{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T06:50:51Z","timestamp":1777704651041,"version":"3.51.4"},"reference-count":48,"publisher":"SAGE Publications","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["IFS"],"published-print":{"date-parts":[[2021,4,22]]},"abstract":"<jats:p>According to a huge interest in implementation of the fuzzy Volterra integral equations, especially the second kind, researchers have been investigating to solve such equations using numerical methods since analytical ones might not be accessible usually. In this research paper, we introduce a new approach based on Fibonacci polynomials collocation method to numerically solve them. Several properties of such polynomials were considered to implement in the collocation method due to approximate the solution of the second kind of fuzzy Volterra integral equations. We approved the existence, uniqueness of the solution, convergence and the error analysis of the proposed method in detail. In order to show the authenticity and applicability of the proposed method, we employed several illustrative examples. The numerical results show that the convergence and precision of the recent method were in a good settlement with the exact solution. Also, the calculations of the suggested method are simple and low computational complexity in respect to other methods as an advantage feature of the presented approach.<\/jats:p>","DOI":"10.3233\/jifs-202523","type":"journal-article","created":{"date-parts":[[2021,2,13]],"date-time":"2021-02-13T21:54:51Z","timestamp":1613253291000},"page":"9899-9914","source":"Crossref","is-referenced-by-count":0,"title":["An efficient method to solve fuzzy Volterra integral equations using Fibonacci polynomials"],"prefix":"10.1177","volume":"40","author":[{"given":"M.","family":"Darabi","sequence":"first","affiliation":[{"name":"Department of Mathematics, Hamedan Branch, Islamic Azad University, Hamedan, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"T.","family":"Allahviranloo","sequence":"additional","affiliation":[{"name":"Faculty of Engineering and Natural Sciences, Bahcesehir University, Istanbul, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","reference":[{"issue":"5","key":"10.3233\/JIFS-202523_ref1","doi-asserted-by":"crossref","first-page":"1039","DOI":"10.1016\/j.chaos.2004.02.051","article-title":"Intuitionistic fuzzy metric spaces","volume":"22","author":"Park","year":"2004","journal-title":"Chaos Solitons & Fractals"},{"issue":"1","key":"10.3233\/JIFS-202523_ref2","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1016\/j.chaos.2004.11.018","article-title":"\u03b8-Compact fuzzy topological spaces","volume":"25","author":"Caldas","year":"2005","journal-title":"Chaos Solitons & Fractals"},{"issue":"2","key":"10.3233\/JIFS-202523_ref3","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1016\/j.chaos.2004.04.013","article-title":"Adaptive control of discrete-time chaotic systems: a fuzzy control approach","volume":"23","author":"Feng","year":"2005","journal-title":"Chaos Solitons & Fractals"},{"issue":"4","key":"10.3233\/JIFS-202523_ref4","doi-asserted-by":"crossref","first-page":"1075","DOI":"10.1016\/j.chaos.2004.09.056","article-title":"H1 variable universe adaptive fuzzy control for chaotic system","volume":"24","author":"Jiang","year":"2005","journal-title":"Chaos Solitons & Fractals"},{"issue":"1","key":"10.3233\/JIFS-202523_ref5","first-page":"117","article-title":"Numerical solution of fuzzy differential equation by Runge-Kutta method","volume":"11","author":"Abbasbandy","year":"2004","journal-title":"Journal of Science Teacher Training University"},{"issue":"10-11","key":"10.3233\/JIFS-202523_ref6","doi-asserted-by":"crossref","first-page":"1633","DOI":"10.1016\/j.camwa.2004.03.009","article-title":"Numerical methods for fuzzy differential inclusions","volume":"48","author":"Abbasbandy","year":"2004","journal-title":"Computers & Mathematics with Applications"},{"issue":"5","key":"10.3233\/JIFS-202523_ref7","doi-asserted-by":"crossref","first-page":"1337","DOI":"10.1016\/j.chaos.2005.03.018","article-title":"Tuning of reachable set in one dimensional fuzzy differential inclusions","volume":"26","author":"Abbasbandy","year":"2005","journal-title":"Chaos Solitons & Fractals"},{"key":"10.3233\/JIFS-202523_ref8","doi-asserted-by":"crossref","first-page":"105312","DOI":"10.1016\/j.cnsns.2020.105312","article-title":"Optimal control of a fractional order model for granular SEIR epidemic with uncertainty","volume":"88","author":"Dong","year":"2020","journal-title":"Communications in Nonlinear Science and Numerical Simulation"},{"key":"10.3233\/JIFS-202523_ref10","doi-asserted-by":"crossref","unstructured":"Son N.T.K. , Long H.V. and Dong N.P. , Fuzzy delay differential equations under granular differentiability with applications. Computational and Applied Mathematics, 38(3) (2019), 107\u2013136.","DOI":"10.1007\/s40314-019-0881-x"},{"issue":"2","key":"10.3233\/JIFS-202523_ref11","doi-asserted-by":"crossref","first-page":"919","DOI":"10.1007\/s00034-019-01248-1","article-title":"On the stabilizability for a class of linear time-invariant systems under uncertainty","volume":"39","author":"Son","year":"2020","journal-title":"Circuits Systems and Signal Processing"},{"key":"10.3233\/JIFS-202523_ref12","doi-asserted-by":"crossref","first-page":"16845","DOI":"10.1007\/s11042-019-7388-8","article-title":"Towards granular calculus of single-valued neutrosophic functions under granular computing","volume":"79","author":"Son","year":"2019","journal-title":"Multimedia Tools and Applications"},{"issue":"1","key":"10.3233\/JIFS-202523_ref13","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1016\/S0960-0779(03)00278-9","article-title":"A review of E infinity theory and the mass spectrum of high energy particle physics","volume":"19","author":"El Naschie","year":"2004","journal-title":"Chaos Solitons & Fractals"},{"issue":"2","key":"10.3233\/JIFS-202523_ref14","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1515\/IJNSNS.2005.6.2.95","article-title":"On a fuzzy K\u00e4hler-like manifold which is consistent with the two slit experiment","volume":"6","author":"El Naschie","year":"2005","journal-title":"International Journal of Nonlinear Sciences and Numerical Simulation"},{"issue":"2","key":"10.3233\/JIFS-202523_ref15","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1016\/j.chaos.2004.02.028","article-title":"The concepts of E infinity: An elementary introduction to the Cantorian-fractal theory of quantum physics","volume":"22","author":"El Naschie","year":"2004","journal-title":"Chaos Solitons & Fractals"},{"issue":"2","key":"10.3233\/JIFS-202523_ref16","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1016\/j.chaos.2004.09.034","article-title":"Chaotic dynamics in the Friedmann equation","volume":"24","author":"Tanaka","year":"2005","journal-title":"Chaos Solitons & Fractals"},{"issue":"2","key":"10.3233\/JIFS-202523_ref17","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1016\/0165-0114(82)90001-X","article-title":"Towards fuzzy differential calculus part 2: Integration on fuzzy intervals","volume":"8","author":"Dubois","year":"1982","journal-title":"Fuzzy Sets and Systems"},{"issue":"1","key":"10.3233\/JIFS-202523_ref18","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/0165-0114(86)90026-6","article-title":"Elementary fuzzy calculus","volume":"18","author":"Goetschel","year":"1986","journal-title":"Fuzzy Sets and Systems"},{"issue":"3","key":"10.3233\/JIFS-202523_ref19","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1016\/S0165-0114(99)00057-3","article-title":"On Henstock integral of fuzzy-number-valued functions (I)","volume":"120","author":"Wu","year":"2001","journal-title":"Fuzzy Sets and Systems"},{"issue":"3","key":"10.3233\/JIFS-202523_ref20","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1016\/S0165-0114(03)00182-9","article-title":"Quadrature rules for integrals of fuzzy-number-valued functions","volume":"145","author":"Bede","year":"2004","journal-title":"Fuzzy Sets and Systems"},{"issue":"3","key":"10.3233\/JIFS-202523_ref21","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1016\/0165-0114(87)90029-7","article-title":"Fuzzy differential equations","volume":"24","author":"Kaleva","year":"1987","journal-title":"Fuzzy Sets and Systems"},{"issue":"3","key":"10.3233\/JIFS-202523_ref22","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1016\/0165-0114(87)90030-3","article-title":"On the fuzzy initial value problem","volume":"24","author":"Seikkala","year":"1987","journal-title":"Fuzzy