{"id":"https://openalex.org/W4316806124","doi":"https://doi.org/10.3390/sym15020258","title":"Numerical Solution of Time-Fractional Emden\u2013Fowler-Type Equations Using the Rational Homotopy Perturbation Method","display_name":"Numerical Solution of Time-Fractional Emden\u2013Fowler-Type Equations Using the Rational Homotopy Perturbation Method","publication_year":2023,"publication_date":"2023-01-17","ids":{"openalex":"https://openalex.org/W4316806124","doi":"https://doi.org/10.3390/sym15020258"},"language":"en","primary_location":{"id":"doi:10.3390/sym15020258","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15020258","pdf_url":"https://www.mdpi.com/2073-8994/15/2/258/pdf?version=1675127407","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/15/2/258/pdf?version=1675127407","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5074497072","display_name":"Kholoud Saad Albalawi","orcid":"https://orcid.org/0000-0002-1204-8568"},"institutions":[{"id":"https://openalex.org/I240666556","display_name":"Imam Mohammad ibn Saud Islamic University","ror":"https://ror.org/05gxjyb39","country_code":"SA","type":"education","lineage":["https://openalex.org/I240666556"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Kholoud Saad Albalawi","raw_affiliation_strings":["Department of Mathematics and Statistics, Imam Mohammad Ibn Saud Islamic University, Riyadh 11566, Saudi Arabia"],"raw_orcid":"https://orcid.org/0000-0002-1204-8568","affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, Imam Mohammad Ibn Saud Islamic University, Riyadh 11566, Saudi Arabia","institution_ids":["https://openalex.org/I240666556"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5066887622","display_name":"Badr Saad T. Alkahtani","orcid":"https://orcid.org/0000-0002-2864-6888"},"institutions":[{"id":"https://openalex.org/I28022161","display_name":"King Saud University","ror":"https://ror.org/02f81g417","country_code":"SA","type":"education","lineage":["https://openalex.org/I28022161"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Badr Saad Alkahtani","raw_affiliation_strings":["Department of Mathematics, College of Science, King Saud University, Riyadh 11989, Saudi Arabia"],"raw_orcid":"https://orcid.org/0000-0002-2864-6888","affiliations":[{"raw_affiliation_string":"Department of Mathematics, College of Science, King Saud University, Riyadh 11989, Saudi Arabia","institution_ids":["https://openalex.org/I28022161"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100325449","display_name":"Ashish Kumar","orcid":"https://orcid.org/0000-0002-2003-6209"},"institutions":[{"id":"https://openalex.org/I190765188","display_name":"Ambedkar University Delhi","ror":"https://ror.org/0039wpa60","country_code":"IN","type":"education","lineage":["https://openalex.org/I190765188"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Ashish Kumar","raw_affiliation_strings":["School of Liberal Studies, Dr B.R. Ambedkar University Delhi, Delhi 110006, India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Liberal Studies, Dr B.R. Ambedkar University Delhi, Delhi 110006, India","institution_ids":["https://openalex.org/I190765188"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5049348187","display_name":"Pranay Goswami","orcid":"https://orcid.org/0000-0003-1205-1975"},"institutions":[{"id":"https://openalex.org/I190765188","display_name":"Ambedkar University Delhi","ror":"https://ror.org/0039wpa60","country_code":"IN","type":"education","lineage":["https://openalex.org/I190765188"]}],"countries":["IN"],"is_corresponding":true,"raw_author_name":"Pranay Goswami","raw_affiliation_strings":["School of Liberal Studies, Dr B.R. Ambedkar University Delhi, Delhi 110006, India"],"raw_orcid":"https://orcid.org/0000-0003-1205-1975","affiliations":[{"raw_affiliation_string":"School of Liberal Studies, Dr B.R. Ambedkar University Delhi, Delhi 110006, India","institution_ids":["https://openalex.org/I190765188"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5049348187"],"corresponding_institution_ids":["https://openalex.org/I190765188"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":1.915,"has_fulltext":true,"cited_by_count":9,"citation_normalized_percentile":{"value":0.83513839,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":96,"max":97},"biblio":{"volume":"15","issue":"2","first_page":"258","last_page":"258"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":1.0,"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":1.0,"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":0.9861000180244446,"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":0.9811999797821045,"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/mathematics","display_name":"Mathematics","score":0.819812536239624},{"id":"https://openalex.org/keywords/banach-fixed-point-theorem","display_name":"Banach fixed-point theorem","score":0.6573764681816101},{"id":"https://openalex.org/keywords/homotopy-analysis-method","display_name":"Homotopy analysis method","score":0.6272392868995667},{"id":"https://openalex.org/keywords/uniqueness","display_name":"Uniqueness","score":0.5312297940254211},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5162577629089355},{"id":"https://openalex.org/keywords/homotopy-perturbation-method","display_name":"Homotopy perturbation method","score":0.48610377311706543},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.4605291485786438},{"id":"https://openalex.org/keywords/homotopy","display_name":"Homotopy","score":0.4559110403060913},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.43304187059402466},{"id":"https://openalex.org/keywords/power-series","display_name":"Power series","score":0.4226812720298767},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.41603997349739075},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.16554543375968933}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.819812536239624},{"id":"https://openalex.org/C2780549194","wikidata":"https://www.wikidata.org/wiki/Q220680","display_name":"Banach