{"id":"https://openalex.org/W2013614753","doi":"https://doi.org/10.1155/2013/157956","title":"Couple of the Variational Iteration Method and Legendre Wavelets for Nonlinear Partial Differential Equations","display_name":"Couple of the Variational Iteration Method and Legendre Wavelets for Nonlinear Partial Differential Equations","publication_year":2013,"publication_date":"2013-01-01","ids":{"openalex":"https://openalex.org/W2013614753","doi":"https://doi.org/10.1155/2013/157956","mag":"2013614753"},"language":"en","primary_location":{"id":"doi:10.1155/2013/157956","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2013/157956","pdf_url":"https://downloads.hindawi.com/journals/jam/2013/157956.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"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"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/2013/157956.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5089866247","display_name":"Fukang Yin","orcid":"https://orcid.org/0000-0002-8353-7000"},"institutions":[{"id":"https://openalex.org/I170215575","display_name":"National University of Defense Technology","ror":"https://ror.org/05d2yfz11","country_code":"CN","type":"education","lineage":["https://openalex.org/I170215575"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Fukang Yin","raw_affiliation_strings":["College of Computer, National University of Defense Technology, Changsha 410073, China","College of Computer, National University of Defense Technology, Changsha, 410073,China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Computer, National University of Defense Technology, Changsha 410073, China","institution_ids":["https://openalex.org/I170215575"]},{"raw_affiliation_string":"College of Computer, National University of Defense Technology, Changsha, 410073,China","institution_ids":["https://openalex.org/I170215575"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101788380","display_name":"Junqiang Song","orcid":"https://orcid.org/0009-0003-2686-566X"},"institutions":[{"id":"https://openalex.org/I170215575","display_name":"National University of Defense Technology","ror":"https://ror.org/05d2yfz11","country_code":"CN","type":"education","lineage":["https://openalex.org/I170215575"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Junqiang Song","raw_affiliation_strings":["College of Computer, National University of Defense Technology, Changsha 410073, China","College of Computer, National University of Defense Technology, Changsha, 410073,China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Computer, National University of Defense Technology, Changsha 410073, China","institution_ids":["https://openalex.org/I170215575"]},{"raw_affiliation_string":"College of Computer, National University of Defense Technology, Changsha, 410073,China","institution_ids":["https://openalex.org/I170215575"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5110714473","display_name":"Xiaoqun Cao","orcid":"https://orcid.org/0000-0002-6135-0712"},"institutions":[{"id":"https://openalex.org/I170215575","display_name":"National University of Defense Technology","ror":"https://ror.org/05d2yfz11","country_code":"CN","type":"education","lineage":["https://openalex.org/I170215575"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xiaoqun Cao","raw_affiliation_strings":["College of Computer, National University of Defense Technology, Changsha 410073, China","College of Computer, National University of Defense Technology, Changsha, 410073,China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Computer, National University of Defense Technology, Changsha 410073, China","institution_ids":["https://openalex.org/I170215575"]},{"raw_affiliation_string":"College of Computer, National University of Defense Technology, Changsha, 410073,China","institution_ids":["https://openalex.org/I170215575"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5064864405","display_name":"Fengshun Lu","orcid":"https://orcid.org/0000-0002-0005-4493"},"institutions":[{"id":"https://openalex.org/I2802505932","display_name":"China Aerodynamics Research and Development Center","ror":"https://ror.org/00jma8s40","country_code":"CN","type":"facility","lineage":["https://openalex.org/I2802505932"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Fengshun Lu","raw_affiliation_strings":["China Aerodynamics Research and Development Center, Mianyang, Sichuan 621000, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"China Aerodynamics Research and Development Center, Mianyang, Sichuan 621000, China","institution_ids":["https://openalex.org/I2802505932"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5089866247"],"corresponding_institution_ids":["https://openalex.org/I170215575"],"apc_list":{"value":1050,"currency":"USD","value_usd":1050},"apc_paid":{"value":1050,"currency":"USD","value_usd":1050},"fwci":1.9848,"has_fulltext":true,"cited_by_count":15,"citation_normalized_percentile":{"value":0.84169342,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"2013","issue":null,"first_page":"1","last_page":"11"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9998999834060669,"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":0.9998999834060669,"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/T12661","display_name":"Iterative