Sets and Systems"},{"issue":"3","key":"10.3233\/JIFS-202523_ref23","doi-asserted-by":"crossref","first-page":"733","DOI":"10.1016\/j.amc.2003.12.071","article-title":"Numerical solution of linear Fredholm fuzzy integral equations of the second kind by Adomian method","volume":"161","author":"Babolian","year":"2005","journal-title":"Applied Mathematics and Computation"},{"issue":"2","key":"10.3233\/JIFS-202523_ref24","first-page":"31","article-title":"The homotopy perturbation method for fuzzy Volterra integral equations","volume":"8","author":"Allahviranloo","year":"2010","journal-title":"International Journal of Computational Cognition"},{"issue":"2","key":"10.3233\/JIFS-202523_ref25","first-page":"73","article-title":"Numerical solution of fuzzy linear Volterra integral equations of the second kind by homotopy analysis method","volume":"2","author":"Ghanbari","year":"2010","journal-title":"International Journal of Industrial Mathematics"},{"issue":"4","key":"10.3233\/JIFS-202523_ref26","first-page":"25","article-title":"Solving linear Fredholm fuzzy integral equations of the second kind by modified trapezoidal method. Journal of Applied Mathematics","volume":"7","author":"Allahviranloo","year":"2011","journal-title":"Islamic Azad University of Lahijan"},{"issue":"10","key":"10.3233\/JIFS-202523_ref27","doi-asserted-by":"crossref","first-page":"1885","DOI":"10.1007\/s00500-013-1173-9","article-title":"The use of airfoil and Chebyshev polynomials methods for solving fuzzy Fredholm integro-differential equations with Cauchy kernel","volume":"18","author":"Allahviranloo","year":"2014","journal-title":"Soft Computing"},{"issue":"2","key":"10.3233\/JIFS-202523_ref28","first-page":"43","article-title":"Linear Volterra fuzzy integral equation solved by modified trapezoidal method","volume":"2","author":"Hussain","year":"2013","journal-title":"International Journal of Applied Mathematics & Statistical Sciences"},{"issue":"3","key":"10.3233\/JIFS-202523_ref29","doi-asserted-by":"crossref","first-page":"148","DOI":"10.4304\/jait.4.3.148-155","article-title":"Solution of fuzzy Volterra integral equations in a Bernstein polynomial basis","volume":"4","author":"Mosleh","year":"2013","journal-title":"Journal of Advances in Information Technology"},{"issue":"1","key":"10.3233\/JIFS-202523_ref30","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1016\/j.asej.2013.09.002","article-title":"Solving systems of linear Fredholm integro-differential equations with Fibonacci polynomials","volume":"5","author":"Mirzaee","year":"2014","journal-title":"Ain Shams Engineering Journal"},{"issue":"2","key":"10.3233\/JIFS-202523_ref31","first-page":"71","article-title":"Fuzzy collocation methods for second-order fuzzy Abel-Volterra integro-differential equations","volume":"11","author":"Behzadi","year":"2014","journal-title":"Iranian Journal of Fuzzy Systems"},{"issue":"2","key":"10.3233\/JIFS-202523_ref32","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1016\/j.bjbas.2014.05.010","article-title":"A Fibonacci collocation method for solving a class of Fredholm-Volterra integral equations in two-dimensional spaces","volume":"3","author":"Mirzaee","year":"2014","journal-title":"Beni-Suef University Journal of Basic and Applied Sciences"},{"issue":"1","key":"10.3233\/JIFS-202523_ref33","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s40064-016-2038-3","article-title":"Homotopy perturbation method: a versatile tool to evaluate linear and nonlinear fuzzy Volterra integral equations of the second kind","volume":"5","author":"Narayanamoorthy","year":"2016","journal-title":"Springer Plus"},{"issue":"1","key":"10.3233\/JIFS-202523_ref35","doi-asserted-by":"crossref","first-page":"146","DOI":"10.3846\/mma.2020.9695","article-title":"Collocation method for fuzzy Volterra integral equations of the second kind","volume":"25","author":"Alijani","year":"2020","journal-title":"Mathematical Modelling and Analysis"},{"issue":"1","key":"10.3233\/JIFS-202523_ref36","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1016\/j.amc.2004.10.043","article-title":"Numerical methods for approximation of fuzzy data","volume":"169","author":"Allahviranloo","year":"2005","journal-title":"Applied Mathematics and Computation"},{"issue":"3","key":"10.3233\/JIFS-202523_ref37","doi-asserted-by":"crossref","first-page":"581","DOI":"10.1016\/j.fss.2004.08.001","article-title":"Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations","volume":"151","author":"Bede","year":"2005","journal-title":"Fuzzy Sets and Systems"},{"key":"10.3233\/JIFS-202523_ref38","unstructured":"Hochstadt H. , Integral equations, John Wiley & Sons (2011), 91."