Efficient Numerical Solution to a Bivariate Nonlinear Fuzzy Fredholm Integral Equation

Abstract

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.