Methods for Nonlinear Equations","score":0.9715999960899353,"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/T10248","display_name":"Nonlinear Waves and Solitons","score":0.9291999936103821,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/legendre-wavelet","display_name":"Legendre wavelet","score":0.919263482093811},{"id":"https://openalex.org/keywords/legendre-polynomials","display_name":"Legendre polynomials","score":0.7492673397064209},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7396619319915771},{"id":"https://openalex.org/keywords/wavelet","display_name":"Wavelet","score":0.652593731880188},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.6468125581741333},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5982040166854858},{"id":"https://openalex.org/keywords/associated-legendre-polynomials","display_name":"Associated Legendre polynomials","score":0.5313737392425537},{"id":"https://openalex.org/keywords/algebraic-equation","display_name":"Algebraic equation","score":0.5006284713745117},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.49096181988716125},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.4899448752403259},{"id":"https://openalex.org/keywords/computation","display_name":"Computation","score":0.46615490317344666},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3603903353214264},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.15827670693397522},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.14632996916770935},{"id":"https://openalex.org/keywords/orthogonal-polynomials","display_name":"Orthogonal polynomials","score":0.12620189785957336},{"id":"https://openalex.org/keywords/wavelet-transform","display_name":"Wavelet transform","score":0.11765912175178528},{"id":"https://openalex.org/keywords/gegenbauer-polynomials","display_name":"Gegenbauer polynomials","score":0.07480087876319885},{"id":"https://openalex.org/keywords/discrete-wavelet-transform","display_name":"Discrete wavelet transform","score":0.0692453682422638}],"concepts":[{"id":"https://openalex.org/C123769847","wikidata":"https://www.wikidata.org/wiki/Q6517888","display_name":"Legendre wavelet","level":5,"score":0.919263482093811},{"id":"https://openalex.org/C111458787","wikidata":"https://www.wikidata.org/wiki/Q215405","display_name":"Legendre polynomials","level":2,"score":0.7492673397064209},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7396619319915771},{"id":"https://openalex.org/C47432892","wikidata":"https://www.wikidata.org/wiki/Q831390","display_name":"Wavelet","level":2,"score":0.652593731880188},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.6468125581741333},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5982040166854858},{"id":"https://openalex.org/C36956377","wikidata":"https://www.wikidata.org/wiki/Q228376","display_name":"Associated Legendre polynomials","level":5,"score":0.5313737392425537},{"id":"https://openalex.org/C23917780","wikidata":"https://www.wikidata.org/wiki/Q50698","display_name":"Algebraic equation","level":3,"score":0.5006284713745117},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.49096181988716125},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.4899448752403259},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.46615490317344666},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3603903353214264},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.15827670693397522},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.14632996916770935},{"id":"https://openalex.org/C10628310","wikidata":"https://www.wikidata.org/wiki/Q619458","display_name":"Orthogonal polynomials","level":2,"score":0.12620189785957336},{"id":"https://openalex.org/C196216189","wikidata":"https://www.wikidata.org/wiki/Q2867","display_name":"Wavelet transform","level":3,"score":0.11765912175178528},{"id":"https://openalex.org/C78540521","wikidata":"https://www.wikidata.org/wiki/Q1498246","display_name":"Gegenbauer polynomials","level":4,"score":0.07480087876319885},{"id":"https://openalex.org/C46286280","wikidata":"https://www.wikidata.org/wiki/Q2414958","display_name":"Discrete wavelet transform","level":4,"score":0.0692453682422638},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"id":"https://openalex.org/C21736991","wikidata":"https://www.wikidata.org/wiki/Q17006917","display_name":"Classical orthogonal polynomials","level":3,"score":0.0},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"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/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","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}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1155/2013/157956","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2013/157956","pdf_url":"https://downloads.hindawi.com/journals/jam/2013/157956.