W2041431731","https://openalex.org/W2047035089","https://openalex.org/W2048207353","https://openalex.org/W2063292380","https://openalex.org/W2065233224","https://openalex.org/W2065522541","https://openalex.org/W2066068669","https://openalex.org/W2067722113","https://openalex.org/W2068116513","https://openalex.org/W2071399586","https://openalex.org/W2071864376","https://openalex.org/W2074078622","https://openalex.org/W2081182772","https://openalex.org/W2084980626","https://openalex.org/W2090561712","https://openalex.org/W2094040825","https://openalex.org/W2095947114","https://openalex.org/W2096119294","https://openalex.org/W2130417219","https://openalex.org/W2156352382","https://openalex.org/W2156601564","https://openalex.org/W2171084460","https://openalex.org/W2171132099","https://openalex.org/W2593968251","https://openalex.org/W2609618425","https://openalex.org/W2612403816","https://openalex.org/W2964076972","https://openalex.org/W3106296080"],"related_works":["https://openalex.org/W2006953814","https://openalex.org/W2041399066","https://openalex.org/W2545357463","https://openalex.org/W2482793298","https://openalex.org/W2117387008","https://openalex.org/W1998585081","https://openalex.org/W2439864307","https://openalex.org/W2118018204","https://openalex.org/W1550587116","https://openalex.org/W2402673101"],"abstract_inverted_index":{"We":[0,136],"present":[1,138],"an":[2,31],"efficient":[3],"spectral":[4,28,50,149],"methods":[5,150],"solver":[6],"for":[7,11,101,145],"the":[8,37,54,92,109,118,171],"Thomas-Fermi":[9],"equation":[10,21],"neutral":[12],"atoms":[13],"in":[14,91,167],"a":[15,27,82,102,123,130,139],"semi-infinite":[16],"domain.":[17],"The":[18,67,85,157],"ordinary":[19],"differential":[20],"has":[22,88],"been":[23],"solved":[24,59,80],"by":[25],"applying":[26],"method":[29,78,87,162],"using":[30,76,81,129,148],"exponential":[32],"basis":[33,110,155],"set.":[34,111],"One":[35],"of":[36,40,49,71,105,127,133,141],"main":[38],"advantages":[39],"this":[41,77,146],"approach,":[42],"when":[43],"compared":[44],"to":[45],"other":[46],"relevant":[47],"applications":[48],"methods,":[51],"is":[52,74,79,163],"that":[53,73,94,107,117,160],"underlying":[55],"integrals":[56],"can":[57,64,96,121],"be":[58,65],"analytically":[60],"and":[61,166],"numerical":[62],"integration":[63],"avoided.":[66],"nonlinear":[68],"algebraic":[69],"system":[70],"equations":[72],"derived":[75],"minimization":[83],"approach.":[84],"presented":[86],"shown":[89],"robustness":[90],"sense":[93],"it":[95],"find":[97],"high":[98,125],"precision":[99],"solution":[100],"wide":[103],"range":[104],"parameters":[106],"define":[108],"In":[112],"our":[113,161],"test,":[114],"we":[115],"show":[116],"new":[119],"approach":[120],"achieve":[122],"very":[124],"rate":[126],"convergence":[128],"small":[131],"number":[132],"bases":[134],"elements.":[135],"also":[137],"comparison":[140,158],"recently":[142],"published":[143],"results":[144],"problem":[147],"based":[151],"on":[152],"several":[153],"different":[154],"sets.":[156],"shows":[159],"highly":[164],"competitive":[165],"many":[168],"aspects":[169],"outperforms":[170],"previous":[172],"work.":[173]},"counts_by_year":[{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":3},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":3}],"updated_date":"2026-08-15T07:11:24.734988","created_date":"2025-10-10T00:00:00"}