{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,6]],"date-time":"2026-03-06T18:20:45Z","timestamp":1772821245361,"version":"3.50.1"},"reference-count":29,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2020,7,30]],"date-time":"2020-07-30T00:00:00Z","timestamp":1596067200000},"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>The main goal of this paper is to define a simple but effective method for approximating solutions of multi-order fractional differential equations relying on Caputo fractional derivative and under supplementary conditions. Our basis functions are based on some original generalization of the Bessel polynomials, which satisfy many properties shared by the classical orthogonal polynomials as given by Hermit, Laguerre, and Jacobi. The main advantages of our polynomials are two-fold: All the coefficients are positive and any collocation matrix of Bessel polynomials at positive points is strictly totally positive. By expanding the unknowns in a (truncated) series of basis functions at the collocation points, the solution of governing differential equation can be easily converted into the solution of a system of algebraic equations, thus reducing the computational complexities considerably. Several practical test problems also with some symmetries are given to show the validity and utility of the proposed technique. Comparisons with available exact solutions as well as with several alternative algorithms are also carried out. The main feature of our approach is the good performance both in terms of accuracy and simplicity for obtaining an approximation to the solution of differential equations of fractional order.<\/jats:p>","DOI":"10.3390\/sym12081260","type":"journal-article","created":{"date-parts":[[2020,7,31]],"date-time":"2020-07-31T04:15:31Z","timestamp":1596168931000},"page":"1260","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":42,"title":["Generalized Bessel Polynomial for Multi-Order Fractional Differential Equations"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6116-4928","authenticated-orcid":false,"given":"Mohammad","family":"Izadi","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman 76169-14111, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7504-0424","authenticated-orcid":false,"given":"Carlo","family":"Cattani","sequence":"additional","affiliation":[{"name":"Engineering School, DEIM, University of Tuscia, 01100 Viterbo, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,7,30]]},"reference":[{"key":"ref_1","unstructured":"Podlubny, I. (1999). Fractional Differential Equations, Academic Press."},{"key":"ref_2","first-page":"33","article-title":"Numerical solution for multi-term fractional (arbitrary) orders differential equations","volume":"23","year":"2004","journal-title":"Comput. Appl. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1016\/S0377-0427(02)00558-7","article-title":"The numerical solution of linear multi-term fractional differential equations: Systems of equations","volume":"148","author":"Edwards","year":"2002","journal-title":"J. Comput. Appl. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"788","DOI":"10.1016\/j.apnum.2010.02.007","article-title":"Analytical and numerical solutions of multi-term nonlinear fractional order differential equations","volume":"60","author":"Ziada","year":"2010","journal-title":"Appl. Numer. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1326","DOI":"10.1016\/j.camwa.2009.07.006","article-title":"A new operational matrix for solving fractional-order differential equations","volume":"59","author":"Saadatmandi","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1023","DOI":"10.1007\/s11071-015-2087-0","article-title":"A review of operational matrices and spectral techniques for fractional calculus","volume":"81","author":"ABhrawy","year":"2015","journal-title":"Nonlinear Dyn."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1369","DOI":"10.1007\/s00521-017-3118-1","article-title":"An operational matrix based on Chelyshkov polynomials for solving multi-order fractional differential equations","volume":"30","author":"Talaei","year":"2018","journal-title":"Neural Comput. Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1007\/s40819-020-00823-4","article-title":"A comparative study of two Legendre-collocation schemes applied to fractional logistic equation","volume":"6","author":"Izadi","year":"2020","journal-title":"Int. J. Appl. Comput. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1149","DOI":"10.1016\/j.cnsns.2011.07.018","article-title":"The construction of operational matrix of fractional derivatives using B-spline functions, Commun","volume":"17","author":"Lakestani","year":"2012","journal-title":"Nonlinear Sci. Numer. Simul."