{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,19]],"date-time":"2026-06-19T04:09:14Z","timestamp":1781842154228,"version":"3.54.5"},"reference-count":38,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"funder":[{"DOI":"10.13039\/501100023674","name":"Deanship of Scientific Research, King Khalid University","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100023674","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100020912","name":"King Faisal University","doi-asserted-by":"publisher","award":["KFU260983"],"award-info":[{"award-number":["KFU260983"]}],"id":[{"id":"10.13039\/501100020912","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Journal of Computational and Applied Mathematics"],"published-print":{"date-parts":[[2026,10]]},"DOI":"10.1016\/j.cam.2026.117557","type":"journal-article","created":{"date-parts":[[2026,2,27]],"date-time":"2026-02-27T00:15:34Z","timestamp":1772151334000},"page":"117557","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":1,"special_numbering":"C","title":["Discrete fractional operators of delta-type with Mittag-Leffler kernels and applications"],"prefix":"10.1016","volume":"485","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5671-8308","authenticated-orcid":false,"given":"Meraa","family":"Arab","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6837-8075","authenticated-orcid":false,"given":"Pshtiwan Othman","family":"Mohammed","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0206-3828","authenticated-orcid":false,"given":"Majeed Ahmad","family":"Yousif","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"78","reference":[{"key":"10.1016\/j.cam.2026.117557_bib0001","doi-asserted-by":"crossref","first-page":"7688","DOI":"10.1002\/mma.7083","article-title":"Discrete generalized fractional operators defined using h-discrete Mittag-Leffler kernels and applications to AB fractional difference systems","volume":"46","author":"Mohammed","year":"2020","journal-title":"Math. Meth. Appl. Sci."},{"key":"10.1016\/j.cam.2026.117557_bib0002","series-title":"Discrete Fractional Calculus","author":"Goodrich","year":"2015"},{"key":"10.1016\/j.cam.2026.117557_bib0003","doi-asserted-by":"crossref","first-page":"852","DOI":"10.1016\/j.aej.2025.06.058","article-title":"Uniqueness property of the generalized laplace transform and computational visualization of electromagnetic waves in plasma via fractional operators","volume":"128","author":"Mehmood","year":"2025","journal-title":"Alex. Eng. J."},{"key":"10.1016\/j.cam.2026.117557_bib0004","series-title":"Innovative techniques to chaotic dynamics and kinetic differ-integral equations","first-page":"2490515","author":"Mehmood","year":"2025"},{"key":"10.1016\/j.cam.2026.117557_bib0005","doi-asserted-by":"crossref","DOI":"10.1142\/S0218348X24501305","article-title":"A new formulation and analytical applications of fractional operators","volume":"33","author":"Mehmood","year":"2025","journal-title":"Fractals"},{"key":"10.1016\/j.cam.2026.117557_bib0006","doi-asserted-by":"crossref","first-page":"168","DOI":"10.1016\/j.nahs.2018.12.001","article-title":"Solutions of systems with the Caputo-Fabrizio fractional delta derivative on time scales","volume":"32","author":"Mozyrska","year":"2019","journal-title":"Nonlinear Anal. Hybrid Syst."},{"key":"10.1016\/j.cam.2026.117557_bib0007","first-page":"313","article-title":"A new approach for modeling with discrete fractional equations. fund","volume":"151","author":"Atici","year":"2017","journal-title":"Inform"},{"key":"10.1016\/j.cam.2026.117557_bib0008","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1515\/cmb-2020-0105","article-title":"Pharmacokinetics and pharmacodynamics models of tumor growth and anticancer effects in discrete time","volume":"8","author":"Atici","year":"2020","journal-title":"Comput. Math. Biophys."},{"key":"10.1016\/j.cam.2026.117557_bib0009","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/j.jmaa.2011.06.022","article-title":"On discrete sequential fractional boundary value problems","volume":"385","author":"Goodrich","year":"2012","journal-title":"J. Math. Anal. Appl."},{"key":"10.1016\/j.cam.2026.117557_bib0010","doi-asserted-by":"crossref","first-page":"880","DOI":"10.1631\/FITEE.2000133","article-title":"Discrete fractional watermark technique","volume":"21","author":"Wang","year":"2020","journal-title":"Front. Inf. Technol. Electron. Eng"},{"key":"10.1016\/j.cam.2026.117557_bib0011","first-page":"317","article-title":"Laplace transforms for the nabla-difference operator and a fractional variation of parameters formula","volume":"16","author":"Ahrendt","year":"2012","journal-title":"Commun. Appl. Anal."