The aim of this paper is to present a new and efficient numerical method to approximate the solutions of two‐dimensional nonlinear fractional Fredholm and Volterra integral equations. For this aim, the two‐variable shifted fractional‐order Jacobi polynomials are introduced and their operational matrices of fractional integration and product are derived. These operational matrices and shifted fractional‐order Jacobi collocation method are utilized to reduce the understudy equations to systems of nonlinear algebraic equations. Then, the arising systems can be solved by the Newton method. Discussion on the convergence analysis and error bound of the proposed method is presented. The efficiency, accuracy, and validity of the presented method are demonstrated by its application to three test examples and by comparing our results with the results obtained by existing numerical methods in the literature recently.

中文翻译：

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基于位移分数阶雅可比运算矩阵的新型高效数值方法，用于求解某些类别的二维非线性分数积分方程

本文的目的是提出一种新的有效的数值方法，以近似求解二维非线性分数阶Fredholm和Volterra积分方程的解。为此，引入了二变量平移分数阶Jacobi多项式，并推导了它们的分数积分和乘积的运算矩阵。这些运算矩阵和移位的分数阶Jacobi配点方法用于将学习不足的方程式简化为非线性代数方程组。然后，可以通过牛顿法求解产生的系统。讨论了该方法的收敛性分析和误差界。效率，准确性，