In this paper, we proposed an iterative procedure based on Landweber iterative methods to solve fuzzy Fredholm integral equations of the first kind. In addition, this research be based on a strictly convex fuzzy number space and the Riemann integral of fuzzy-number-valued function which is taken value in the space. The error estimation of the proposed method in terms of uniform and partial modulus of continuity was given. The generalized difference of fuzzy numbers was controlled and estimated reasonably. And it was avoided to compute the generalized difference of fuzzy numbers during the iterations. Finally, two illustrative examples are included in order to demonstrate the accuracy and the convergence of the proposed method.
AllahviranlooT. and BehzadiSh.S., The use of airfoil and Chebyshev polynomials methods for solving fuzzy Fredholm integro-differential equations with Cauchy kernel, Soft Computing18 (2014), 1885–1897.
3.
BehzadiSh.S., AllahviranlooT. and AbbasbandyS., Solving fuzzy second-order nonlinear VolterräCFredholm integro-differential equations by using Picard method, Neural Comput & Applic21(Suppl 1) (2012), S337–S346.
4.
BehzadiSh.S., AllahviranlooT. and AbbasbandyS., The use of fuzzy expansion method for solving fuzzy linear Volterra-Fredholm integral equations, Journal of Intelligent & Fuzzy Systems26 (2014), 1817–1822.
5.
BalachandranK. and KanagarajanK., Existence of solutions of general nonlinear fuzzy VolterräCFredholm integral equations, Journal of Applied Mathematics and Stochastic Analysis3 (2005), 333–343.
6.
BalachandranK. and PrakashP., Existence of solutions of nonlinear fuzzy VolterräCfredholm integral equations, Indian Journal of Pure and Applied Mathematics33 (2002), 329–343.
7.
BalachandranK. and PrakashP., Existence of solutions of nonlinear fuzzy integral equations in Banach spaces, Libertas Mathematica21 (2001), 91–97.
8.
BedeB. and GalS.G., Quadrature rules for integrals of fuzzy-number-valued functions, Fuzzy Sets and Systems145 (2004), 359–380.
9.
BedeB. and StefaninibL., Generalized differentiability of fuzzy-valued functions, Fuzzy Sets Syst230 (2013), 119–141.
10.
BicaA.M., Algebraic structures for fuzzy numbers from categorial point of view, Soft Computing11 (2007), 1099–1105.
11.
BicaA.M. and PopescuC., Numerical solutions of the nonlinear fuzzy Hammerstein-Volterra delay integral equations, Information Sciences223 (2013), 236–255.
12.
BicaA.M., One-sided fuzzy numbers and applications to integral equations from epidemiology, Fuzzy Sets and Systems219(16) (2013), 27–48.
13.
BaghmishehM. and EzzatiR., Error estimation and numerical solution of nonlinear fuzzy Fredholm integral equations of the second kind using triangular functions, Journal of Intelligent & Fuzzy Systems30 (2016), 639–649.
14.
BicaA.M. and PopescuC., Iterative numerical method for nonlinear fuzzy Volterra integral equations, Journal of Intelligent & Fuzzy Systems32 (2017), 1639–1648.
15.
HadamardJ., Lectures on Cauchy Problems in Linear PDE, New Haven: Yale University Press, 1923.
16.
ChenM.H., A new fuzzy analysis [in China], Science Press, Beijing. 2009.
17.
DiamondP. and KloedenP., Characterization of compact subsets of fuzzy sets, Fuzzy Sets and Systems29 (1989), 341–348.
18.
EzzatiR. and ZiariS., Numerical solution of nonlinear fuzzy Fredholm integral equations using iterative method, Applied Mathematics and Computation225 (2013), 33–42.
19.
FriedmanM., MaM. and KandelA., On fuzzy integral equations, Fundamenta Informaticae37 (1999), 89–99.
20.
FriedmanM., MaM. and KandelA., Solutions to fuzzy integral equations with arbitrary kernels, International Journal of Approximate Reasoning20 (1999), 249–262.
21.
GoetschelR. and VoxmanW., Elementary fuzzy calculus, Fuzzy Sets and Systems18 (1986), 31–43.
22.
GongZ. and WuC., Bounded variation absolute continuity and absolute integrability for fuzzynumber-valued functions, Fuzzy Sets and Systems129 (2002), 83–94.
23.
GholamA.M., EzzatiR. and AllahviranlooT., Numerical solution of two-dimensional nonlinear fuzzy Fredholm integral equations via quadrature iterative method, Journal of Intelligent & Fuzzy Systems36 (2019), 661–674.
24.
HamoudA., AzeezA. and GhadleK., A study of some iterative methods for solving fuzzy Volterra-Fredholm integral equations, Indonesian J Elec Eng & Comp Sci11(3) (2018), 1228–1235.
25.
HamoudA. and GhadleK., Modified Adomian decomposition method for solving fuzzy Volterra-Fredholm integral equations, J Indian Math Soc85(1-2) (2018), 52–69.
26.
KalevaO., Fuzzy differential equations, Fuzzy Sets and Systems24 (1987), 301–317.
27.
KhorramiN., ShamlooA.S. and MoghaddamB.P., Nystrom method for solution of Fredholm integral equations of the second kind under interval data, Journal of Intelligent & Fuzzy Systems36 (2019), 2807–2816.
28.
MoloodpourA.K. and JafarianA., Solving the First Kind Fuzzy Integral Equations Using a Hybrid Regularization Method and stein Polynomials, Modern Applied Science10(9) (2016), 22–35Bern.
29.
MahalehV.S.K. and EzzatiR., Numerical solution of linear fuzzy Fredholm integral equations of second kind using iterative method and midpoint quadrature formula, Journal of Intelligent & Fuzzy Systems33 (2017), 1293–1302.
30.
MaY., HuangJ., LiuH. and WangC., Numerical solution of two-dimensional nonlinear fuzzy Fredholm integral equations based on Gauss quadrature rule, Journal of Intelligent & Fuzzy Systems35 (2018), 2281–2291.
31.
ParkJ.Y., LeeS.Y. and JeongJ.U., The approximate solution of fuzzy functional integral equations, Fuzzy Sets and Systems110 (2000), 79–90.
32.
ParkJ.Y. and JeongJ.U., On the existence and uniqueness of solutions of fuzzy VoltteräCfredholm integral equations, Fuzzy Sets and Systems115 (2000), 425–431.
33.
ParkJ.Y. and JeongJ.U., A note on fuzzy integral equations, Fuzzy Sets and Systems108 (1999), 193–200.
34.
ParkJ.Y. and HanH.K., Existence and uniqueness theorem for a solution of fuzzy Volterra integral equations, Fuzzy Sets and Systems105 (1999), 481–488.
35.
SadatrasoulS.M. and EzzatiR., Iterative method for numerical solution of two-dimensional nonlinear fuzzy integral equations, Fuzzy Sets and Systems280 (2015), 91–106.
36.
SeikkalaS., On the fuzzy initial value problem, Fuzzy Sets Syst24 (1987), 319C330.
37.
CongxinW. and CongW., The supremum and infimum of the set of fuzzy numbers and its applications, J Math Anal Appl210 (1997), 499–511.
38.
WuC. and GongZ., On Henstock integral of fuzzy-number-valued functions (I), Fuzzy sets Syst120 (2001), 523–532.
39.
YangH. and GongZ., Ill-posedness for fuzzy Fredholm integral equations of the first kind and regularization methods, Fuzzy sets Syst358 (2019), 132–149.
