{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,31]],"date-time":"2025-10-31T14:35:19Z","timestamp":1761921319045,"version":"3.38.0"},"reference-count":34,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2025,1,15]],"date-time":"2025-01-15T00:00:00Z","timestamp":1736899200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,1,15]],"date-time":"2025-01-15T00:00:00Z","timestamp":1736899200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Universitat de Valencia"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2025,4]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>We study a Volterra convolution equation of the second kind, based on a combination of Riemann\u2013Liouville integrals. The problem can be reformulated involving the Caputo fractional derivative, hence the equation becomes of differintegral type. The modeling interpretation is based on a non-Markovian state function, where the Riemann\u2013Liouville multi-orders are memory coefficients that decrease hazard risks of change. We prove the validity of the reformulations with fractional-calculus theory, local existence with fixed-point tools, and global uniqueness with a Gronwall-type argumentation. We show some examples and their associated physics. We also solve the general linear equation by means of the algebraic formalism of Mikusi\u0144ski operational calculus, which is superior to Laplace transforms or Picard\u2019s iterations. Multivariate Mittag\u2013Leffler functions play a key role. We relate the emerging closed-form solution with the fractional power series that one may expect for these types of models.<\/jats:p>","DOI":"10.1007\/s40314-024-03072-z","type":"journal-article","created":{"date-parts":[[2025,1,15]],"date-time":"2025-01-15T12:33:28Z","timestamp":1736944408000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A general Volterra integral problem of Riemann\u2013Liouville type"],"prefix":"10.1007","volume":"44","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0748-3730","authenticated-orcid":false,"given":"Marc","family":"Jornet","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,1,15]]},"reference":[{"issue":"9","key":"3072_CR1","doi-asserted-by":"publisher","first-page":"710","DOI":"10.1080\/10652469.2020.1833194","volume":"32","author":"M Al-Kandari","year":"2021","unstructured":"Al-Kandari M, Hanna LA, Luchko Y (2021) On an extension of the Mikusi\u0144ski type operational calculus for the Caputo fractional derivative. Integral Transform Spec Funct 32(9):710\u2013725","journal-title":"Integral Transform Spec Funct"},{"key":"3072_CR2","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2024.115263","volume":"186","author":"J Calatayud","year":"2024","unstructured":"Calatayud J, Jornet M, Pinto CMA (2024) On the interpretation of Caputo fractional compartmental models. Chaos Solitons Fractals 186:115263","journal-title":"Chaos Solitons Fractals"},{"key":"3072_CR3","unstructured":"Cao Labora D (2025) An extension of the Cartwright\u2013McMullen theorem in fractional calculus for the Stieltjes case. J Integral Equ Appl. Accepted for publication, available at arXiv:2309.07148"},{"issue":"2","key":"3072_CR4","doi-asserted-by":"publisher","first-page":"289","DOI":"10.3390\/math8020289","volume":"8","author":"D Cao Labora","year":"2020","unstructured":"Cao Labora D, Tenreiro Machado JA (2020) Existence of bounded solutions to a modified version of the Bagley\u2013Torvik equation. Mathematics 8(2):289","journal-title":"Mathematics"},{"issue":"1","key":"3072_CR5","doi-asserted-by":"publisher","first-page":"275","DOI":"10.1016\/j.net.2021.07.026","volume":"54","author":"CA Cruz-L\u00f3pez","year":"2022","unstructured":"Cruz-L\u00f3pez CA, Espinosa-Paredes G (2022) Fractional radioactive decay law and Bateman equations. Nucl Eng Technol 54(1):275\u2013282","journal-title":"Nucl Eng Technol"},{"key":"3072_CR6","doi-asserted-by":"publisher","DOI":"10.1016\/j.cpc.2021.108268","volume":"273","author":"CA Cruz-L\u00f3pez","year":"2022","unstructured":"Cruz-L\u00f3pez CA, Espinosa-Paredes G, Fran\u00e7ois JL (2022) Development of the general Bateman solution using fractional calculus: a theoretical and algorithmic approach. Comput Phys Commun 273:108268","journal-title":"Comput Phys Commun"},{"key":"3072_CR7","doi-asserted-by":"crossref","unstructured":"Diethelm K (2010) The analysis of fractional differential equations. An application-oriented exposition using differential operators of caputo type. Lecture Notes in Mathematics. Springer, Berlin","DOI":"10.1007\/978-3-642-14574-2"},{"issue":"2","key":"3072_CR8","doi-asserted-by":"publisher","first-page":"304","DOI":"10.2478\/s13540-012-0022-3","volume":"15","author":"K Diethelm","year":"2012","unstructured":"Diethelm K (2012) The mean value theorems and a Nagumo-type uniqueness theorem for Caputo\u2019s fractional calculus. Fract Calculus Appl Anal 15(2):304\u2013313","journal-title":"Fract Calculus Appl Anal"},{"key":"3072_CR9","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2024.128642","volume":"18","author":"K