Abstract
In this paper, a new numerical approach based on Fibonacci polynomials is proposed for solving first-order fuzzy Fredholm–Volterra integro-differential equations of the second kind. The existence, uniqueness and convergence of the solution under generalized differentiability are proved. Because of reducing the problem into a system of linear algebraic equations, this simple method is computationally attractive. The reliability and accuracy of the presented scheme are demonstrated by several numerical experiments.
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Abd-Elhameed WM, Youssri YH, El-Sissi N, Sadek M (2017) New hypergeometric connection formulae between Fibonacci and Chebyshev polynomials. Ramanujan J 42:347–361. https://doi.org/10.1007/s11139-015-9712-x
Allahviranloo T, Behzadi SS (2014) The use of airfoil and Chebyshev polynomials methods for solving fuzzy Fredholm integro-differential equations with Cauchy kernel. Soft Comput 18:1885–1897. https://doi.org/10.1007/s00500-013-1173-9
Babolian E, Goghary HS, Abbasbandy S (2005) Numerical solution of linear Fredholm fuzzy integral equations of the second kind by Adomian method. Appl Math Comput 161:733–744. https://doi.org/10.1016/j.amc.2003.12.071
Balachandran K, Kanagarajan K (2005) Existence of solutions of general nonlinear fuzzy Volterra-Fredholm integral equations. J Appl Math Stoch Anal 2005:333–343. https://doi.org/10.1155/JAMSA.2005.333
Balachandran K, Prakash P (2002) Existence of solutions of nonlinear fuzzy Volterra-Fredholm integral equations. Indian J Pure Appl Math 33:329–343
Bede B, Gal SG (2004) Quadrature rules for integrals of fuzzy-number-valued functions. Fuzzy Sets Syst 145:359–380. https://doi.org/10.1016/S0165-0114(03)00182-9
Bede B, Gal SG (2005) Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations. Fuzzy Sets Syst 151:581–599. https://doi.org/10.1016/j.fss.2004.08.001
Behzadi SS, Allahviranloo T, Abbasbandy S (2012) Solving fuzzy second-order nonlinear Volterra-Fredholm integro-differential equations by using Picard method. Neural Comput Appl 21:337–346. https://doi.org/10.1007/s00521-012-0926-1
Behzadi S, Allahviranloo T, Abbasbandy S (2014a) The use of fuzzy expansion method for solving fuzzy linear Volterra-Fredholm integral equations. J Intell Fuzzy Syst 26:1817–1822. https://doi.org/10.3233/IFS-130861
Behzadi SS, Allahviranloo T, Abbasbandy S (2014b) Fuzzy collocation methods for second-order fuzzy Abel-Volterra integro-differential equations. Iran J Fuzzy Syst 11:71–88
Bica AM (2008) Error estimation in the approximation of the solution of nonlinear fuzzy Fredholm integral equations. Inf Sci (NY) 178:1279–1292. https://doi.org/10.1016/j.ins.2007.10.021
Bica AM, Ziari S (2017) Iterative numerical method for fuzzy Volterra linear integral equations in two dimensions. Soft Comput 21:1097–1108. https://doi.org/10.1007/s00500-016-2085-2
Bica AM, Ziari S (2018) Open fuzzy cubature rule with application to nonlinear fuzzy Volterra integral equations in two dimensions. Fuzzy Sets Syst 1:1–24. https://doi.org/10.1016/j.fss.2018.04.010
Chang SSL, Zadeh LA (1972) On fuzzy mapping and control. IEEE Trans Syst Man Cybern SMC 2:30–34. https://doi.org/10.1109/TSMC.1972.5408553
Diamond P (2002) Theory and applications of fuzzy Volterra integral equations. IEEE Trans Fuzzy Syst 10:97–102
Dubois D, Prade H (1982) Towards fuzzy differential calculus. Fuzzy Sets Syst 8:1–7
Ezzati R, Sadatrasoul SM (2016) On numerical solution of two-dimensional nonlinear Urysohn fuzzy integral equations based on fuzzy Haar wavelets. Fuzzy Sets Syst 1:1–20. https://doi.org/10.1016/j.fss.2016.08.005
Ezzati R, Ziari S (2011) Numerical solution and error estimation of fuzzy Fredholm integral equation using fuzzy Bernstein polynomials. Aust J Basic Appl Sci 5:2072–2082
