In this paper, we consider a bivariate nonlinear Fredholm integral equation of the second kind. Then an approximate solution for some of these problems is investigated. To determine the aimed solution, the hybrid of 2D block-pulse functions with Chebyshev polynomials basis with the operational matrices is applied. In this work, we generalize the operational matrices stated by Behbahani (J Basic Appl 4:131–141, 2015), from one-dimensional to 2D space. Comparing this technique with other works, we can distinguish the advantage of developing the solution with fewer runtime and computations with more satisfactory approximations. In this procedure, using operational matrices, the nonlinear Fredholm integral equations are reduced to a system of nonlinear algebraic equations. Furthermore, the convergence analysis for this numerical method is investigated. Numerical examples are presented to describe the performance and rectitude of this method.

在本文中，我们考虑第二类二元非线性Fredholm积分方程。然后研究其中一些问题的近似解决方案。为了确定目标解决方案，将二维块脉冲函数与切比雪夫多项式与运算矩阵的混合应用。在这项工作中，我们概括了Behbahani（J Basic Appl 4：131–141，2015）所述的从一维到二维空间的运算矩阵。通过将该技术与其他作品进行比较，我们可以区分以更少的运行时间和更令人满意的近似值进行计算开发解决方案的优势。在此过程中，使用运算矩阵，将非线性Fredholm积分方程简化为一个非线性代数方程组。此外，研究了该数值方法的收敛性分析。