Diethelm","year":"2024","unstructured":"Diethelm K, Hashemishahraki S, Thai HD, Tuan HT (2024) A constructive approach for investigating the stability of incommensurate fractional differential systems. J Math Anal Appl 18:128642","journal-title":"J Math Anal Appl"},{"key":"3072_CR10","doi-asserted-by":"publisher","first-page":"507","DOI":"10.1007\/s10928-010-9170-4","volume":"37","author":"A Dokoumetzidis","year":"2010","unstructured":"Dokoumetzidis A, Magin R, Macheras P (2010) Fractional kinetics in multi-compartmental systems. J Pharmacokinet Pharmacodyn 37:507\u2013524","journal-title":"J Pharmacokinet Pharmacodyn"},{"issue":"10","key":"3072_CR11","doi-asserted-by":"publisher","first-page":"503","DOI":"10.3390\/sym10100503","volume":"10","author":"J Duan","year":"2018","unstructured":"Duan J, Chen L (2018) Solution of fractional differential equation systems and computation of matrix Mittag\u2013Leffler functions. Symmetry 10(10):503","journal-title":"Symmetry"},{"issue":"3","key":"3072_CR12","doi-asserted-by":"publisher","first-page":"441","DOI":"10.37193\/CJM.2021.03.07","volume":"37","author":"H Fazli","year":"2021","unstructured":"Fazli H, Sun H, Aghchi S, Nieto JJ (2021) On a class of nonlinear nonlocal fractional differential equations. Carpath J Math 37(3):441\u2013448","journal-title":"Carpath J Math"},{"issue":"1","key":"3072_CR13","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1007\/s40590-023-00494-3","volume":"29","author":"A Fernandez","year":"2023","unstructured":"Fernandez A (2023) Mikusi\u0144ski\u2019s operational calculus for general conjugated fractional derivatives. Bolet\u00edn de la Sociedad Matematica Mexicana 29(1):25","journal-title":"Bolet\u00edn de la Sociedad Matematica Mexicana"},{"issue":"12","key":"3072_CR14","doi-asserted-by":"publisher","first-page":"143","DOI":"10.1016\/j.ifacol.2024.08.180","volume":"58","author":"A Fernandez","year":"2024","unstructured":"Fernandez A (2024) On complex orders in fractional calculus: floors, ceilings, and analytic continuation. IFAC-PapersOnLine 58(12):143\u2013148","journal-title":"IFAC-PapersOnLine"},{"issue":"2","key":"3072_CR15","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1007\/s43037-023-00258-1","volume":"17","author":"A Fernandez","year":"2023","unstructured":"Fernandez A, Rani N, Tomovski \u017d (2023) An operational calculus approach to Hilfer\u2013Prabhakar fractional derivatives. Banach J Math Anal 17(2):33","journal-title":"Banach J Math Anal"},{"issue":"5","key":"3072_CR16","doi-asserted-by":"publisher","first-page":"1085","DOI":"10.1016\/j.cam.2010.07.008","volume":"235","author":"R Garrappa","year":"2011","unstructured":"Garrappa R, Popolizio M (2011) On accurate product integration rules for linear fractional differential equations. J Comput Appl Math 235(5):1085\u20131097","journal-title":"J Comput Appl Math"},{"key":"3072_CR17","volume":"390","author":"IT Huseynov","year":"2021","unstructured":"Huseynov IT, Ahmadova A, Fernandez A, Mahmudov NI (2021) Explicit analytical solutions of incommensurate fractional differential equation systems. Appl Math Comput 390:125590","journal-title":"Appl Math Comput"},{"key":"3072_CR18","doi-asserted-by":"publisher","first-page":"623","DOI":"10.1016\/j.cjph.2024.10.002","volume":"92","author":"M Jornet","year":"2024","unstructured":"Jornet M (2024a) Analysis of the multi-term fractional Bateman equations in radioactive decay by means of Mikusi\u0144ski algebraic calculus. Chin J Phys 92:623\u2013630","journal-title":"Chin J Phys"},{"issue":"11","key":"3072_CR19","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1140\/epjp\/s13360-024-05772-1","volume":"139","author":"M Jornet","year":"2024","unstructured":"Jornet M (2024b) Closed-form solution for a mathematical extension of the multi-term fractional Bateman equations via Mikusi\u0144ski operational method. Eur Phys J Plus 139(11):1\u20137","journal-title":"Eur Phys J Plus"},{"key":"3072_CR20","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2024.134139","volume":"462","author":"M Jornet","year":"2024","unstructured":"Jornet M (2024c) On the Cauchy\u2013Kovalevskaya theorem for Caputo fractional differential equations. Physica D 462:134139","journal-title":"Physica D"},{"issue":"7","key":"3072_CR21","doi-asserted-by":"publisher","first-page":"411","DOI":"10.3390\/fractalfract8070411","volume":"8","author":"M Jornet","year":"2024","unstructured":"Jornet M (2024d) Theory on linear L-fractional differential equations and a new Mittag\u2013Leffler-type function. Fractal Fract 8(7):411","journal-title":"Fractal Fract"},{"issue":"11","key":"3072_CR22","doi-asserted-by":"publisher","first-page":"665","DOI":"10.3390\/fractalfract8110665","volume":"8","author":"M Jornet","year":"2024","unstructured":"Jornet M (2024e) Theory on new fractional operators using normalization and probability tools. Fractal Fract 8(11):665","journal-title":"Fractal Fract"},{"key":"3072_CR23","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2024.109085","volume":"154","author":"M Jornet","year":"2024","unstructured":"Jornet