{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T15:52:53Z","timestamp":1772553173443,"version":"3.50.1"},"reference-count":47,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2024,7,1]],"date-time":"2024-07-01T00:00:00Z","timestamp":1719792000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of Jiangsu Province, China","award":["BK20190578"],"award-info":[{"award-number":["BK20190578"]}]},{"name":"Jiangsu Province Colleges and Universities Undergraduate Scientific Research Innovative Program","award":["BK20190578"],"award-info":[{"award-number":["BK20190578"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we present a general theory for fractional-order sequential differential equations with Riemann\u2013Liouville nabla derivatives and Caputo nabla derivatives on time scales. The explicit solution, in the case of constant coefficients, for both the homogeneous and the non-homogeneous problems, are given using the \u2207-Mittag-Leffler function, Laplace transform method, operational method and operational decomposition method. In addition, we also provide some results about a solution to a new class of fractional-order sequential differential equations with convolutional-type variable coefficients using the Laplace transform method.<\/jats:p>","DOI":"10.3390\/axioms13070447","type":"journal-article","created":{"date-parts":[[2024,7,2]],"date-time":"2024-07-02T05:08:34Z","timestamp":1719896914000},"page":"447","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Fractional-Order Sequential Linear Differential Equations with Nabla Derivatives on Time Scales"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7395-0427","authenticated-orcid":false,"given":"Cheng-Cheng","family":"Zhu","sequence":"first","affiliation":[{"name":"School of Science, Jiangnan University, Wuxi 214122, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2722-7665","authenticated-orcid":false,"given":"Jiang","family":"Zhu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,7,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/j.camwa.2024.01.010","article-title":"A novel numerical inverse technique for multi-parameter time fractional radially symmetric anomalous diffusion problem with initial singularity","volume":"158","author":"Fan","year":"2024","journal-title":"Comput. Math. Appl."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Li, Z., and Zhang, Z. (2023). Stabilization Control for a Class of Fractional-Order HIV-1 Infection Model with Time Delays. Axioms, 12.","DOI":"10.3390\/axioms12070695"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2771","DOI":"10.1002\/asjc.2645","article-title":"Controllability of singular dynamic systems on time scales","volume":"24","author":"Malik","year":"2022","journal-title":"Asian J. Control"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"321","DOI":"10.2478\/s13540-014-0171-7","article-title":"Optimal random search, fractional dynamics and fractional calculus","volume":"17","author":"Zeng","year":"2014","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1017\/S0305004100045060","article-title":"Certain fractional q-integrals and q-derivatives","volume":"66","author":"Agarwal","year":"1969","journal-title":"Proc. Camb. Philos. Soc."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1090\/S0025-5718-1974-0346352-5","article-title":"Differences of fractional order","volume":"28","author":"Diaz","year":"1974","journal-title":"Math. Comput."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1016\/j.sigpro.2014.02.022","article-title":"Chaos synchronization of the discrete fractional logistic map","volume":"102","author":"Guo","year":"2014","journal-title":"Signal Process."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1697","DOI":"10.1007\/s11071-014-1250-3","article-title":"Discrete chaos in fractional delayed logistic maps","volume":"80","author":"Guo","year":"2015","journal-title":"Nonlinear Dyn."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1007\/s11071-013-1065-7","article-title":"Discrete fractional logistic map and its chaos","volume":"75","author":"Guo","year":"2014","journal-title":"Nonlinear Dyn."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"484","DOI":"10.1016\/j.physleta.2013.12.010","article-title":"Discrete chaos in fractional sine and standard maps","volume":"378","author":"Guo","year":"2014","journal-title":"Phys. Lett. A"},{"key":"ref_11","first-page":"165","article-title":"A Transform Method in Discrete Fractional Calculus","volume":"2","author":"Atici","year":"2007","journal-title":"Inter. J. Diff. Equ."