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"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"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/1394807863","is_oa":false,"landing_page_url":"https://projecteuclid.org/euclid.jam/1394807863","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:157956","is_oa":false,"landing_page_url":"http://downloads.hindawi.com/journals/JAM/2013/157956.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:576fff6ecd124f4187b0e4f2ff49b838","is_oa":true,"landing_page_url":"https://doaj.org/article/576fff6ecd124f4187b0e4f2ff49b838","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 2013 (2013)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.1155/2013/157956","is_oa":true,"landing_page_url":"https://doi.org/10.1155/2013/157956","pdf_url":"https://downloads.hindawi.com/journals/jam/2013/157956.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"],"host_organization_lineage_names":["Hindawi Publishing Corporation"],"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":[{"id":"https://openalex.org/G163740092","display_name":"\u4e91\u6c34\u89c2\u6d4b\u8d44\u6599\u540c\u5316\u4e2d\u5148\u8fdb\u80cc\u666f\u8bef\u5dee\u534f\u65b9\u5dee\u6a21\u578b\u7814\u7a76","funder_award_id":"41105063","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2013614753.pdf","grobid_xml":"https://content.openalex.org/works/W2013614753.grobid-xml"},"referenced_works_count":58,"referenced_works":["https://openalex.org/W1970467391","https://openalex.org/W1974767027","https://openalex.org/W1975517304","https://openalex.org/W1980737983","https://openalex.org/W1984853762","https://openalex.org/W1988251003","https://openalex.org/W1994328718","https://openalex.org/W1997290136","https://openalex.org/W1997858083","https://openalex.org/W1999776604","https://openalex.org/W2000448984","https://openalex.org/W2002725408","https://openalex.org/W2004591828","https://openalex.org/W2006367435","https://openalex.org/W2007579839","https://openalex.org/W2012645543","https://openalex.org/W2022099021","https://openalex.org/W2025245367","https://openalex.org/W2030204179","https://openalex.org/W2030928359","https://openalex.org/W2033413333","https://openalex.org/W2033462292","https://openalex.org/W2035249396","https://openalex.org/W2051091468","https://openalex.org/W2054016546","https://openalex.org/W2054459595","https://openalex.org/W2058986223","https://openalex.org/W2065733349","https://openalex.org/W2067548886","https://openalex.org/W2069821908","https://openalex.org/W2070454382","https://openalex.org/W2073033699","https://openalex.org/W2073806305","https://openalex.org/W2080699423","https://openalex.org/W2084264130","https://openalex.org/W2087226000","https://openalex.org/W2092419561","https://openalex.org/W2093228677","https://openalex.org/W2101070138","https://openalex.org/W2104518667","https://openalex.org/W2109617339","https://openalex.org/W2138262300","https://openalex.org/W2141858378","https://openalex.org/W2144399742","https://openalex.org/W2149569043","https://openalex.org/W2158267260","https://openalex.org/W2159676772","https://openalex.org/W2164357122","https://openalex.org/W2165917005","https://openalex.org/W2168358956","https://openalex.org/W2170606157","https://openalex.org/W2173900448","https://openalex.org/W2181858789","https://openalex.org/W2182408342","https://openalex.org/W2282892142","https://openalex.org/W2314090441","https://openalex.org/W2315501962","https://openalex.org/W3176191004"],"related_works":["https://openalex.org/W2545227774","https://openalex.org/W1607269076","https://openalex.org/W2953361964","https://openalex.org/W2736737644","https://openalex.org/W4313180762","https://openalex.org/W2462438656","https://openalex.org/W2357310454","https://openalex.org/W2288349649","https://openalex.org/W2789913125","https://openalex.org/W2089945891"],"abstract_inverted_index":{"This":[0],"paper":[1],"develops":[2],"a":[3,39,47],"modified":[4],"variational":[5],"iteration":[6],"method":[7,72,101],"coupled":[8],"with":[9],"the":[10,18,36,57,63,70,79,87,99],"Legendre":[11,58],"wavelets,":[12],"which":[13],"can":[14,76],"be":[15],"used":[16,54],"for":[17],"efficient":[19],"numerical":[20],"solution":[21],"of":[22,31,38,62,69],"nonlinear":[23,64,80],"partial":[24],"differential":[25],"equations":[26],"(PDEs).":[27],"The":[28,66,95],"approximate":[29],"solutions":[30],"PDEs":[32],"are":[33,43,53],"calculated":[34],"in":[35],"form":[37,90],"series":[40],"whose":[41],"components":[42],"computed":[44],"by":[45],"applying":[46],"recursive":[48],"relation.":[49],"Block":[50],"pulse":[51],"functions":[52],"to":[55,107],"calculate":[56],"wavelets":[59],"coefficient":[60],"matrices":[61],"terms.":[65],"main":[67],"advantage":[68],"new":[71],"is":[73,102],"that":[74,98],"it":[75,92],"avoid":[77],"solving":[78],"algebraic":[81],"system":[82],"and":[83,105],"symbolic":[84],"computation.":[85],"Furthermore,":[86],"developed":[88],"vector-matrix":[89],"makes":[91],"computationally":[93],"efficient.":[94],"results":[96],"show":[97],"proposed":[100],"very":[103],"effective":[104],"easy":[106],"implement.":[108]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2014,"cited_by_count":3},{"year":2013,"cited_by_count":5}],"updated_date":"2026-07-31T08:31:51.225901","created_date":"2025-10-10T00:00:00"}