{"id":"https://openalex.org/W4311051460","doi":"https://doi.org/10.3390/sym14122539","title":"The Exact Solutions of Fractional Differential Systems with n Sinusoidal Terms under Physical Conditions","display_name":"The Exact Solutions of Fractional Differential Systems with n Sinusoidal Terms under Physical Conditions","publication_year":2022,"publication_date":"2022-12-01","ids":{"openalex":"https://openalex.org/W4311051460","doi":"https://doi.org/10.3390/sym14122539"},"language":"en","primary_location":{"id":"doi:10.3390/sym14122539","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14122539","pdf_url":"https://www.mdpi.com/2073-8994/14/12/2539/pdf?version=1669864816","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/2073-8994/14/12/2539/pdf?version=1669864816","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5112369379","display_name":"Laila F. Seddek","orcid":null},"institutions":[{"id":"https://openalex.org/I142608572","display_name":"Prince Sattam Bin Abdulaziz University","ror":"https://ror.org/04jt46d36","country_code":"SA","type":"education","lineage":["https://openalex.org/I142608572"]},{"id":"https://openalex.org/I192398990","display_name":"Zagazig University","ror":"https://ror.org/053g6we49","country_code":"EG","type":"education","lineage":["https://openalex.org/I192398990"]}],"countries":["EG","SA"],"is_corresponding":true,"raw_author_name":"Laila F. Seddek","raw_affiliation_strings":["Department of Engineering Mathematics and Physics, Faculty of Engineering, Zagazig University, Zagazig 44519, Egypt","Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdul-Aziz University, Al-Kharj 11942, Saudi Arabia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Engineering Mathematics and Physics, Faculty of Engineering, Zagazig University, Zagazig 44519, Egypt","institution_ids":["https://openalex.org/I192398990"]},{"raw_affiliation_string":"Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdul-Aziz University, Al-Kharj 11942, Saudi Arabia","institution_ids":["https://openalex.org/I142608572"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5110608891","display_name":"Essam R. El\u2010Zahar","orcid":"https://orcid.org/0000-0001-7266-1893"},"institutions":[{"id":"https://openalex.org/I142608572","display_name":"Prince Sattam Bin Abdulaziz University","ror":"https://ror.org/04jt46d36","country_code":"SA","type":"education","lineage":["https://openalex.org/I142608572"]},{"id":"https://openalex.org/I63601056","display_name":"Menoufia University","ror":"https://ror.org/05sjrb944","country_code":"EG","type":"education","lineage":["https://openalex.org/I63601056"]}],"countries":["EG","SA"],"is_corresponding":false,"raw_author_name":"Essam R. El-Zahar","raw_affiliation_strings":["Department of Basic Engineering Science, Faculty of Engineering, Menofia University, Shebin El-Kom 32511, Egypt","Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdul-Aziz University, Al-Kharj 11942, Saudi Arabia"],"raw_orcid":"https://orcid.org/0000-0001-7266-1893","affiliations":[{"raw_affiliation_string":"Department of Basic Engineering Science, Faculty of Engineering, Menofia University, Shebin El-Kom 32511, Egypt","institution_ids":["https://openalex.org/I63601056"]},{"raw_affiliation_string":"Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdul-Aziz University, Al-Kharj 11942, Saudi Arabia","institution_ids":["https://openalex.org/I142608572"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5075207817","display_name":"Abdelhalim Ebaid","orcid":"https://orcid.org/0000-0002-1122-6297"},"institutions":[{"id":"https://openalex.org/I72264486","display_name":"University of Tabuk","ror":"https://ror.org/04yej8x59","country_code":"SA","type":"education","lineage":["https://openalex.org/I72264486"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Abdelhalim Ebaid","raw_affiliation_strings":["Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia"],"raw_orcid":"https://orcid.org/0000-0002-1122-6297","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia","institution_ids":["https://openalex.org/I72264486"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5112369379"],"corresponding_institution_ids":["https://openalex.org/I142608572","https://openalex.org/I192398990"],"apc_list":{"value":1800,"currency":"CHF","value_usd":1885},"apc_paid":{"value":1800,"currency":"CHF","value_usd":1885},"fwci":0.1286,"has_fulltext":false,"cited_by_count":2,"citation_normalized_percentile":{"value":0.39968141,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"14","issue":"12","first_page":"2539","last_page":"2539"},"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":0.996999979019165,"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/T12396","display_name":"Advanced Differential Equations and Dynamical Systems","score":0.9955999851226807,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/fractional-calculus","display_name":"Fractional calculus","score":0.8192317485809326},{"id":"https://openalex.org/keywords/laplace-transform","display_name":"Laplace transform","score":0.7375836372375488},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6746734976768494},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5618356466293335},{"id":"https://openalex.org/keywords/exponential-function","display_name":"Exponential function","score":0.48648378252983093},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.42941558361053467},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3754729926586151}],"concepts":[{"id":"https://openalex.org/C154249771","wikidata":"https://www.wikidata.org/wiki/Q1339058","display_name":"Fractional