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.

中文翻译：

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二元非线性模糊Fredholm积分方程的高效数值解


本文提出了一种新的求解二元非线性模糊Fredholm积分方程的迭代数值方法。该方法结合了两种经过充分验证的方法——逐次逼近以及用于数值积分的混合梯形和中点规则。这两种方法都是针对模糊值函数而详细阐述的。该方法的主要优点是，通过针对特定方程，在共同约束下解决了另一个从属问题。我们指的是模糊积分数值方法的发展。因此，与两种单独且独立的方法的任何其他机械组合相比，所提出的方法更加准确。我们给出解的存在性和唯一性的条件，并估计所获得的近似值的误差。我们证明了该方法的稳定性，并对测试问题进行了验证，以验证我们的理论结果；将数值结果与文献中现有方法的结果进行比较，以证实所提出方法的准确性和效率。