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"981","DOI":"10.1090\/S0002-9939-08-09626-3","article-title":"Initial value problems in discrete fractional calculus","volume":"137","author":"Atici","year":"2009","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jmaa.2010.02.009","article-title":"Modeling with fractional difference equations","volume":"369","author":"Atici","year":"2010","journal-title":"J. Math. Anal. Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"153","DOI":"10.4067\/S0719-06462011000300009","article-title":"Sum and Difference Compositions in Discrete Fractional Calculus","volume":"13","author":"Holm","year":"2011","journal-title":"CUBO Math. J."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"562","DOI":"10.1016\/j.mcm.2009.11.006","article-title":"Nabla discrete fractional calculus and nabla inequalities","volume":"51","author":"Anastassiou","year":"2010","journal-title":"Math. Comput. Model."},{"key":"ref_16","first-page":"1","article-title":"Discrecte fractional caculus with the nabla operator","volume":"2009","author":"Atici","year":"2009","journal-title":"Electron. J. Qual. Theory Differ. Equ. Spec. Ed. I"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1007\/BF03323153","article-title":"Analysis on measure chains a unified approach to continuous and discrete calculas","volume":"18","author":"Hilger","year":"1990","journal-title":"Results Math."},{"key":"ref_18","unstructured":"Williams, P.A. (2012). Unifying Fractional Calculus with Time Scales. [Doctoral Thesis, The University of Melbourne]."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"3750","DOI":"10.1016\/j.camwa.2010.03.072","article-title":"Foundations of nabla fractional calculus on time scales and inequalities","volume":"59","author":"Anastassiou","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_20","unstructured":"Bastos, N. (2012). Fractional Calculus on Time Scales. [Doctoral Thesis, The University of Aveiro]."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"556","DOI":"10.1016\/j.mcm.2010.03.055","article-title":"Principles of delta fractional calculus on time scales and inequalities","volume":"52","author":"Anastassiou","year":"2010","journal-title":"Math. Comput. Model."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"513","DOI":"10.1016\/j.sigpro.2010.05.001","article-title":"Discrete-time fractional variational problems","volume":"91","author":"Bastos","year":"2011","journal-title":"Sign. Proc."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1591","DOI":"10.1016\/j.camwa.2011.04.019","article-title":"The Laplace transform in discrete fractional calculus","volume":"62","author":"Holm","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"795701","DOI":"10.1155\/2013\/795701","article-title":"Fractional Cauchy problem with Riemann-Liouville derivative on time scales","volume":"2013","author":"Wu","year":"2013","journal-title":"Abstr. Appl. Anal."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"486054","DOI":"10.1155\/2015\/486054","article-title":"Fractional Cauchy Problem with Caputo Nabla Derivative on Time Scales","volume":"2015","author":"Zhu","year":"2015","journal-title":"Abstr. Appl. Anal."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"401596","DOI":"10.1155\/2013\/401596","article-title":"Fractional Cauchy Problem with Riemann-Liouville Fractional Delta Derivative on Time Scales","volume":"2013","author":"Zhu","year":"2013","journal-title":"Abstr. Appl. Anal."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1186\/s13662-022-03678-9","article-title":"Relative asymptotic equivalence of dynamic equations on time scales","volume":"2022","author":"Duque","year":"2022","journal-title":"Adv. Contin. Discret. Models"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"166186","DOI":"10.1016\/j.ijleo.2020.166186","article-title":"Discrete fractional soliton dynamics of the fractional Ablowitz-Ladik model","volume":"228","author":"Fang","year":"2021","journal-title":"Optik"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1007\/s00009-020-01629-w","article-title":"On dynamical systems with Nabla half derivative on time scales","volume":"17","author":"Kisela","year":"2020","journal-title":"Mediterr. J. Math."},{"key":"ref_30","first-page":"2555","article-title":"Modelling and analysis of dynamic systems on time-space scales and application in burgers equation","volume":"12","author":"Liu","year":"2022","journal-title":"J. Appl. Anal. Comput."},{"key":"ref_31","first-page":"215","article-title":"Qualitative analysis of dynamic equations on time scales using Lyapunov functions","volume":"14","author":"Messina","year":"2022","journal-title":"Diff. Equ. Appl."