{"id":"https://openalex.org/W2996765885","doi":"https://doi.org/10.3390/e22010028","title":"An Investigation of Fractional Bagley\u2013Torvik Equation","display_name":"An Investigation of Fractional Bagley\u2013Torvik Equation","publication_year":2019,"publication_date":"2019-12-24","ids":{"openalex":"https://openalex.org/W2996765885","doi":"https://doi.org/10.3390/e22010028","mag":"2996765885","pmid":"https://pubmed.ncbi.nlm.nih.gov/33285803"},"language":"en","primary_location":{"id":"doi:10.3390/e22010028","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e22010028","pdf_url":"https://www.mdpi.com/1099-4300/22/1/28/pdf?version=1577353605","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"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":"Entropy","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj","pubmed"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/1099-4300/22/1/28/pdf?version=1577353605","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5017356241","display_name":"Azhar Ali Zafar","orcid":"https://orcid.org/0000-0003-3825-0446"},"institutions":[{"id":"https://openalex.org/I188884621","display_name":"Lodz University of Technology","ror":"https://ror.org/00s8fpf52","country_code":"PL","type":"education","lineage":["https://openalex.org/I188884621"]},{"id":"https://openalex.org/I43879129","display_name":"Government College University, Lahore","ror":"https://ror.org/040gec961","country_code":"PK","type":"education","lineage":["https://openalex.org/I43879129"]}],"countries":["PK","PL"],"is_corresponding":true,"raw_author_name":"Azhar Ali Zafar","raw_affiliation_strings":["Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, 1/15 Stefanowskiego St., 90-924 Lodz, Poland","Department of Mathematics, Government College University, Lahore 54000, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, 1/15 Stefanowskiego St., 90-924 Lodz, Poland","institution_ids":["https://openalex.org/I188884621"]},{"raw_affiliation_string":"Department of Mathematics, Government College University, Lahore 54000, Pakistan","institution_ids":["https://openalex.org/I43879129"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5082398705","display_name":"Grzegorz Kudra","orcid":"https://orcid.org/0000-0003-0209-4664"},"institutions":[{"id":"https://openalex.org/I188884621","display_name":"Lodz University of Technology","ror":"https://ror.org/00s8fpf52","country_code":"PL","type":"education","lineage":["https://openalex.org/I188884621"]}],"countries":["PL"],"is_corresponding":false,"raw_author_name":"Grzegorz Kudra","raw_affiliation_strings":["Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, 1/15 Stefanowskiego St., 90-924 Lodz, Poland"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, 1/15 Stefanowskiego St., 90-924 Lodz, Poland","institution_ids":["https://openalex.org/I188884621"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5025074757","display_name":"Jan Awrejcewicz","orcid":"https://orcid.org/0000-0003-0387-921X"},"institutions":[{"id":"https://openalex.org/I188884621","display_name":"Lodz University of Technology","ror":"https://ror.org/00s8fpf52","country_code":"PL","type":"education","lineage":["https://openalex.org/I188884621"]}],"countries":["PL"],"is_corresponding":false,"raw_author_name":"Jan Awrejcewicz","raw_affiliation_strings":["Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, 1/15 Stefanowskiego St., 90-924 Lodz, Poland"],"raw_orcid":"https://orcid.org/0000-0003-0387-921X","affiliations":[{"raw_affiliation_string":"Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, 1/15 Stefanowskiego St., 90-924 Lodz, Poland","institution_ids":["https://openalex.org/I188884621"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5017356241"],"corresponding_institution_ids":["https://openalex.org/I188884621","https://openalex.org/I43879129"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.681,"has_fulltext":false,"cited_by_count":16,"citation_normalized_percentile":{"value":0.69394033,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"22","issue":"1","first_page":"28","last_page":"28"},"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/T10541","display_name":"Nonlinear Differential Equations Analysis","score":0.9865000247955322,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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":0.9833999872207642,"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/fractional-calculus","display_name":"Fractional calculus","score":0.7275094985961914},{"id":"https://openalex.org/keywords/operator","display_name":"Operator (biology)","score":0.47791749238967896},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.47384458780288696},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4548817276954651},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.45473983883857727},{"id":"https://openalex.org/keywords/block","display_name":"Block (permutation group theory)","score":0.4149627685546875},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.38634824752807617}],"concepts":[{"id":"https://openalex.org/C154249771","wikidata":"https://www.wikidata.org/wiki/Q1339058","display_name":"Fractional calculus","level":2,"score":0.7275094985961914},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.47791749238967896},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.47384458780288696},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4548817276954651},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.45473983883857727},{"id":"https://openalex.org/C2777210771","wikidata":"https://www.wikidata.org/wiki/Q4927124","display_name":"Block (permutation group theory)","level":2,"score":0.4149627685546875},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.38634824752807617},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","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},{"id":"https://openalex.org/C86339819","wikidata":"https://www.wikidata.org/wiki/Q407384","display_name":"Transcription factor","level":3,"score":0.0},{"id":"https://openalex.org/C158448853","wikidata":"https://www.wikidata.org/wiki/Q425218","display_name":"Repressor","level":4,"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/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.3390/e22010028","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e22010028","pdf_url":"https://www.mdpi.com/1099-4300/22/1/28/pdf?version=1577353605","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"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":"Entropy","raw_type":"journal-article"},{"id":"pmid:33285803","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/33285803","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy (Basel, Switzerland)","raw_type":null},{"id":"pmh:oai:doaj.org/article:9e4a8274718a44eba794f57bab1462c7","is_oa":false,"landing_page_url":"https://doaj.org/article/9e4a8274718a44eba794f57bab1462c7","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":"Entropy, Vol 22, Iss 1, p 28 (2019)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/1099-4300/22/1/28/","is_oa":true,"landing_page_url":"http://dx.doi.org/10.3390/e22010028","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":"Entropy","raw_type":"Text"},{"id":"pmh:oai:pubmedcentral.nih.gov:7516449","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/7516449","pdf_url":null,"source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"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":"Entropy (Basel)","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/e22010028","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e22010028","pdf_url":"https://www.mdpi.com/1099-4300/22/1/28/pdf?version=1577353605","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"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":"Entropy","raw_type":"journal-article"},"sustainable_development_goals":[{"display_name":"Peace, Justice and strong institutions","id":"https://metadata.un.org/sdg/16","score":0.7699999809265137}],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2996765885.pdf","grobid_xml":"https://content.openalex.org/works/W2996765885.grobid-xml"},"referenced_works_count":42,"referenced_works":["https://openalex.org/W1559244587","https://openalex.org/W1965396783","https://openalex.org/W1966014675","https://openalex.org/W1969950423","https://openalex.org/W1984533905","https://openalex.org/W1990421301","https://openalex.org/W2019463338","https://openalex.org/W2023477691","https://openalex.org/W2038765999","https://openalex.org/W2043095423","https://openalex.org/W2045950404","https://openalex.org/W2057395226","https://openalex.org/W2060477697","https://openalex.org/W2070188088","https://openalex.org/W2083559699","https://openalex.org/W2087337573","https://openalex.org/W2088328305","https://openalex.org/W2140318154","https://openalex.org/W2143130504","https://openalex.org/W2152152827","https://openalex.org/W2167191107","https://openalex.org/W2181758453","https://openalex.org/W2269344468","https://openalex.org/W2467934420","https://openalex.org/W2492855992","https://openalex.org/W2586052349","https://openalex.org/W2610804809","https://openalex.org/W2780851474","https://openalex.org/W2793288245","https://openalex.org/W2895157544","https://openalex.org/W2911462820","https://openalex.org/W2930359366","https://openalex.org/W2948870349","https://openalex.org/W2985207885","https://openalex.org/W2996765885","https://openalex.org/W3216942329","https://openalex.org/W6642479713","https://openalex.org/W6655500276","https://openalex.org/W6685887091","https://openalex.org/W6755477414","https://openalex.org/W6761309880","https://openalex.org/W6804267759"],"related_works":["https://openalex.org/W2082446734","https://openalex.org/W4313560640","https://openalex.org/W4246784705","https://openalex.org/W3121927021","https://openalex.org/W2253351617","https://openalex.org/W4390948881","https://openalex.org/W1607068974","https://openalex.org/W2021368388","https://openalex.org/W2035671297","https://openalex.org/W2742971749"],"abstract_inverted_index":{"function.":[0],"Further,":[1],"we":[2,38],"will":[3],"compare":[4],"the":[5,13,25,28,35,59,62],"obtained":[6],"results":[7,11],"with":[8,58],"other":[9],"available":[10],"in":[12,42],"literature":[14],"to":[15,23,61],"validate":[16],"their":[17],"usefulness.":[18],"Furthermore,":[19],"examples":[20],"are":[21],"included":[22],"highlight":[24],"control":[26],"of":[27,34,44,48,54],"fractional":[29],"parameters":[30],"on":[31],"he":[32],"dynamics":[33],"model.":[36],"Moreover,":[37],"use":[39],"this":[40],"equation":[41],"modelling":[43],"real":[45],"free":[46],"oscillations":[47],"a":[49,55,72],"one-degree-of-freedom":[50],"mechanical":[51],"system":[52],"composed":[53],"cart":[56],"connected":[57],"springs":[60],"support":[63],"and":[64],"moving":[65],"via":[66],"linear":[67],"rolling":[68],"bearing":[69],"block":[70],"along":[71],"rail.":[73]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2024,"cited_by_count":6},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":1}],"updated_date":"2026-05-06T08:25:59.206177","created_date":"2025-10-10T00:00:00"}