calculus","level":2,"score":0.8192317485809326},{"id":"https://openalex.org/C97937538","wikidata":"https://www.wikidata.org/wiki/Q199691","display_name":"Laplace transform","level":2,"score":0.7375836372375488},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6746734976768494},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5618356466293335},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.48648378252983093},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.42941558361053467},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3754729926586151},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/sym14122539","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14122539","pdf_url":"https://www.mdpi.com/2073-8994/14/12/2539/pdf?version=1669864816","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:5028669ec65441d290542917a87250de","is_oa":true,"landing_page_url":"https://doaj.org/article/5028669ec65441d290542917a87250de","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":"Symmetry, Vol 14, Iss 12, p 2539 (2022)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/sym14122539","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14122539","pdf_url":"https://www.mdpi.com/2073-8994/14/12/2539/pdf?version=1669864816","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":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4311051460.pdf"},"referenced_works_count":49,"referenced_works":["https://openalex.org/W433794350","https://openalex.org/W1897281038","https://openalex.org/W1983854366","https://openalex.org/W1986990081","https://openalex.org/W1991873179","https://openalex.org/W1992828457","https://openalex.org/W2032878796","https://openalex.org/W2053356441","https://openalex.org/W2071593170","https://openalex.org/W2144366324","https://openalex.org/W2146343323","https://openalex.org/W2166879152","https://openalex.org/W2336986620","https://openalex.org/W2475247823","https://openalex.org/W2565653510","https://openalex.org/W2579529287","https://openalex.org/W2590074270","https://openalex.org/W2596239748","https://openalex.org/W2751260878","https://openalex.org/W2866352888","https://openalex.org/W2908577864","https://openalex.org/W2914217916","https://openalex.org/W2943979749","https://openalex.org/W2957148326","https://openalex.org/W2985137733","https://openalex.org/W2995978642","https://openalex.org/W3004586170","https://openalex.org/W3022888366","https://openalex.org/W3040980323","https://openalex.org/W3042218216","https://openalex.org/W3043531732","https://openalex.org/W3092280592","https://openalex.org/W3104090932","https://openalex.org/W3107423797","https://openalex.org/W3165412603","https://openalex.org/W4200517655","https://openalex.org/W4210614331","https://openalex.org/W4214566274","https://openalex.org/W4224297236","https://openalex.org/W4225664730","https://openalex.org/W4281621686","https://openalex.org/W4291007011","https://openalex.org/W4293195138","https://openalex.org/W4296699763","https://openalex.org/W6721617068","https://openalex.org/W6730937772","https://openalex.org/W6732359853","https://openalex.org/W6743692589","https://openalex.org/W6811179486"],"related_works":["https://openalex.org/W2082446734","https://openalex.org/W4313560640","https://openalex.org/W2253351617","https://openalex.org/W4317627639","https://openalex.org/W4313396013","https://openalex.org/W3024206168","https://openalex.org/W3005643231","https://openalex.org/W2806181810","https://openalex.org/W2369988800","https://openalex.org/W4287905284"],"abstract_inverted_index":{"This":[0,64],"paper":[1,108],"considers":[2],"the":[3,6,23,32,51,78,122,138,146,162,168,172,176,185,188,191],"classes":[4],"of":[5,16,30,85,114,121,145,154,161,171,175,190],"first-order":[7],"fractional":[8,20,25,42,56,71,80,112,124,156,178],"differential":[9,43],"systems":[10,72,113,125,179],"containing":[11],"a":[12,28,67,95,133,155],"finite":[13],"number":[14],"n":[15],"sinusoidal":[17],"terms.":[18],"The":[19,82,100,118,158],"derivative":[21],"employs":[22],"Riemann\u2013Liouville":[24,79],"definition.":[26,81],"As":[27],"method":[29,47],"solution,":[31],"Laplace":[33],"transform":[34],"is":[35,88,103],"an":[36,115],"efficient":[37],"tool":[38],"to":[39,49,69,89,109],"solve":[40,70,110],"linear":[41,111],"equations.":[44],"However,":[45],"this":[46,86,107],"requires":[48],"express":[50],"initial":[52,128],"conditions":[53,75,129],"in":[54,106,132,143],"certain":[55],"forms":[57],"which":[58],"have":[59],"no":[60],"physical":[61,127],"meaning":[62],"currently.":[63],"issue":[65],"formulated":[66],"challenge":[68],"under":[73,126],"real/physical":[74],"when":[76],"applying":[77],"principal":[83],"incentive":[84],"work":[87],"overcome":[90],"such":[91],"difficulties":[92],"via":[93],"presenting":[94],"simple":[96],"but":[97],"effective":[98],"approach.":[99],"proposed":[101],"approach":[102],"successfully":[104],"applied":[105],"oscillatory":[116],"nature.":[117],"exact":[119,169],"solutions":[120,140,164,170,193],"present":[123,192],"are":[130,141,165,180,194],"derived":[131],"straightforward":[134],"manner.":[135],"In":[136,183],"addition,":[137,184],"obtained":[139,163],"given":[142],"terms":[144],"entire":[147],"exponential":[148],"and":[149,187,196],"periodic":[150],"functions":[151],"with":[152],"arguments":[153],"order.":[157],"symmetric/asymmetric":[159],"behaviors/properties":[160],"illustrated.":[166],"Moreover,":[167],"classical/ordinary":[173],"versions":[174],"undertaken":[177],"determined":[181],"smoothly.":[182],"properties":[186],"behaviors":[189],"discussed":[195],"interpreted.":[197]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2023,"cited_by_count":1}],"updated_date":"2026-08-28T12:50:07.497085","created_date":"2025-10-10T00:00:00"}