{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,23]],"date-time":"2026-01-23T07:20:53Z","timestamp":1769152853102,"version":"3.49.0"},"reference-count":30,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2021,11,24]],"date-time":"2021-11-24T00:00:00Z","timestamp":1637712000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we introduce the nabla fractional derivative and fractional integral on time scales in the Riemann\u2013Liouville sense. We also introduce the nabla fractional derivative in Gr\u00fcnwald\u2013Letnikov sense. Some of the basic properties and theorems related to nabla fractional calculus are discussed.<\/jats:p>","DOI":"10.3390\/axioms10040317","type":"journal-article","created":{"date-parts":[[2021,11,25]],"date-time":"2021-11-25T04:01:28Z","timestamp":1637812888000},"page":"317","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Nabla Fractional Derivative and Fractional Integral on Time Scales"],"prefix":"10.3390","volume":"10","author":[{"given":"Bikash","family":"Gogoi","sequence":"first","affiliation":[{"name":"Department of Basic and Applied Science, National Institute of Technology, Jote 791113, Arunachal Pradesh, India"}]},{"given":"Utpal Kumar","family":"Saha","sequence":"additional","affiliation":[{"name":"Department of Basic and Applied Science, National Institute of Technology, Jote 791113, Arunachal Pradesh, India"}]},{"given":"Bipan","family":"Hazarika","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Gauhati University, Guwahati 781014, Assam, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8641-2505","authenticated-orcid":false,"given":"Delfim F. M.","family":"Torres","sequence":"additional","affiliation":[{"name":"Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5438-5407","authenticated-orcid":false,"given":"Hijaz","family":"Ahmad","sequence":"additional","affiliation":[{"name":"Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, 00186 Roma, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2021,11,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"753601","DOI":"10.1155\/S0161171203301486","article-title":"Recent application of fractional calculus to science and engineering","volume":"2003","author":"Debnath","year":"2003","journal-title":"Int. J. Math. Math. Sci."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Garrappa, K., Kaslik, E., and Popolizio, M. (2019). Evaluation of fractional integral and derivative of elementary function: Overview and Tutorial. Mathematics, 7.","DOI":"10.3390\/math7050407"},{"key":"ref_3","unstructured":"Miller, K.S., and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, Inc.. A Wiley-Interscience Publication."},{"key":"ref_4","unstructured":"Oldham, K.B., and Spanier, J. (1974). The Fractional Calculus, Academic Press."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1007\/BF03323153","article-title":"Analysis on measure chains, a unified approach to continuous and discrete calculus","volume":"18","author":"Hilger","year":"1990","journal-title":"Results Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2683","DOI":"10.1016\/S0362-546X(96)00204-0","article-title":"Differential and difference calculus, unified","volume":"30","author":"Hilger","year":"1997","journal-title":"Nonlinear Anal."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Bohner, M., and Peterson, A. (2004). Advances in Dynamic Equations on Time Scales, Birkh\u00e4user.","DOI":"10.1007\/978-0-8176-8230-9"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Bohner, M., and Peterson, A. (2001). Dynamic Equations on Time Scales: An Introduction with Applications, Birkh\u00e4uster.","DOI":"10.1007\/978-1-4612-0201-1"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"718","DOI":"10.1016\/j.mcm.2005.08.014","article-title":"An application of time scales to economics","volume":"43","author":"Atici","year":"2006","journal-title":"Math. Comput. Model."},{"key":"ref_10","first-page":"1","article-title":"Fractional Derivatives and Integrals on Time Scales via the Inverse Generalized Laplace Transform","volume":"11","author":"Bastos","year":"2011","journal-title":"Int. J. Math. Comput."},{"key":"ref_11","first-page":"486054","article-title":"Fractional Cauchy problem with Caputo nabla derivative on time scales","volume":"23","author":"Zhu","year":"2015","journal-title":"Abst. Appl. Anal."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1007\/s00498-013-0106-6","article-title":"Linear positive, control system on time scales controllability","volume":"25","author":"Bartosiewiez","year":"2013","journal-title":"Math. Control Signals Syst."