{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,12]],"date-time":"2026-01-12T22:30:31Z","timestamp":1768257031553,"version":"3.49.0"},"reference-count":21,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2021,3,5]],"date-time":"2021-03-05T00:00:00Z","timestamp":1614902400000},"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>In this article, we begin by introducing two classes of lacunary fractional spline functions by using the Liouville\u2013Caputo fractional Taylor expansion. We then introduce a new higher-order lacunary fractional spline method. We not only derive the existence and uniqueness of the method, but we also provide the error bounds for approximating the unique positive solution. As applications of our fundamental findings, we offer some Liouville\u2013Caputo fractional differential equations (FDEs) to illustrate the practicability and effectiveness of the proposed method. Several recent developments on the the theory and applications of FDEs in (for example) real-life situations are also indicated.<\/jats:p>","DOI":"10.3390\/sym13030422","type":"journal-article","created":{"date-parts":[[2021,3,5]],"date-time":"2021-03-05T11:46:09Z","timestamp":1614944769000},"page":"422","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Some Higher-Degree Lacunary Fractional Splines in the Approximation of Fractional Differential Equations"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9277-8092","authenticated-orcid":false,"given":"Hari Mohan","family":"Srivastava","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada"},{"name":"Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan"},{"name":"Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, AZ1007 Baku, Azerbaijan"},{"name":"Section of Mathematics, International Telematic University Uninettuno, I-00186 Rome, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6837-8075","authenticated-orcid":false,"given":"Pshtiwan Othman","family":"Mohammed","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Education, University of Sulaimani, Sulaimani 46001, Kurdistan Region, Iraq"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2788-809X","authenticated-orcid":false,"given":"Juan L. G.","family":"Guirao","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica Aplicada y Estad\u00edstica, Universidad Polit\u00e9cnica de Cartagena, Campus de la Muralla, 30203 Cartagena, Murcia, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0365-0282","authenticated-orcid":false,"given":"Y. S.","family":"Hamed","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,3,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Mainardi, F. (2010). Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press.","DOI":"10.1142\/9781848163300"},{"key":"ref_2","unstructured":"Magin, R.L. (2006). Fractional Calculus in Bioengineering, Begell House Publishers Incorporated."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"43","DOI":"10.5539\/jmr.v11n4p43","article-title":"A Generalized Uncertain Fractional Forward Difference Equations of Riemann\u2013Liouville Type","volume":"11","author":"Mohammed","year":"2019","journal-title":"J. Math. Res."},{"key":"ref_4","unstructured":"Mohammed, P.O., and Abdeljawad, T. (2020). Discrete generalized fractional operators defined using h-discrete Mittag-Leffler kernels and applications to AB fractional difference systems. Math. Meth. Appl. Sci., 1\u201326."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"6598682","DOI":"10.1155\/2020\/6598682","article-title":"Existence and uniqueness of uncertain fractional backward difference equations of Riemann\u2013Liouville type","volume":"2020","author":"Mohammed","year":"2020","journal-title":"Math. Prob. Eng."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1250970","DOI":"10.1155\/2020\/1250970","article-title":"Solution of singular integral equations via Riemann\u2013Liouville fractional integrals","volume":"2020","author":"Alqudah","year":"2020","journal-title":"Math. Prob. Eng."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"341","DOI":"10.46793\/KgJMat2203.341M","article-title":"Non-conformable fractional Laplace transform","volume":"46","author":"Martinez","year":"2020","journal-title":"Kragujevac J. Math."},{"key":"ref_8","first-page":"67","article-title":"Generalized quartic fractional spline interpolation with applications","volume":"8","author":"Hamasalh","year":"2015","journal-title":"Internat. J. Open Probl. Compt. Math."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Tassaddiq1, A., Yaseen, M., Yousaf, A., and Srivastava, R. (2021). A cubic B-spline collocation method with new approximation for the numerical treatment of the heat equation with classical and non-classical boundary conditions. Phys. Scr., 96, 045212.","DOI":"10.1088\/1402-4896\/abe066"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"51","DOI":"10.11648\/j.pamj.20170601.17","article-title":"2017 Cubic B-spline collocation method for one-dimensional heat equation","volume":"6","author":"Khabir","year":"2017","journal-title":"Pure Appl. Math."},{"key":"ref_11","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier Sci. B.V.. North-Holland Mathematics Studies."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Hilfer, R. (2000). Applications of Fractional Calculus in Physics, World Scientific Publishing Company.","DOI":"10.1142\/9789812817747"},{"key":"ref_13","unstructured":"Podlubny, I. (1999). Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press. Mathematics in Science and Engineering."},{"key":"ref_14","first-page":"73","article-title":"Fractional-order derivatives and integrals: Introductory overview and recent developments","volume":"60","author":"Srivastava","year":"2020","journal-title":"Kyungpook Math. J."},{"key":"ref_15","unstructured":"Usero, D. (2018, March 02). Fractional Taylor Series for Caputo Fractional Derivatives: Construction of Numerical Schemes. Available online: http:\/\/www.fdi.ucm.es\/profesor\/lvazquez\/calcfrac\/docs\/paper_Usero.pdf."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1186\/s13662-018-1543-9","article-title":"The mean value theorem and Taylor\u2019s theorem for fractional derivatives with Mittag\u2013Leffler kernel","volume":"2018","author":"Fernandez","year":"2018","journal-title":"Adv. Differ. Equ."},{"key":"ref_17","unstructured":"Steffens, K.G. (2006). The History of Approximation Theory, Birkh\u00e4user."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.23880\/phoa-16000163","article-title":"Diabetes and its resulting complications: Mathematical modeling via fractional calculus","volume":"4","author":"Srivastava","year":"2020","journal-title":"Public Health Open Access"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"101345","DOI":"10.1016\/j.jksus.2021.101345","article-title":"An efficient semi-analytical method for solving the generalized regularized long wave equations with a new fractional derivative operator","volume":"33","author":"Srivastava","year":"2021","journal-title":"J. King Saud Univ. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"103722","DOI":"10.1016\/j.rinp.2020.103722","article-title":"Numerical simulation and stability analysis for the fractional-order dynamics of COVID-19","volume":"20","author":"Singh","year":"2021","journal-title":"Results Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"280","DOI":"10.3389\/fphy.2020.00280","article-title":"A correlation between solutions of uncertain fractional forward difference equations and their paths","volume":"8","author":"Srivastava","year":"2020","journal-title":"Front. Phys."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/3\/422\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:33:31Z","timestamp":1760160811000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/3\/422"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,3,5]]},"references-count":21,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2021,3]]}},"alternative-id":["sym13030422"],"URL":"https:\/\/doi.org\/10.3390\/sym13030422","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,3,5]]}}}