},{"key":"10.3233\/JIFS-202523_ref39","first-page":"11","article-title":"On the integrals, series and integral equations of fuzzy set valued functions","volume":"21","author":"Congxin","year":"1990","journal-title":"Journal of Harbin Institute of Technology"},{"issue":"1","key":"10.3233\/JIFS-202523_ref40","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1007\/BF02239501","article-title":"Stability of numerical methods for Volterra integro-differential equations","volume":"12","author":"Brunner","year":"1974","journal-title":"Computing"},{"issue":"4","key":"10.3233\/JIFS-202523_ref41","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1007\/BF02238797","article-title":"Global stability analysis of the Runge-Kutta methods for Volterra integral and integro-differential equations with degenerate kernels","volume":"45","author":"Crisci","year":"1990","journal-title":"Computing"},{"issue":"3","key":"10.3233\/JIFS-202523_ref42","doi-asserted-by":"crossref","first-page":"1005","DOI":"10.1016\/j.chaos.2007.03.007","article-title":"On k-Fibonacci sequences and polynomials and their derivatives","volume":"39","author":"Falcon","year":"2009","journal-title":"Chaos Solitons & Fractals"},{"key":"10.3233\/JIFS-202523_ref43","unstructured":"Rainville E.D. , Special functions, Chelsea Publishing Company (1971)."},{"key":"10.3233\/JIFS-202523_ref44","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1016\/j.fss.2016.04.004","article-title":"A new Bernoulli wavelet method for accurate solutions of nonlinear fuzzy Hammerstein-Volterra delay integral equations","volume":"309","author":"Sahu","year":"2017","journal-title":"Fuzzy Sets and Systems"},{"issue":"2","key":"10.3233\/JIFS-202523_ref45","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1016\/S0165-0114(96)00270-9","article-title":"Fuzzy linear systems","volume":"96","author":"Friedman","year":"1998","journal-title":"Fuzzy Sets and Systems"},{"issue":"1","key":"10.3233\/JIFS-202523_ref46","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/s00500-009-0537-7","article-title":"Solving fuzzy linear systems","volume":"15","author":"Ezzati","year":"2011","journal-title":"Soft Computing"},{"issue":"2","key":"10.3233\/JIFS-202523_ref47","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1007\/s11139-015-9712-x","article-title":"New hypergeometric connection formulae between Fibonacci and Chebyshev polynomials","volume":"42","author":"Abd-Elhameed","year":"2017","journal-title":"The Ramanujan Journal"},{"issue":"1","key":"10.3233\/JIFS-202523_ref48","first-page":"99","article-title":"The autobiography of leonardo pisano","volume":"11","author":"Grimm","year":"1973","journal-title":"The Fibonacci Quarterly"},{"key":"10.3233\/JIFS-202523_ref49","doi-asserted-by":"crossref","first-page":"1","DOI":"10.5899\/2014\/jfsva-00190","article-title":"An application of a semi-analytical method on linear fuzzy Volterra integral equations","volume":"2014","author":"Allahviranloo","year":"2014","journal-title":"Journal of Fuzzy Set Valued Analysis"},{"key":"10.3233\/JIFS-202523_ref50","doi-asserted-by":"crossref","unstructured":"Wazwaz A.M. , Linear and nonlinear integral equations, Springer, Berlin, Heidelberg, (2011), 639.","DOI":"10.1007\/978-3-642-21449-3"}],"container-title":["Journal of Intelligent &amp; Fuzzy Systems"],"original-title":[],"link":[{"URL":"https:\/\/content.iospress.com\/download?id=10.3233\/JIFS-202523","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T09:41:42Z","timestamp":1777455702000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/full\/10.3233\/JIFS-202523"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4,22]]},"references-count":48,"journal-issue":{"issue":"5"},"URL":"https:\/\/doi.org\/10.3233\/jifs-202523","relation":{},"ISSN":["1064-1246","1875-8967"],"issn-type":[{"value":"1064-1246","type":"print"},{"value":"1875-8967","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,4,22]]}}}