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"230","DOI":"10.1016\/j.sigpro.2014.05.026","article-title":"A fractional calculus on arbitrary time scales: Fractional differentiation and fractional Integration","volume":"107","author":"Benkhettou","year":"2015","journal-title":"Signal Process."},{"key":"ref_14","unstructured":"Duke, E.R. (2006). Solving Higher Order Dynamic Equation on Time Scales as First Order Systems. [Master\u2019s Thesis, Marshall University]."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1016\/S0022-247X(03)00361-5","article-title":"Integration on Time Scales","volume":"285","author":"Guseinov","year":"2003","journal-title":"J. Math. Anal. Appl."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Rogosin, S., and Dubatovskaya, M. (2018). Letnikov vs. Marchaud: A survey on Two Prominent Constructions of Fractional Derivatives. Mathematics, 6.","DOI":"10.3390\/math6010003"},{"key":"ref_17","first-page":"1","article-title":"A new fractional derivative on time scales","volume":"11","author":"Zhao","year":"2016","journal-title":"Adv. Appl. Math. Anal."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"255","DOI":"10.14403\/jcms.2016.29.2.255","article-title":"On delta alpha derivative on time scales","volume":"29","author":"Zhao","year":"2016","journal-title":"J. Chungcheong Math. Soc."},{"key":"ref_19","first-page":"401","article-title":"Fractional Cauchy problem with Riemann-Liouville fractional delta derivative on time scales","volume":"19","author":"Zhu","year":"2013","journal-title":"Abst. Appl. Anal."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/j.jksus.2015.08.001","article-title":"Existence and Uniqueness of Solution for a Fractional Riemann-Liouville Initial Value Problem on Time Scales","volume":"28","author":"Benkhettou","year":"2016","journal-title":"J. King Saud Univ. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"107407","DOI":"10.1016\/j.aml.2021.107407","article-title":"Cauchy\u2019s formula on nonempty closed sets and a new notion of Riemann-Liouville fractional integral on time scales","volume":"121","author":"Torres","year":"2021","journal-title":"Appl. Math. Lett."},{"key":"ref_22","first-page":"202","article-title":"A Nabla Conformable Fractional Calculus on Time Scales","volume":"7","author":"Bendouma","year":"2019","journal-title":"Electr. J. Math. Anal. Appl."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"238","DOI":"10.1186\/s13662-021-03385-x","article-title":"A new conformable nabla derivative and its application on arbitrary time scales","volume":"2021","author":"Rahmat","year":"2021","journal-title":"Adv. Diff. Equ."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Wang, C., Agarwal, R.P., O\u2019Regan, D., and Sakthivel, R. (2020). Theory of Translation Closedness for Time Scales: With Applications in Translation Functions and Dynamic Equations, Springer Nature.","DOI":"10.1007\/978-3-030-38644-3"},{"key":"ref_25","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.I. (2006). Theory and Application of Fractional Differential Equation, North-Holand Mathematics Studies, Elsevier Science B. V.. pp. xvi+523."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1006\/jmaa.2000.7194","article-title":"Analysis of Fractional Differential Equation","volume":"265","author":"Diethelm","year":"2002","journal-title":"J. Math. Anal. Appl."},{"key":"ref_27","unstructured":"Koning, D.E., Sterk, A.E., and Trantelman, H.L. (2005). Fractional Calculus. [Bachelor\u2019s Thesis, University of Groningen]."},{"key":"ref_28","unstructured":"Podlubny, I. (1999). Fractional Differential Equation, Academic Press."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1016\/j.cnsns.2016.04.006","article-title":"A fractal derivative model for the characterization of anomalous diffusion in magnetic resonance imaging","volume":"39","author":"Liang","year":"2016","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1016\/j.chaos.2017.03.066","article-title":"New methodologies in fractional and fractal derivatives modeling","volume":"102","author":"Chen","year":"2017","journal-title":"Chaos Solitons Fractals"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/10\/4\/317\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:35:00Z","timestamp":1760168100000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/10\/4\/317"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,11,24]]},"references-count":30,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2021,12]]}},"alternative-id":["axioms10040317"],"URL":"https:\/\/doi.org\/10.3390\/axioms10040317","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,11,24]]}}}