Efficient Numerical Solution to a Bivariate Nonlinear Fuzzy Fredholm Integral Equation

In this article, a new iterative numerical method for solving bivariate nonlinear fuzzy Fredholm integral equations is proposed. The method combines two well-proven approaches-successive approximations and mixed trapezoidal and midpoint rules for the numerical integration. Both approaches are elaborated for fuzzy-valued functions. The main advantage of the proposed approach is that, by targeting the particular equation, another subordinate problem was solved under the common constraints. By this, we mean the development of a numerical method for fuzzy integrals. As a result, the proposed method is more accurate in comparison with any other mechanical combination of two separate and independent methods. We give conditions for the existence and uniqueness of a solution and estimate the error of the obtained approximation. We prove the stability and the method is performed on test problems to verify our theoretical results; numerical results are compared with those from existing methods in the literature to confirm the accuracy and efficiency of the proposed method.

I. Introduction

There are many distinguished equations in applied mathematics that characterize various physical processes in the real world. In most cases, they are expressed in the language of (partial) differential or integral equations. Particularly, fuzzy integral equations play an important role, as they are applied in fuzzy economic and fuzzy control systems [1], [2]. In the field of fuzzy mathematics, fuzzy integral equations were first considered in the works of Kaleva [3] and Seikkala [4], where they were linked with the initial value problem for first-order fuzzy differential equations.