},{"key":"10.1016\/j.cam.2026.117557_bib0012","article-title":"Discrete fractional Bihari inequality and uniqueness theorem of solutions of nabla fractional difference equations with non-Lipschitz nonlinearities","volume":"367","author":"Wang","year":"2020","journal-title":"Appl. Math. Comput."},{"key":"10.1016\/j.cam.2026.117557_bib0013","doi-asserted-by":"crossref","DOI":"10.1016\/j.chaos.2023.114098","article-title":"Approximation of sequential fractional systems of Liouville-Caputo type by discrete delta difference operators","volume":"176","author":"Almusawa","year":"2023","journal-title":"Chaos Soliton. Fract."},{"key":"10.1016\/j.cam.2026.117557_bib0014","series-title":"On convexity analysis for discrete delta Riemann-Liouville fractional differences analytically and numerically","author":"Baleanu","year":"2004"},{"key":"10.1016\/j.cam.2026.117557_bib0015","doi-asserted-by":"crossref","first-page":"7461","DOI":"10.1002\/mma.5869","article-title":"Ulam-Hyers stability of Caputo fractional difference equations","volume":"42","author":"Chen","year":"2019","journal-title":"Math. Meth. Appl. Sci."},{"key":"10.1016\/j.cam.2026.117557_bib0016","doi-asserted-by":"crossref","first-page":"55","DOI":"10.3390\/fractalfract9010055","article-title":"Discrete fractional-Order modeling of recurrent childhood diseases using the caputo difference operator","volume":"9","author":"Madani","year":"2025","journal-title":"Fractal Fract."},{"key":"10.1016\/j.cam.2026.117557_bib0017","doi-asserted-by":"crossref","first-page":"7321","DOI":"10.3934\/math.2024355","article-title":"Analysis of a nonlinear problem involving discrete and proportional delay with application to houseflies model","volume":"9","author":"Shah","year":"2024","journal-title":"Aims Math."},{"key":"10.1016\/j.cam.2026.117557_bib0018","article-title":"Approximation of sequential fractional systems of Liouville-Caputo type by discrete delta difference operators","volume":"204","author":"Mohammed","year":"2026","journal-title":"Chaos Solit. Fractals"},{"key":"10.1016\/j.cam.2026.117557_bib0019","series-title":"An introduction to the fractional calculus and fractional differential equations","author":"Miller","year":"1993"},{"key":"10.1016\/j.cam.2026.117557_bib0020","first-page":"31","article-title":"Generalized special functions in the description of fractional diffusive equations","volume":"10","author":"Cesarano","year":"2019","journal-title":"Commun. Appl. Ind. Math."},{"key":"10.1016\/j.cam.2026.117557_bib0021","doi-asserted-by":"crossref","DOI":"10.1016\/j.cam.2022.114476","article-title":"Analysis and some applications of a regularized \u03a8-Hilfer fractional derivative","volume":"415","author":"Jajarmi","year":"2022","journal-title":"J. Comput. Appl. Math."},{"key":"10.1016\/j.cam.2026.117557_bib0022","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1007\/s11071-018-4367-y","article-title":"On the nonlinear dynamical systems within the generalized fractional derivatives with Mittag-Leffler Kernel","volume":"94","author":"Baleanu","year":"2018","journal-title":"Nonlinear Dyn."},{"key":"10.1016\/j.cam.2026.117557_bib0023","doi-asserted-by":"crossref","DOI":"10.1063\/1.5096159","article-title":"A new fractional model and optimal control of a tumor-immune surveillance with non-singular derivative operator","volume":"29","author":"Baleanu","year":"2019","journal-title":"Chaos"},{"key":"10.1016\/j.cam.2026.117557_bib0024","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1007\/s44198-024-00215-y","article-title":"Enhancing water pollution management through a comprehensive fractional modeling framework and optimal control techniques","author":"Ebrahimzadeh","year":"2024","journal-title":"J. Nonlinear Math. Phys."},{"key":"10.1016\/j.cam.2026.117557_bib0025","doi-asserted-by":"crossref","first-page":"196","DOI":"10.1177\/14613484231187439","article-title":"Fractional investigation of time-dependent mass pendulum","volume":"43","author":"Baleanu","year":"2023","journal-title":"J. low freq. noise vib. act. control"},{"key":"10.1016\/j.cam.2026.117557_bib0026","doi-asserted-by":"crossref","DOI":"10.1016\/j.chaos.2022.112990","article-title":"Stability analysis and system properties of Nipah virus transmission: a fractional calculus case study","volume":"166","author":"Baleanu","year":"2023","journal-title":"Chaos Solit. Fract."