{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T19:20:40Z","timestamp":1775503240618,"version":"3.50.1"},"reference-count":20,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2019,11,4]],"date-time":"2019-11-04T00:00:00Z","timestamp":1572825600000},"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>This paper deals with the initial value problem for linear systems of fractional differential equations (FDEs) with variable coefficients involving Riemann\u2013Liouville and Caputo derivatives. Some basic properties of fractional derivatives and antiderivatives, including their non-symmetry w.r.t. each other, are discussed. The technique of the generalized Peano\u2013Baker series is used to obtain the state-transition matrix. Explicit solutions are derived both in the homogeneous and inhomogeneous case. The theoretical results are supported by examples.<\/jats:p>","DOI":"10.3390\/sym11111366","type":"journal-article","created":{"date-parts":[[2019,11,4]],"date-time":"2019-11-04T10:49:07Z","timestamp":1572864547000},"page":"1366","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":22,"title":["Analytical Solution of Linear Fractional Systems with Variable Coefficients Involving Riemann\u2013Liouville and Caputo Derivatives"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3808-5882","authenticated-orcid":false,"given":"Ivan","family":"Matychyn","sequence":"first","affiliation":[{"name":"Faculty of Mathematics and Computer Science, University of Warmia and Mazury, S\u0142oneczna 54, 10-710 Olsztyn, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,11,4]]},"reference":[{"key":"ref_1","unstructured":"Podlubny, I. (1998). Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"501","DOI":"10.2478\/s13540-013-0031-x","article-title":"What Euler could further write, or the unnoticed \u201cbig bang\u201d of the fractional calculus","volume":"16","author":"Podlubny","year":"2013","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1810","DOI":"10.1016\/j.camwa.2009.08.019","article-title":"Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag\u2013Leffler stability","volume":"59","author":"Li","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"350","DOI":"10.1016\/j.sigpro.2010.08.003","article-title":"On the fractional signals and systems","volume":"91","author":"Magin","year":"2011","journal-title":"Signal Process."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"218","DOI":"10.1016\/j.jmaa.2008.10.018","article-title":"Maximum principle for the generalized time-fractional diffusion equation","volume":"351","author":"Luchko","year":"2009","journal-title":"J. Math. Anal. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1515\/fca-2018-0015","article-title":"Complex spatio-temporal solutions in fractional reaction-diffusion systems near a bifurcation point","volume":"21","author":"Datsko","year":"2018","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Datsko, B., Podlubny, I., and Povstenko, Y. (2019). Time-Fractional Diffusion-Wave Equation with Mass Absorption in a Sphere under Harmonic Impact. Mathematics, 7.","DOI":"10.3390\/math7050433"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1007\/BF02732983","article-title":"Generalized Mittag-Leffler matrix functions in game problems for evolutionary equations of fractional order","volume":"36","author":"Chikrii","year":"2000","journal-title":"Cybern. Syst. Anal."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1615\/JAutomatInfScien.v40.i6.10","article-title":"Presentation of solutions of linear systems with fractional derivatives in the sense of Riemann\u2013Liouville, Caputo, and Miller\u2013Ross","volume":"40","author":"Chikrii","year":"2008","journal-title":"J. Autom. Inf. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"687","DOI":"10.1515\/fca-2015-0042","article-title":"Time-optimal control of fractional-order linear systems","volume":"18","author":"Matychyn","year":"2015","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1515\/fca-2018-0009","article-title":"Optimal control of linear systems with fractional derivatives","volume":"21","author":"Matychyn","year":"2018","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1016\/j.cam.2017.10.016","article-title":"On time-optimal control of fractional-order systems","volume":"339","author":"Matychyn","year":"2018","journal-title":"J. Comput. Appl. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"170","DOI":"10.1515\/fca-2019-0011","article-title":"Optimal control of linear systems of arbitrary fractional order","volume":"22","author":"Matychyn","year":"2019","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1109\/LCSYS.2018.2852600","article-title":"Solution of Time-Variant Fractional Differential Equations With a Generalized Peano-Baker Series","volume":"3","author":"Eckert","year":"2019","journal-title":"IEEE Control Syst. Lett."},{"key":"ref_15","first-page":"249","article-title":"Initialized fractional calculus","volume":"3","author":"Lorenzo","year":"2000","journal-title":"Int. J. Appl. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1134\/S0081543811080098","article-title":"The Peano-Baker series","volume":"275","author":"Baake","year":"2011","journal-title":"Proc. Steklov Inst. Math."},{"key":"ref_17","unstructured":"Samko, S.G., Kilbas, A.A., and Marichev, O.I. (1987). Integrals and Derivatives of Fractional Order and Some of Their Applications, Nauka i Tekhnika."},{"key":"ref_18","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Diethelm, K. (2010). The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer.","DOI":"10.1007\/978-3-642-14574-2"},{"key":"ref_20","unstructured":"Chen, W. (2003). Introduction to Lebesgue Integration, Imperial College."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/11\/1366\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:31:44Z","timestamp":1760189504000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/11\/1366"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,11,4]]},"references-count":20,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2019,11]]}},"alternative-id":["sym11111366"],"URL":"https:\/\/doi.org\/10.3390\/sym11111366","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,11,4]]}}}