{"id":"https://openalex.org/W2562801437","doi":"https://doi.org/10.1007/s11075-016-0248-y","title":"On the utility of the homotopy analysis method for non-analytic and global solutions to nonlinear differential equations","display_name":"On the utility of the homotopy analysis method for non-analytic and global solutions to nonlinear differential equations","publication_year":2016,"publication_date":"2016-12-26","ids":{"openalex":"https://openalex.org/W2562801437","doi":"https://doi.org/10.1007/s11075-016-0248-y","mag":"2562801437"},"language":"en","primary_location":{"id":"doi:10.1007/s11075-016-0248-y","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s11075-016-0248-y","pdf_url":"https://link.springer.com/content/pdf/10.1007%2Fs11075-016-0248-y.pdf","source":{"id":"https://openalex.org/S117494069","display_name":"Numerical Algorithms","issn_l":"1017-1398","issn":["1017-1398","1572-9265"],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Numerical Algorithms","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007%2Fs11075-016-0248-y.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5056297542","display_name":"Robert A. Van Gorder","orcid":"https://orcid.org/0000-0002-8506-3961"},"institutions":[{"id":"https://openalex.org/I40120149","display_name":"University of Oxford","ror":"https://ror.org/052gg0110","country_code":"GB","type":"education","lineage":["https://openalex.org/I40120149"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Robert A. Van Gorder","raw_affiliation_strings":["Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK"],"affiliations":[{"raw_affiliation_string":"Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK","institution_ids":["https://openalex.org/I40120149"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5056297542"],"corresponding_institution_ids":["https://openalex.org/I40120149"],"apc_list":{"value":2390,"currency":"EUR","value_usd":2990},"apc_paid":{"value":2390,"currency":"EUR","value_usd":2990},"fwci":1.4066,"has_fulltext":true,"cited_by_count":12,"citation_normalized_percentile":{"value":0.82637134,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":97},"biblio":{"volume":"76","issue":"1","first_page":"151","last_page":"162"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9998000264167786,"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":0.9998000264167786,"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":0.9944999814033508,"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/T11416","display_name":"Numerical methods for differential equations","score":0.9488999843597412,"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/mathematics","display_name":"Mathematics","score":0.8635584712028503},{"id":"https://openalex.org/keywords/taylor-series","display_name":"Taylor series","score":0.8435657620429993},{"id":"https://openalex.org/keywords/homotopy-analysis-method","display_name":"Homotopy analysis method","score":0.7419351935386658},{"id":"https://openalex.org/keywords/power-series","display_name":"Power series","score":0.7224470973014832},{"id":"https://openalex.org/keywords/radius-of-convergence","display_name":"Radius of convergence","score":0.6602305769920349},{"id":"https://openalex.org/keywords/homotopy","display_name":"Homotopy","score":0.5652527809143066},{"id":"https://openalex.org/keywords/series","display_name":"Series (stratigraphy)","score":0.5406718850135803},{"id":"https://openalex.org/keywords/taylors-theorem","display_name":"Taylor's theorem","score":0.5328819155693054},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.5234394073486328},{"id":"https://openalex.org/keywords/equivalence","display_name":"Equivalence (formal languages)","score":0.5228786468505859},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4923311471939087},{"id":"https://openalex.org/keywords/domain","display_name":"Domain (mathematical analysis)","score":0.47689512372016907},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.46306082606315613},{"id":"https://openalex.org/keywords/analytic-function","display_name":"Analytic