ABSTRACT
The study of fuzzy integral equations and fuzzy differential equations is an emerging area of research for many authors. Originally, the concept of fuzzy sets was first introduced by Zadeh [225, 50]. The development of fuzzy integral equations was first invented by Kaleva [90] and Seikkala [191]. In recent years, many researchers have focused their interest on this field and published many articles which are available in the literature. Many analytical methods like the Adomian decomposition method [23], homotopy analysis method [144], and homotopy perturbation method [10] have been used to solve fuzzy integral equations. There are available many numerical techniques to solve fuzzy integral equations. The method of successive approximations
interpolation [30], Legendre wavelet method [185], Sinc function [97], the residual minimization method [87], the fuzzy transforms method [60], and the Galerkin method [116] have been applied to solve fuzzy integral equations numerically. Recently, Sadatrasoul et al. [170] have solved nonlinear fuzzy integral equations by applying the iterative method. Many theories related to fuzzy fractional functional integral and differential equations have been included in [149] and convergence in measure theorem for nonlinear integral functionals has been provided in [94]. The existence of solutions to fuzzy differential equations with generalized Hukuhara derivative via contractive-like mapping principles has been presented by Villamizar-Roa et al. [202]. A classical solution of the fuzzy boundary value problem has been given in [65]. The Cauchy problem for complex fuzzy differential equations has been solved in [93]. Hybrid block-pulse functions and the Taylor series method [25] have been applied to solve nonlinear fuzzy Fredholm integral equations of the second kind and also linear Fredholm fuzzy integral equations of the second kind have been solved by artificial neural networks [62]. Also, there are available many works related to fuzzy integrodifferential equations [105, 106, 8, 2, 145] in the literature. The learned researcher Abbasbandy et al. have solved fuzzy integro-differential equations by the homotopy analysis method [2]. Fuzzy Fredholm integro-differential equations have been solved by the Newton-cotes method [145]. The existence and uniqueness of the solutions of fuzzy integro-differential equations have been presented in [105, 8, 154]. In this chapter, we have solved Hammerstein fuzzy integral equations, fuzzy Hammerstein Volterra delay integral equations and fuzzy integro-differential equations. In Section 9.2, we discuss the preliminaries of fuzzy calculus. In Section 9.3, a nonlinear fuzzy Hammerstein integral equation has been solved by Bernstein polynomials and Legendre wavelets, and then compared with the homotopy analysis method. In Section 9.4, we have solved nonlinear fuzzy Hammerstein Volterra integral equations with constant delay by the Bernoulli wavelet method and then compared with the B-spline wavelet method. In Section 9.5, the fuzzy integro-differential equation has been solved by Legendre wavelet method and compared with the homotopy analysis method, and Section 9.6 describes the concluding remarks.