},{"key":"ref_10","first-page":"189","article-title":"Numerical solution of multi-order fractional differential equations using generalized triangular function operational matrices","volume":"263","author":"Damarla","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1090\/S0002-9947-1949-0028473-1","article-title":"A new class of orthogonal polynomials: The Bessel polynomials","volume":"65","author":"Krall","year":"1949","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1544","DOI":"10.1016\/j.ejc.2007.12.002","article-title":"Combinatorial proofs of inverse relations and log-concavity for Bessel numbers","volume":"29","author":"Han","year":"2008","journal-title":"Eur. J. Combin."},{"key":"ref_13","first-page":"627","article-title":"The Bessel numbers and Bessel matrices","volume":"31","author":"Yang","year":"2011","journal-title":"J. Math. Res. Exp."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1137\/0507009","article-title":"The Bessel polynomials and the student t distribution","volume":"7","author":"Ismail","year":"1976","journal-title":"SIAM J. Math. Anal."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"730","DOI":"10.1007\/BF01180560","article-title":"\u00dcber Sturm-Liouvillesche polynomsysteme","volume":"29","author":"Bochner","year":"1929","journal-title":"Math. Zeit."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1090\/S0002-9947-1951-0053280-2","article-title":"On some algebraic properties of the Bessel polynomials","volume":"71","author":"Grosswald","year":"1951","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Grosswald, E. (1978). Bessel Polynomials, Lecture Notes in Math. Vol. 698, Springer.","DOI":"10.1007\/BFb0063135"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1264","DOI":"10.1007\/s10915-019-00975-6","article-title":"Accurate algorithms for Bessel matrices","volume":"80","author":"Delgado","year":"2019","journal-title":"J. Sci. Comput."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"637","DOI":"10.21136\/AM.2019.0279-18","article-title":"Fractional-order Bessel functions with various applications","volume":"64","author":"Dehestani","year":"2019","journal-title":"Appl. Math."},{"key":"ref_20","first-page":"286","article-title":"Generalized Taylor\u2019s formula","volume":"186","author":"Odibat","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"5498","DOI":"10.1016\/j.apm.2012.10.026","article-title":"Fractional-order Legendre functions for solving fractional-order differential equations","volume":"37","author":"Kazem","year":"2013","journal-title":"Appl. Math. Model."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"894","DOI":"10.1016\/j.apm.2011.07.045","article-title":"A numerical method for solving boundary value problems for fractional differential equations","volume":"36","author":"Rehman","year":"2012","journal-title":"Appl. Math. Model."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"061025","DOI":"10.1115\/1.4029785","article-title":"On a numerical approach to solve multi order fractional differential equations with initial\/boundary conditions","volume":"10","author":"Firoozjaee","year":"2015","journal-title":"J. Comput. Nonlinear Dyn."},{"key":"ref_24","first-page":"4798","article-title":"Local discontinuous Galerkin approximations to fractional Bagley-Torvik equation","volume":"43","author":"Izadi","year":"2020","journal-title":"Math. Meth. Appl. Sci."},{"key":"ref_25","first-page":"1541","article-title":"Uniqueness and approximation of solution for fractional Bagley-Torvik equations with variable coefficients","volume":"94","author":"Wei","year":"2016","journal-title":"Int. J. Comput. Math."},{"key":"ref_26","first-page":"123","article-title":"Fractional polynomial approximations to the solution of fractional Riccati equation","volume":"51","author":"Izadi","year":"2019","journal-title":"Punjab Univ. J. Math."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Izadi, M. (2020). Comparison of Various Fractional Basis Functions for Solving Fractional-Order Logistic Population Model, University of Ni\u0161. Facta Univ. Ser. Math. Inform.","DOI":"10.22190\/FUMI2004181I"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1186\/s13662-015-0558-8","article-title":"Numerical solutions of fractional differential equations of Lane-Emden type by an accurate technique","volume":"2015","author":"Akgul","year":"2015","journal-title":"Adv. Differ. Equ."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"851","DOI":"10.4236\/am.2012.38126","article-title":"Numerical study of fractional differential equations of Lane-Emden type by method of collocation","volume":"3","author":"Mechee","year":"2012","journal-title":"Appl. Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/8\/1260\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:53:09Z","timestamp":1760176389000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/8\/1260"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,30]]},"references-count":29,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2020,8]]}},"alternative-id":["sym12081260"],"URL":"https:\/\/doi.org\/10.3390\/sym12081260","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,7,30]]}}}