Numerical solution of nonlinear fuzzy Fredholm integral equations of the second kind using hybrid of block-pulse functions and Taylor series

Mahdy Baghmisheh,Reza Ezzati

doi:10.1186/s13662-015-0389-7

Abstract

The main purpose of this paper is to introduce a method of successive approximations in terms of a hybrid of Taylor series and a block-pulse function, which is given for solving nonlinear fuzzy Fredholm integral equations of the second kind and the error estimate of the approximation solution. Finally, two numerical examples are presented to show the accuracy of the proposed method.

Highlights

The study of fuzzy integral equations (FIE) from both theoretical and numerical points of view has been developed in recent years after a distinct study of the existence of a unique solution for fuzzy Fredholm integral equations had been carried out in [ ]
The Adomian decomposition method is used in [ – ], as well as quadrature rules, fuzzy Bernstein polynomials, fuzzy wavelet, and Nystorm techniques were applied to fuzzy integral equations of the second kind in [ – ]
Ezzati and Ziari [ ] proved the convergence of the method of successive approximations for solving nonlinear fuzzy Fredholm integral equations of the second kind, and they proposed an iterative procedure based on the trapezoidal quadrature

Summary

Introduction

The study of fuzzy integral equations (FIE) from both theoretical and numerical points of view has been developed in recent years after a distinct study of the existence of a unique solution for fuzzy Fredholm integral equations had been carried out in [ ]. Ezzati and Ziari [ ] proved the convergence of the method of successive approximations for solving nonlinear fuzzy Fredholm integral equations of the second kind, and they proposed an iterative procedure based on the trapezoidal quadrature. [ ] A fuzzy number is a function u : R → [ , ] satisfying the following properties:. For < r ≤ , we denote [u]r = {x ∈ R : u(x) ≥ r} the r-level (or the r-cut) set of u which is a closed interval (see [ ]) and [u]r = [ur, ur], ∀r ∈ [ , ] This leads to the usual parametric representation of a fuzzy number. For any fuzzy-number-valued function f : I ⊂ R −→ RF we can define the functions f These functions are the left and right r-level functions of f.

RF if ur

Let f

We say that

Nj j!

The exact solution is

Conclusion