fixed-point theorem","level":3,"score":0.6573764681816101},{"id":"https://openalex.org/C173636693","wikidata":"https://www.wikidata.org/wiki/Q17030668","display_name":"Homotopy analysis method","level":3,"score":0.6272392868995667},{"id":"https://openalex.org/C2777021972","wikidata":"https://www.wikidata.org/wiki/Q22976830","display_name":"Uniqueness","level":2,"score":0.5312297940254211},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5162577629089355},{"id":"https://openalex.org/C91328119","wikidata":"https://www.wikidata.org/wiki/Q17030668","display_name":"Homotopy perturbation method","level":3,"score":0.48610377311706543},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.4605291485786438},{"id":"https://openalex.org/C5961521","wikidata":"https://www.wikidata.org/wiki/Q746083","display_name":"Homotopy","level":2,"score":0.4559110403060913},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.43304187059402466},{"id":"https://openalex.org/C73905626","wikidata":"https://www.wikidata.org/wiki/Q206925","display_name":"Power series","level":2,"score":0.4226812720298767},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.41603997349739075},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.16554543375968933},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","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":3,"locations":[{"id":"doi:10.3390/sym15020258","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15020258","pdf_url":"https://www.mdpi.com/2073-8994/15/2/258/pdf?version=1675127407","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:1f93511a9b6d40639decad7de9962c35","is_oa":false,"landing_page_url":"https://doaj.org/article/1f93511a9b6d40639decad7de9962c35","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 15, Iss 2, p 258 (2023)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/15/2/258/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym15020258","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"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":"Symmetry","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym15020258","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15020258","pdf_url":"https://www.mdpi.com/2073-8994/15/2/258/pdf?version=1675127407","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G4532692949","display_name":null,"funder_award_id":"RP-21-09-06","funder_id":"https://openalex.org/F4320325415","funder_display_name":"Al-Imam Muhammad Ibn Saud Islamic University"}],"funders":[{"id":"https://openalex.org/F4320325415","display_name":"Al-Imam Muhammad Ibn Saud Islamic University","ror":null}],"has_content":{"pdf":true,"grobid_xml":false},"content_urls":{"pdf":"https://content.openalex.org/works/W4316806124.pdf"},"referenced_works_count":33,"referenced_works":["https://openalex.org/W1964751776","https://openalex.org/W1966816682","https://openalex.org/W1998535670","https://openalex.org/W2001916833","https://openalex.org/W2009343239","https://openalex.org/W2029384065","https://openalex.org/W2034094163","https://openalex.org/W2037611934","https://openalex.org/W2054872013","https://openalex.org/W2061039074","https://openalex.org/W2063619180","https://openalex.org/W2066648070","https://openalex.org/W2066824297","https://openalex.org/W2068705444","https://openalex.org/W2071734762","https://openalex.org/W2127616341","https://openalex.org/W2134848150","https://openalex.org/W2291097739","https://openalex.org/W2317731414","https://openalex.org/W2322116852","https://openalex.org/W2570022981","https://openalex.org/W3106433977","https://openalex.org/W3136995544","https://openalex.org/W3164442843","https://openalex.org/W3172135037","https://openalex.org/W3194714386","https://openalex.org/W4220765430","https://openalex.org/W4221011689","https://openalex.org/W4293207314","https://openalex.org/W6642060049","https://openalex.org/W6680245841","https://openalex.org/W6800453214","https://openalex.org/W6809619232"],"related_works":["https://openalex.org/W2088506180","https://openalex.org/W2362256999","https://openalex.org/W2188605998","https://openalex.org/W2087729225","https://openalex.org/W1964997874","https://openalex.org/W1545358442","https://openalex.org/W4239956528","https://openalex.org/W1497843313","https://openalex.org/W2127579732","https://openalex.org/W2151283633"],"abstract_inverted_index":{"The":[0,38,50,101,118],"integral-order":[1],"derivative":[2,14],"is":[3,41,107,123],"not":[4],"suitable":[5,75],"where":[6],"infinite":[7],"variances":[8],"are":[9,57,95],"expected,":[10],"and":[11,28,52,77,86,115],"the":[12,32,55,60,67,80,83,104,111,128],"fractional":[13],"manages":[15],"to":[16,126],"consider":[17],"effects":[18],"with":[19,70],"more":[20],"precision;":[21],"therefore,":[22],"we":[23,65],"considered":[24],"timefractional":[25],"Emden\u2013Fowler-type":[26],"equations":[27],"solved":[29],"them":[30],"using":[31,59,82,97],"rational":[33,48],"homotopy":[34],"perturbation":[35],"method":[36,40,85],"(RHPM).":[37],"RHPM":[39],"based":[42],"on":[43],"two":[44],"power":[45],"series":[46],"in":[47],"form.":[49],"existence":[51],"uniqueness":[53],"of":[54,73,103,113,120],"equation":[56],"proved":[58],"Banach":[61],"fixed-point":[62],"theorem.":[63],"Furthermore,":[64],"approximate":[66,89,116],"term":[68],"h(z)":[69],"a":[71,74],"polynomial":[72],"degree":[76],"then":[78],"solve":[79],"system":[81],"proposed":[84,99,105],"obtain":[87],"an":[88],"symmetric":[90],"solution.":[91],"Two":[92],"numerical":[93],"examples":[94],"investigated":[96],"this":[98],"approach.":[100],"effectiveness":[102],"approach":[106],"checked":[108],"by":[109],"representing":[110],"graphs":[112],"exact":[114],"solutions.":[117],"table":[119],"absolute":[121],"error":[122],"also":[124],"presented":[125],"understand":[127],"method\u2032s":[129],"accuracy.":[130]},"counts_by_year":[{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":3}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}