Ezzati R, Ziari S (2013) Numerical solution of nonlinear fuzzy Fredholm integral equations using iterative method. Appl Math Comput 225:33–42. https://doi.org/10.1016/j.amc.2013.09.020
Falcón S, Plaza Á (2009) On k-Fibonacci sequences and polynomials and their derivatives. Chaos Solitons Fractals 39:1005–1019. https://doi.org/10.1016/j.chaos.2007.03.007
Friedman M, Ma M, Kandel A (1996) Numerical methods for calculating the fuzzy integral. Fuzzy Sets Syst 83:57–62. https://doi.org/10.1016/0165-0114(95)00307-X
Friedman M, Ma M, Kandel A (1999a) Solutions to fuzzy integral equations with arbitrary kernels. Int J Approx Reason 20:249–262
Friedman M, Ma M, Kandel A (1999b) Numerical solutions of fuzzy differential and integral equations. Fuzzy Sets Syst 106:35–48. https://doi.org/10.1016/S0165-0114(98)00355-8
Goetschel R, Voxman W (1986) Elementary fuzzy calculus. Fuzzy Sets Syst 18:31–43
Hüllermeier E (1997) An approach to modelling and simulation of uncertain dynamical systems. Int J Uncertain Fuzziness Knowl Based Syst 2:117–137
Hussain EA, Ali AW (2013) Homotopy analysis method for solving fuzzy integro-differential equations. Mod Appl Sci 7:15–25. https://doi.org/10.5539/mas.v7n3p15
Kaleva O (1987) Fuzzy differential equations. Fuzzy Sets Syst 24:301–317
Lotfi T, Mahdiani K (2011) Fuzzy Galerkin method for solving Fredholm. Int J Ind Math 3:237–249
Mordeson J, Newman W (1995) Fuzzy integral equations. Inf Sci 87:215–229
Nanda S (1989) On integration of fuzzy mappings. Fuzzy Sets Syst 32:95–101. https://doi.org/10.1016/0165-0114(89)90090-0
Park JY, Han HK (1999) Existence and uniqueness theorem for a solution of fuzzy Volterra integral equations. Fuzzy Sets Syst 105:481–488
Park JY, Jeong JU (1999) A note on fuzzy integral equations. Fuzzy Sets Syst 108:193–200
Park JY, Jeong JU (2000) On the existence and uniqueness of solutions of fuzzy Volterra-Fredholm integral equations. Fuzzy Sets Syst 115:425–431
Puri ML, Ralescu DA (1983) Differentials of fuzzy functions. J Math Anal Appl 91:552–558. https://doi.org/10.1016/0022-247X(83)90169-5
Rainville ED (1960) Special functions. The Macmillan Co, New York, p 1960
Ralescu D, Adams G (1980) The fuzzy integral. J Math Anal Appl 75:562–570
Sahu P, Saha Ray S (2017) A new Bernoulli wavelet method for accurate solutions of nonlinear fuzzy Hammerstein-Volterra delay integral equations. Fuzzy Sets Syst 309:131–144. https://doi.org/10.1016/j.fss.2016.04.004
Seikkala S (1987) On the fuzzy initial value problem. Fuzzy Sets Syst 24:319–330. https://doi.org/10.1016/0165-0114(87)90030-3
Wang Z (1984) The autocontinuity of set function and the fuzzy integral. J Math Anal Appl 99:195–218. https://doi.org/10.1016/0022-247X(84)90243-9
Ziari S (2018a) Towards the accuracy of iterative numerical methods for fuzzy Hammerstein-Fredholm integral equations. Fuzzy Sets Syst. https://doi.org/10.1016/j.fss.2018.09.006
Ziari S (2018b) Iterative method for solving two-dimensional nonlinear fuzzy integral equations using fuzzy bivariate block-pulse functions with error estimation. Iran J Fuzzy Syst 15:55–76
Ziari S, Bica AM (2016) New error estimate in the iterative numerical method for nonlinear fuzzy Hammerstein-Fredholm integral equations. Fuzzy Sets Syst 295:136–152. https://doi.org/10.1016/j.fss.2015.09.021
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The authors would like to extend their sincere thanks to the referees’ comments which lead to considerable improvements of this work.
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Seifi, A., Lotfi, T. & Allahviranloo, T. A new efficient method using Fibonacci polynomials for solving of first-order fuzzy Fredholm–Volterra integro-differential equations. Soft Comput 23, 9777–9791 (2019). https://doi.org/10.1007/s00500-019-04031-1
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DOI: https://doi.org/10.1007/s00500-019-04031-1