},{"key":"10.1016\/j.cam.2026.117557_bib0027","doi-asserted-by":"crossref","first-page":"1602","DOI":"10.1016\/j.camwa.2011.03.036","article-title":"On Riemann and Caputo fractional differences","volume":"62","author":"Abdeljawad","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"10.1016\/j.cam.2026.117557_bib0028","series-title":"Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels","first-page":"232","author":"Abdeljawad","year":"2016"},{"key":"10.1016\/j.cam.2026.117557_bib0029","doi-asserted-by":"crossref","first-page":"146","DOI":"10.1016\/j.chaos.2018.09.022","article-title":"Different type kernel h-fractional differences and their fractional h-sums","volume":"116","author":"Abdeljawad","year":"2018","journal-title":"Chaos Solit. Fract."},{"key":"10.1016\/j.cam.2026.117557_bib0030","series-title":"Arbitrary order fractional difference operators with discrete exponential kernels and applications","author":"Abdeljawad","year":"2017"},{"key":"10.1016\/j.cam.2026.117557_bib0031","doi-asserted-by":"crossref","first-page":"1303","DOI":"10.3390\/math9111303","article-title":"On Riemann-Liouville and Caputo fractional forward difference monotonicity analysis","author":"Mohammed","year":"2021","journal-title":"Mathematics"},{"key":"10.1016\/j.cam.2026.117557_bib0032","series-title":"Solutions comparative for a novel model of fractional delta difference type","first-page":"2482560","author":"Mohammed","year":"2025"},{"key":"10.1016\/j.cam.2026.117557_bib0033","first-page":"1","article-title":"Discrete Mittag-Leffler function and its applications","volume":"1302","author":"Nagai","year":"2003","journal-title":"Publ. Res. Inst. Math. Sci. Kyoto Univ"},{"key":"10.1016\/j.cam.2026.117557_bib0034","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/j.jksus.2015.06.004","article-title":"Mittag-Leffler function for discrete fractional modelling","volume":"28","author":"Wu","year":"2016","journal-title":"J. King Saud Univ. Sci."},{"key":"10.1016\/j.cam.2026.117557_bib0035","doi-asserted-by":"crossref","first-page":"254","DOI":"10.3390\/fractalfract7030254","article-title":"Mittag-Leffler functions in discrete time","volume":"7","author":"Atici","year":"2023","journal-title":"Fractal Fract"},{"key":"10.1016\/j.cam.2026.117557_bib0036","doi-asserted-by":"crossref","DOI":"10.1016\/j.chaos.2021.111182","article-title":"Discrete Prabhakar fractional difference and sum operators","volume":"150","author":"Mohammed","year":"2021","journal-title":"Chaos Soliton. Fract."},{"key":"10.1016\/j.cam.2026.117557_bib0037","doi-asserted-by":"crossref","first-page":"1367","DOI":"10.1080\/00207160.2021.1976762","article-title":"The calculation of the Mittag-Leffler function","volume":"99","author":"Saenko","year":"2022","journal-title":"Int. J. Comput. Math."},{"key":"10.1016\/j.cam.2026.117557_bib0038","doi-asserted-by":"crossref","DOI":"10.1016\/j.cam.2025.116565","article-title":"On the delta Mittag-Leffler functions and its application in monotonic analysis","volume":"465","author":"Mohammed","year":"2025","journal-title":"J. Comput. Appl. Math."}],"container-title":["Journal of Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0377042726002104?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0377042726002104?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,5,29]],"date-time":"2026-05-29T00:37:59Z","timestamp":1780015079000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0377042726002104"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,10]]},"references-count":38,"alternative-id":["S0377042726002104"],"URL":"https:\/\/doi.org\/10.1016\/j.cam.2026.117557","relation":{},"ISSN":["0377-0427"],"issn-type":[{"value":"0377-0427","type":"print"}],"subject":[],"published":{"date-parts":[[2026,10]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Discrete fractional operators of delta-type with Mittag-Leffler kernels and applications","name":"articletitle","label":"Article Title"},{"value":"Journal of Computational and Applied Mathematics","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.cam.2026.117557","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright","label":"Copyright"}],"article-number":"117557"}}