function","score":0.4586505889892578},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.13988065719604492}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8635584712028503},{"id":"https://openalex.org/C158946198","wikidata":"https://www.wikidata.org/wiki/Q131187","display_name":"Taylor series","level":2,"score":0.8435657620429993},{"id":"https://openalex.org/C173636693","wikidata":"https://www.wikidata.org/wiki/Q17030668","display_name":"Homotopy analysis method","level":3,"score":0.7419351935386658},{"id":"https://openalex.org/C73905626","wikidata":"https://www.wikidata.org/wiki/Q206925","display_name":"Power series","level":2,"score":0.7224470973014832},{"id":"https://openalex.org/C155568369","wikidata":"https://www.wikidata.org/wiki/Q1428097","display_name":"Radius of convergence","level":3,"score":0.6602305769920349},{"id":"https://openalex.org/C5961521","wikidata":"https://www.wikidata.org/wiki/Q746083","display_name":"Homotopy","level":2,"score":0.5652527809143066},{"id":"https://openalex.org/C143724316","wikidata":"https://www.wikidata.org/wiki/Q312468","display_name":"Series (stratigraphy)","level":2,"score":0.5406718850135803},{"id":"https://openalex.org/C192523876","wikidata":"https://www.wikidata.org/wiki/Q1137206","display_name":"Taylor's theorem","level":3,"score":0.5328819155693054},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5234394073486328},{"id":"https://openalex.org/C2780069185","wikidata":"https://www.wikidata.org/wiki/Q7977945","display_name":"Equivalence (formal languages)","level":2,"score":0.5228786468505859},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4923311471939087},{"id":"https://openalex.org/C36503486","wikidata":"https://www.wikidata.org/wiki/Q11235244","display_name":"Domain (mathematical analysis)","level":2,"score":0.47689512372016907},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.46306082606315613},{"id":"https://openalex.org/C205979905","wikidata":"https://www.wikidata.org/wiki/Q215084","display_name":"Analytic function","level":2,"score":0.4586505889892578},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.13988065719604492},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C151730666","wikidata":"https://www.wikidata.org/wiki/Q7205","display_name":"Paleontology","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/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1007/s11075-016-0248-y","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s11075-016-0248-y","pdf_url":"https://link.springer.com/content/pdf/10.1007%2Fs11075-016-0248-y.pdf","source":{"id":"https://openalex.org/S117494069","display_name":"Numerical Algorithms","issn_l":"1017-1398","issn":["1017-1398","1572-9265"],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Numerical Algorithms","raw_type":"journal-article"},{"id":"pmh:oai:ora.ox.ac.uk:uuid:d31aab08-516c-4fcb-84f3-9e00de20f1f0","is_oa":true,"landing_page_url":"https://ora.ox.ac.uk/objects/uuid:d31aab08-516c-4fcb-84f3-9e00de20f1f0","pdf_url":null,"source":{"id":"https://openalex.org/S4306402636","display_name":"Oxford University Research Archive (ORA) (University of Oxford)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I40120149","host_organization_name":"University of Oxford","host_organization_lineage":["https://openalex.org/I40120149"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Journal article"}],"best_oa_location":{"id":"doi:10.1007/s11075-016-0248-y","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s11075-016-0248-y","pdf_url":"https://link.springer.com/content/pdf/10.1007%2Fs11075-016-0248-y.pdf","source":{"id":"https://openalex.org/S117494069","display_name":"Numerical Algorithms","issn_l":"1017-1398","issn":["1017-1398","1572-9265"],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Numerical