This paper proposes a method based on two-dimensional Euler polynomials combined with Gauss-Jacobi quadrature formula. The method is used to solve two-dimensional Volterra integral equations with fractional order weakly singular kernels. Firstly, we prove the existence and uniqueness of the original equation by Gronwall inequality and mathematical induction method. Secondly, we use two-dimensional Euler polynomials to approximate the unknown function of the original equation, and the Gauss-Jacobi quadrature formula is used to approximate the integrals in the original equation. Thirdly, we prove the existence and uniqueness of the solution of approximate equation, and the error analysis of the proposed method is given. Finally, some numerical examples illustrate the efficiency of the method.

提出了一种基于二维欧拉多项式结合高斯-雅各比正交公式的方法。该方法用于求解具有分数阶弱奇异核的二维Volterra积分方程。首先，我们通过Gronwall不等式和数学归纳法证明了原始方程的存在性和唯一性。其次，我们使用二维Euler多项式逼近原始方程的未知函数，并使用Gauss-Jacobi正交公式逼近原始方程中的积分。第三，证明了近似方程解的存在性和唯一性，并给出了所提出方法的误差分析。最后，一些数值例子说明了该方法的有效性。