{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T18:41:16Z","timestamp":1777401676831,"version":"3.51.4"},"reference-count":30,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2021,11,28]],"date-time":"2021-11-28T00:00:00Z","timestamp":1638057600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this study, we consider regular eigenvalue problems formulated by using the left and right standard fractional derivatives and extend the notion of a fractional Sturm\u2013Liouville problem to the regular Prabhakar eigenvalue problem, which includes the left and right Prabhakar derivatives. In both cases, we study the spectral properties of Sturm\u2013Liouville operators on function space restricted by homogeneous Dirichlet boundary conditions. Fractional and fractional Prabhakar Sturm\u2013Liouville problems are converted into the equivalent integral ones. Afterwards, the integral Sturm\u2013Liouville operators are rewritten as Hilbert\u2013Schmidt operators determined by kernels, which are continuous under the corresponding assumptions. In particular, the range of fractional order is here restricted to interval (1\/2,1]. Applying the spectral Hilbert\u2013Schmidt theorem, we prove that the spectrum of integral Sturm\u2013Liouville operators is discrete and the system of eigenfunctions forms a basis in the corresponding Hilbert space. Then, equivalence results for integral and differential versions of respective eigenvalue problems lead to the main theorems on the discrete spectrum of differential fractional and fractional Prabhakar Sturm\u2013Liouville operators.<\/jats:p>","DOI":"10.3390\/sym13122265","type":"journal-article","created":{"date-parts":[[2021,12,1]],"date-time":"2021-12-01T03:12:40Z","timestamp":1638328360000},"page":"2265","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Spectrum of Fractional and Fractional Prabhakar Sturm\u2013Liouville Problems with Homogeneous Dirichlet Boundary Conditions"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1029-1863","authenticated-orcid":false,"given":"Malgorzata","family":"Klimek","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Mechanical Engineering and Computer Science, Czestochowa University of Technology, 42-201 Czestochowa, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,11,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1016\/j.chaos.2007.07.041","article-title":"An efficient method for solving fractional Sturm-Liouville problems","volume":"40","year":"2009","journal-title":"Chaos Solitons Fractals"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2837","DOI":"10.1080\/00207160802562549","article-title":"On the numerical solution of fractional Sturm-Liouville problems","volume":"87","year":"2010","journal-title":"Int. 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