M, Nieto JJ (2024) Power-series solution of the L-fractional logistic equation. Appl Math Lett 154:109085","journal-title":"Appl Math Lett"},{"key":"3072_CR24","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2024.134400","volume":"470","author":"M Jornet","year":"2024","unstructured":"Jornet M, Nieto JJ (2024) On a nonlinear stochastic fractional differential equation of fluid dynamics. Physica D 470:134400","journal-title":"Physica D"},{"issue":"12","key":"3072_CR25","doi-asserted-by":"publisher","first-page":"2019","DOI":"10.1016\/j.aml.2011.05.035","volume":"24","author":"L Kexue","year":"2011","unstructured":"Kexue L, Jigen P (2011) Laplace transform and fractional differential equations. Appl Math Lett 24(12):2019\u20132023","journal-title":"Appl Math Lett"},{"issue":"1","key":"3072_CR26","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1186\/s13661-023-01769-4","volume":"2023","author":"P Kumar","year":"2023","unstructured":"Kumar P, Govindaraj V, Murillo-Arcila M (2023) The existence, uniqueness, and stability results for a nonlinear coupled system using $$\\psi $$-Caputo fractional derivatives. Boundary Value Probl 2023(1):75","journal-title":"Boundary Value Probl"},{"key":"3072_CR27","volume":"443","author":"C K\u00fcrt","year":"2023","unstructured":"K\u00fcrt C, Fernandez A, \u00d6zarslan MA (2023) Two unified families of bivariate Mittag\u2013Leffler functions. Appl Math Comput 443:127785","journal-title":"Appl Math Comput"},{"issue":"4","key":"3072_CR28","first-page":"463","volume":"2","author":"YF Luchko","year":"1999","unstructured":"Luchko YF (1999) Operational method in fractional calculus. Fract Calculus Appl Anal 2(4):463\u2013488","journal-title":"Fract Calculus Appl Anal"},{"issue":"2","key":"3072_CR29","first-page":"207","volume":"24","author":"Y Luchko","year":"1999","unstructured":"Luchko Y, Gorenflo R (1999) An operational method for solving fractional differential equations with the Caputo derivatives. Acta Math Vietnam 24(2):207\u2013233","journal-title":"Acta Math Vietnam"},{"key":"3072_CR30","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2023.113164","volume":"168","author":"VF Morales-Delgado","year":"2023","unstructured":"Morales-Delgado VF, Taneco-Hern\u00e1ndez MA, Vargas-De-Le\u00f3n C, G\u00f3mez-Aguilar JF (2023) Exact solutions to fractional pharmacokinetic models using multivariate Mittag\u2013Leffler functions. Chaos Solitons Fractals 168:113164","journal-title":"Chaos Solitons Fractals"},{"issue":"2","key":"3072_CR31","doi-asserted-by":"publisher","first-page":"247","DOI":"10.1080\/00207160.2021.1906869","volume":"99","author":"MA \u00d6zarslan","year":"2022","unstructured":"\u00d6zarslan MA, Fernandez A (2022) On the fractional calculus of multivariate Mittag\u2013Leffler functions. Int J Comput Math 99(2):247\u2013273","journal-title":"Int J Comput Math"},{"key":"3072_CR32","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2024.108249","volume":"138","author":"N Rani","year":"2024","unstructured":"Rani N, Fernandez A (2024) Mikusi\u0144ski\u2019s operational calculus for multi-dimensional fractional operators with applications to fractional PDEs. Commun Nonlinear Sci Numer Simul 138:108249","journal-title":"Commun Nonlinear Sci Numer Simul"},{"issue":"1","key":"3072_CR33","doi-asserted-by":"publisher","first-page":"312","DOI":"10.1016\/j.mbs.2006.10.008","volume":"208","author":"E White","year":"2007","unstructured":"White E, Comiskey C (2007) Heroin epidemics, treatment and ODE modelling. Math Biosci 208(1):312\u2013324","journal-title":"Math Biosci"},{"issue":"2","key":"3072_CR34","first-page":"195","volume":"12","author":"Y Zhou","year":"2009","unstructured":"Zhou Y (2009) Existence and uniqueness of solutions for a system of fractional differential equations. Fract Calculus Appl Anal 12(2):195\u2013204","journal-title":"Fract Calculus Appl Anal"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-024-03072-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-024-03072-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-024-03072-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,3,14]],"date-time":"2025-03-14T06:07:28Z","timestamp":1741932448000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-024-03072-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,15]]},"references-count":34,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,4]]}},"alternative-id":["3072"],"URL":"https:\/\/doi.org\/10.1007\/s40314-024-03072-z","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"type":"print","value":"2238-3603"},{"type":"electronic","value":"1807-0302"}],"subject":[],"published":{"date-parts":[[2025,1,15]]},"assertion":[{"value":"24 August 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 December 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 December 2024","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 January 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The author states that there is no Conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"124"}}