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1007\/s11071-014-1867-2","article-title":"Discrete fractional diffusion equation","volume":"80","author":"Wu","year":"2015","journal-title":"Nonlinear Dyn."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1007\/s41478-023-00597-0","article-title":"Existence of solution of a nonlinear fractional dynamic equation with initial and boundary conditions on time scales","volume":"32","author":"Gogoi","year":"2024","journal-title":"J. Anal."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"228","DOI":"10.1007\/s00025-023-02007-0","article-title":"Periodic boundary value problems for fractional dynamic equations on time scales","volume":"78","author":"Gogoi","year":"2023","journal-title":"Results Math."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Gogoi, B., Saha, U.K., Hazarika, B., Torres, D.F., and Ahmad, H. (2021). Nabla Fractional Derivative and Fractional Integral on Time Scales. Axioms, 10.","DOI":"10.3390\/axioms10040317"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"5934","DOI":"10.3934\/math.2023299","article-title":"Sequential fractional order Neutral functional Integro differential equations on time scales with Caputo fractional operator over Banach spaces","volume":"8","author":"Morsy","year":"2023","journal-title":"AIMS Math."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1186\/1687-1847-2013-259","article-title":"Power functions and essentials of frational calculus on isolated time scales","volume":"2013","author":"Kisela","year":"2013","journal-title":"Adv. Diff. Equ."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Bohner, M., and Peterson, A. (2003). Advances in Dynamic Equations on Time Scales, Birkh\u00e4user.","DOI":"10.1007\/978-0-8176-8230-9"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1311","DOI":"10.1007\/s40808-021-01158-9","article-title":"The influence of awareness campaigns on the spread of an infectious disease: Aqualitative analysis of a fractional epidemic model","volume":"8","author":"Akdim","year":"2022","journal-title":"Model. Earth Syst. Environ."},{"key":"ref_40","first-page":"100524","article-title":"Dynamics of a time fractional order spatio-temporal SIR with vaccination and temporary immunity","volume":"7","author":"Bounkaicha","year":"2023","journal-title":"Partial Diff. Equ. Appl. Math."},{"key":"ref_41","first-page":"100216","article-title":"Dynamics of SIQR epidemic model with fractional order derivative","volume":"5","author":"Paul","year":"2022","journal-title":"Partial Diff. Equ. Appl. Math."},{"key":"ref_42","first-page":"100593","article-title":"Fractional-order SIR epidemic model with treatment cure rate","volume":"8","author":"Sadki","year":"2023","journal-title":"Partial Diff. Equ. Appl. Math."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1007\/s40314-021-01456-z","article-title":"The memory effect on fractional calculus: An application in the spread of COVID-19","volume":"40","author":"Barros","year":"2021","journal-title":"Comput. Appl. Math."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"142","DOI":"10.1007\/s40819-021-01086-3","article-title":"Fractional Model and Numerical Algorithms for Predicting COVID-19 with Isolation and Quarantine Strategies","volume":"7","author":"Hamou","year":"2021","journal-title":"Int. J. Appl. Comput. Math."},{"key":"ref_45","first-page":"100470","article-title":"Studying of COVID-19 fractional model: Stability analysis","volume":"7","author":"Khalaf","year":"2023","journal-title":"Partial Diff. Equ. Appl. Math."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"1707","DOI":"10.1016\/j.camwa.2017.02.014","article-title":"Traveling waves in a nonlocal dispersal SIRH model with relapse","volume":"73","author":"Zhu","year":"2017","journal-title":"Comput. Math. Appl."},{"key":"ref_47","first-page":"147","article-title":"Traveling waves of a reaction-diffusion SIRQ epidemic model with relapse","volume":"7","author":"Zhu","year":"2017","journal-title":"J. Appl. Anal. Comput."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/7\/447\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:08:50Z","timestamp":1760108930000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/7\/447"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,7,1]]},"references-count":47,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2024,7]]}},"alternative-id":["axioms13070447"],"URL":"https:\/\/doi.org\/10.3390\/axioms13070447","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,7,1]]}}}