{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:21:21Z","timestamp":1760145681699,"version":"build-2065373602"},"reference-count":42,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2024,8,21]],"date-time":"2024-08-21T00:00:00Z","timestamp":1724198400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"University of Oradea, Romania"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this study, to approximate nabla sequential differential equations of fractional order, a class of discrete Liouville\u2013Caputo fractional operators is discussed. First, some special functions are re-called that will be useful to make a connection with the proposed discrete nabla operators. These operators exhibit inherent symmetrical properties which play a crucial role in ensuring the consistency and stability of the method. Next, a formula is adopted for the solution of the discrete system via binomial coefficients and analyzing the Riemann\u2013Liouville fractional sum operator. The symmetry in the binomial coefficients contributes to the precise approximation of the solutions. Based on this analysis, the solution of its corresponding continuous case is obtained when the step size p0 tends to 0. The transition from discrete to continuous domains highlights the symmetrical nature of the fractional operators. Finally, an example is shown to testify the correctness of the presented theoretical results. We discuss the comparison of the solutions of the operators along with the numerical example, emphasizing the role of symmetry in the accuracy and reliability of the numerical method.<\/jats:p>","DOI":"10.3390\/axioms13080570","type":"journal-article","created":{"date-parts":[[2024,8,23]],"date-time":"2024-08-23T12:58:07Z","timestamp":1724417887000},"page":"570","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Theoretical Investigation of Fractional Estimations in Liouville\u2013Caputo Operators of Mixed Order with Applications"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6837-8075","authenticated-orcid":false,"given":"Pshtiwan Othman","family":"Mohammed","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Education, University of Sulaimani, Sulaymaniyah 46001, Iraq"},{"name":"Research and Development Center, University of Sulaimani, Sulaymaniyah 46001, Iraq"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2855-7535","authenticated-orcid":false,"given":"Alina Alb","family":"Lupas","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, University of Oradea, 410087 Oradea, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0634-2370","authenticated-orcid":false,"given":"Ravi P.","family":"Agarwal","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Systems Engineering, Florida Institute of Technology, Melbourne, FL 32901, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0206-3828","authenticated-orcid":false,"given":"Majeed A.","family":"Yousif","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Education, University of Zakho, Duhok 42001, Iraq"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0223-4711","authenticated-orcid":false,"given":"Eman","family":"Al-Sarairah","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Khalifa University of Science and Technology, Abu Dhabi P.O. Box 127788, United Arab Emirates"},{"name":"Department of Mathematics, Al-Hussein Bin Talal University, P.O. Box 20, Ma\u2019an 71111, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9896-6692","authenticated-orcid":false,"given":"Mohamed","family":"Abdelwahed","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,21]]},"reference":[{"key":"ref_1","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). 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