{"id":"https://openalex.org/W3215422660","doi":"https://doi.org/10.3390/sym13122265","title":"Spectrum of Fractional and Fractional Prabhakar Sturm\u2013Liouville Problems with Homogeneous Dirichlet Boundary Conditions","display_name":"Spectrum of Fractional and Fractional Prabhakar Sturm\u2013Liouville Problems with Homogeneous Dirichlet Boundary Conditions","publication_year":2021,"publication_date":"2021-11-28","ids":{"openalex":"https://openalex.org/W3215422660","doi":"https://doi.org/10.3390/sym13122265","mag":"3215422660"},"language":"en","primary_location":{"id":"doi:10.3390/sym13122265","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13122265","pdf_url":"https://www.mdpi.com/2073-8994/13/12/2265/pdf?version=1638084855","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/2073-8994/13/12/2265/pdf?version=1638084855","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5001274870","display_name":"Ma\u0142gorzata Klimek","orcid":"https://orcid.org/0000-0002-1029-1863"},"institutions":[{"id":"https://openalex.org/I130294970","display_name":"Cz\u0119stochowa University of Technology","ror":"https://ror.org/046awyn59","country_code":"PL","type":"education","lineage":["https://openalex.org/I130294970"]}],"countries":["PL"],"is_corresponding":true,"raw_author_name":"Malgorzata Klimek","raw_affiliation_strings":["Department of Mathematics, Faculty of Mechanical Engineering and Computer Science, Czestochowa University of Technology, 42-201 Czestochowa, Poland"],"raw_orcid":"https://orcid.org/0000-0002-1029-1863","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Mechanical Engineering and Computer Science, Czestochowa University of Technology, 42-201 Czestochowa, Poland","institution_ids":["https://openalex.org/I130294970"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5001274870"],"corresponding_institution_ids":["https://openalex.org/I130294970"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":4.0934,"has_fulltext":false,"cited_by_count":11,"citation_normalized_percentile":{"value":0.94930959,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":98},"biblio":{"volume":"13","issue":"12","first_page":"2265","last_page":"2265"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11022","display_name":"Spectral Theory in Mathematical Physics","score":0.9986000061035156,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T11022","display_name":"Spectral Theory in Mathematical Physics","score":0.9986000061035156,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T12589","display_name":"Differential Equations and Boundary Problems","score":0.9973000288009644,"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/T10541","display_name":"Nonlinear Differential Equations Analysis","score":0.996399998664856,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/sturm\u2013liouville-theory","display_name":"Sturm\u2013Liouville theory","score":0.8663100004196167},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8194162845611572},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.689485490322113},{"id":"https://openalex.org/keywords/eigenfunction","display_name":"Eigenfunction","score":0.667346715927124},{"id":"https://openalex.org/keywords/fractional-calculus","display_name":"Fractional calculus","score":0.5811129808425903},{"id":"https://openalex.org/keywords/hilbert-space","display_name":"Hilbert space","score":0.5445889830589294},{"id":"https://openalex.org/keywords/spectrum","display_name":"Spectrum (functional analysis)","score":0.5162996649742126},{"id":"https://openalex.org/keywords/differential-operator","display_name":"Differential operator","score":0.47566312551498413},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.45980727672576904},{"id":"https://openalex.org/keywords/dirichlet-boundary-condition","display_name":"Dirichlet boundary condition","score":0.4581524431705475},{"id":"https://openalex.org/keywords/boundary-value-problem","display_name":"Boundary value problem","score":0.44386523962020874},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.39047175645828247},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.34441936016082764}],"concepts":[{"id":"https://openalex.org/C22892174","wikidata":"https://www.wikidata.org/wiki/Q1154082","display_name":"Sturm\u2013Liouville theory","level":3,"score":0.8663100004196167},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8194162845611572},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.689485490322113},{"id":"https://openalex.org/C128803854","wikidata":"https://www.wikidata.org/wiki/Q1307821","display_name":"Eigenfunction","level":3,"score":0.667346715927124},{"id":"https://openalex.org/C154249771","wikidata":"https://www.wikidata.org/wiki/Q1339058","display_name":"Fractional calculus","level":2,"score":0.5811129808425903},{"id":"https://openalex.org/C62799726","wikidata":"https://www.wikidata.org/wiki/Q190056","display_name":"Hilbert space","level":2,"score":0.5445889830589294},{"id":"https://openalex.org/C156778621","wikidata":"https://www.wikidata.org/wiki/Q1365748","display_name":"Spectrum (functional analysis)","level":2,"score":0.5162996649742126},{"id":"https://openalex.org/C70915906","wikidata":"https://www.wikidata.org/wiki/Q1058681","display_name":"Differential operator","level":2,"score":0.47566312551498413},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.45980727672576904},{"id":"https://openalex.org/C110167270","wikidata":"https://www.wikidata.org/wiki/Q1193699","display_name":"Dirichlet boundary condition","level":3,"score":0.4581524431705475},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.44386523962020874},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.39047175645828247},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.34441936016082764},{"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/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym13122265","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13122265","pdf_url":"https://www.mdpi.com/2073-8994/13/12/2265/pdf?version=1638084855","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:1873f0abf0b24dc5a5a9c04b9d258758","is_oa":true,"landing_page_url":"https://doaj.org/article/1873f0abf0b24dc5a5a9c04b9d258758","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 13, Iss 12, p 2265 (2021)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/13/12/2265/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym13122265","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/sym13122265","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13122265","pdf_url":"https://www.mdpi.com/2073-8994/13/12/2265/pdf?version=1638084855","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/W3215422660.pdf"},"referenced_works_count":31,"referenced_works":["https://openalex.org/W69681244","https://openalex.org/W591434585","https://openalex.org/W1531510807","https://openalex.org/W1975765037","https://openalex.org/W1983461269","https://openalex.org/W1992614405","https://openalex.org/W1998819932","https://openalex.org/W2067313050","https://openalex.org/W2072995473","https://openalex.org/W2075267567","https://openalex.org/W2082817084","https://openalex.org/W2118288723","https://openalex.org/W2132997731","https://openalex.org/W2190158699","https://openalex.org/W2308190093","https://openalex.org/W2754379126","https://openalex.org/W2797154165","https://openalex.org/W2884442081","https://openalex.org/W2901117020","https://openalex.org/W2921956839","https://openalex.org/W2924154063","https://openalex.org/W2962785683","https://openalex.org/W3006903434","https://openalex.org/W3043289495","https://openalex.org/W3043366963","https://openalex.org/W3084956730","https://openalex.org/W3105737454","https://openalex.org/W3148115524","https://openalex.org/W6602848315","https://openalex.org/W6667333371","https://openalex.org/W6698152579"],"related_works":["https://openalex.org/W2386085043","https://openalex.org/W4399515674","https://openalex.org/W2512523565","https://openalex.org/W3160358654","https://openalex.org/W4301821448","https://openalex.org/W2964166133","https://openalex.org/W2061441441","https://openalex.org/W2129740403","https://openalex.org/W2366885346","https://openalex.org/W2041581680"],"abstract_inverted_index":{"In":[0,41,94],"this":[1],"study,":[2],"we":[3,44,112],"consider":[4],"regular":[5,29],"eigenvalue":[6,31,146],"problems":[7,66,147],"formulated":[8],"by":[9,56,85],"using":[10],"the":[11,20,28,35,46,70,75,91,96,108,115,124,132,150,154],"left":[12,36],"and":[13,18,37,62,123,141,160],"right":[14,38],"standard":[15],"fractional":[16,24,63,99,159,161],"derivatives":[17],"extend":[19],"notion":[21],"of":[22,49,98,117,126,144,157],"a":[23,129],"Sturm\u2013Liouville":[25,50,65,77,119,163],"problem":[26],"to":[27,104,149],"Prabhakar":[30,39,64,162],"problem,":[32],"which":[33,87],"includes":[34],"derivatives.":[40],"both":[42],"cases,":[43],"study":[45],"spectral":[47,109],"properties":[48],"operators":[51,78,83,120],"on":[52,153],"function":[53],"space":[54],"restricted":[55,103],"homogeneous":[57],"Dirichlet":[58],"boundary":[59],"conditions.":[60],"Fractional":[61],"are":[67,79,88],"converted":[68],"into":[69],"equivalent":[71],"integral":[72,76,118,140],"ones.":[73],"Afterwards,":[74],"rewritten":[80],"as":[81],"Hilbert\u2013Schmidt":[82,110],"determined":[84],"kernels,":[86],"continuous":[89],"under":[90],"corresponding":[92,133],"assumptions.":[93],"particular,":[95],"range":[97],"order":[100],"is":[101,121],"here":[102],"interval":[105],"(1/2,1].":[106],"Applying":[107],"theorem,":[111],"prove":[113],"that":[114],"spectrum":[116,156],"discrete":[122,155],"system":[125],"eigenfunctions":[127],"forms":[128],"basis":[130],"in":[131],"Hilbert":[134],"space.":[135],"Then,":[136],"equivalence":[137],"results":[138],"for":[139],"differential":[142,158],"versions":[143],"respective":[145],"lead":[148],"main":[151],"theorems":[152],"operators.":[164]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":3}],"updated_date":"2026-05-22T06:13:13.366637","created_date":"2025-10-10T00:00:00"}