Algorithms","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2562801437.pdf","grobid_xml":"https://content.openalex.org/works/W2562801437.grobid-xml"},"referenced_works_count":21,"referenced_works":["https://openalex.org/W634523187","https://openalex.org/W1483773443","https://openalex.org/W1963804126","https://openalex.org/W1983178975","https://openalex.org/W1997308351","https://openalex.org/W2000189075","https://openalex.org/W2003806194","https://openalex.org/W2013095468","https://openalex.org/W2019469368","https://openalex.org/W2021591459","https://openalex.org/W2043652197","https://openalex.org/W2050602884","https://openalex.org/W2059062496","https://openalex.org/W2077817453","https://openalex.org/W2095464875","https://openalex.org/W2119794696","https://openalex.org/W2144522870","https://openalex.org/W2160411620","https://openalex.org/W2170031978","https://openalex.org/W2480770084","https://openalex.org/W4254879547"],"related_works":["https://openalex.org/W2377017989","https://openalex.org/W2059145352","https://openalex.org/W2485604075","https://openalex.org/W1530517392","https://openalex.org/W2349056005","https://openalex.org/W4388514619","https://openalex.org/W2051542710","https://openalex.org/W3021454176","https://openalex.org/W1646458645","https://openalex.org/W2562801437"],"abstract_inverted_index":{"In":[0,35,61,125],"recent":[1],"work":[2],"on":[3,139,147],"the":[4,10,21,30,36,55,63,69,78,93,101,115,123,127,140,151,172,182,190,207,213,225,238,243,250,260,264],"area":[5],"of":[6,12,96,150,181,186,200],"approximation":[7],"methods":[8],"for":[9,223,274],"solution":[11,70,209,227],"nonlinear":[13,284],"differential":[14,285],"equations,":[15],"it":[16,233],"has":[17,82],"been":[18,83],"suggested":[19],"that":[20,41,114,249],"so-called":[22],"generalized":[23,116,173,239,251],"Taylor":[24,117,174,192,240,252],"series":[25,76,118,175,193,241,253],"approach":[26,119,254],"is":[27,45,57,65,71,120,210,228,255,266],"equivalent":[28,121],"to":[29,122,132,159,283],"homotopy":[31,97,208],"analysis":[32],"method":[33],"(HAM).":[34],"present":[37],"paper,":[38],"we":[39,110,247],"demonstrate":[40,178],"such":[42],"a":[43,74,197,220],"view":[44],"only":[46,66],"valid":[47,67,211],"in":[48,53,77],"very":[49],"special":[50],"cases,":[51],"and":[52,99,231,262],"general,":[54],"HAM":[56,128,154,183,265],"far":[58],"more":[59,267],"robust.":[60],"particular,":[62,126],"equivalence":[64],"when":[68,100,280],"represented":[72],"as":[73,257,259],"power":[75,108],"independent":[79],"variable.":[80],"As":[81],"shown":[84],"many":[85],"times,":[86],"alternative":[87],"basis":[88],"functions":[89,103],"can":[90,129,155,167],"greatly":[91],"improve":[92],"error":[94],"properties":[95,180],"solutions,":[98,162],"base":[102],"are":[104,136,278],"not":[105,168,229,235,256],"polynomials":[106],"or":[107],"functions,":[109],"no":[111],"longer":[112],"have":[113,196,271],"HAM.":[124],"be":[130,157,169],"used":[131,158],"obtain":[133,160],"solutions":[134,277,282],"which":[135,163,224],"global":[137],"(defined":[138,146],"whole":[141],"domain)":[142],"rather":[143],"than":[144],"local":[145],"some":[148],"restriction":[149],"domain).":[152],"The":[153],"also":[156],"non-analytic":[161],"by":[164,184],"their":[165],"nature":[166],"expressed":[170],"through":[171],"approach.":[176],"We":[177,217],"these":[179],"consideration":[185],"an":[187],"example":[188,222],"where":[189],"generalizes":[191],"must":[194],"always":[195],"finite":[198],"radius":[199],"convergence":[201],"(and":[202],"hence":[203],"limited":[204],"applicability),":[205],"while":[206],"over":[212,242],"entire":[214],"infinite":[215],"domain.":[216,244],"then":[218],"give":[219],"second":[221],"exact":[226],"analytic,":[230],"hence,":[232,263],"will":[234],"agree":[236],"with":[237],"Doing":[245],"so,":[246],"show":[248],"robust":[258],"HAM,":[261],"general.":[268],"Such":[269],"results":[270],"important":[272],"implications":[273],"how":[275],"iterative":[276],"calculated":[279],"approximating":[281],"equations.":[286]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":3},{"year":2019,"cited_by_count":3},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":1}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}
