Solving the First Kind Fuzzy Integral Equations Using a Hybrid Regularization Method and Bernstein Polynomials

A K Moloodpour,A Jafarian

doi:10.5539/mas.v10n9p22

Abstract

A hybrid of regularization methodand Bernstein polynomials is used to solve the first kind fuzzy integral equations. In this paper, first the regularization method applied to convert the first kind fuzzy integral equation into the second kind fuzzy integral equation. Then by approximating Bernstein polynomials, the obtained second kind fuzzy integral equation is solved. In this method, one parameter was created in the second kind equation. When this parameter tends to zero, the solutions of integral equation of the second kind tend to solutions of the integral equation of the first kind. The obtained solutions are comparable to the solutions of the other similar methods. Performance of the mentioned method is illustrated by considering some example.

Highlights

Integral equations are employed in numerical Computations and in modeling of engineering science and other sciences (Jerri, 1999; Tikhonov, 1963; Wazwaz, 2011) as well
We present some examples of non-fuzzy and fuzzy integral equations so that we can study the efficiency of the hybrid method
In this paper the fuzzy integral equations of the first kind was solved by a hybrid method including regularization method and Bernstein polynomials approximation

Summary

Introduction

Integral equations are employed in numerical Computations and in modeling of engineering science and other sciences (Jerri, 1999; Tikhonov, 1963; Wazwaz, 2011) as well. Adomian decomposition method is employed to solve the Fredholm fuzzy integral equations of the second by Babolian, et al (Babolian, Sadeghi & Goghary, 2005). Another approach to solve the integral equations of the first kind is to transform them to the integral equations of second kind by regularization method. In (Wazwaz, 2011) the first kind integral equations have been solved by hybrid regularization and Homotopy method Another type of the first kind equations is the Abel integral equation which is presented in Fuzzy form.

The Regularization Method

Abel Fuzzy Integral Equations

Findings

Conclusions and Recommendations