Weakly singular Volterra integral equations of the second kind typically have nonsmooth solutions near the initial point of the interval of integration, which seriously affects the accuracy of spectral methods. We present Jacobi spectral-collocation method to solve two-dimensional weakly singular Volterra-Hammerstein integral equations based on smoothing transformation and implicitly linear method. The solution of the smoothed equation is much smoother than the original one after smoothing transformation and the spectral method can be used. For the nonlinear Hammerstein term, the implicitly linear method is applied to simplify the calculation and improve the accuracy. The weakly singular integral term is discretized by Jacobi Gauss quadrature formula which can absorb the weakly singular kernel function into the quadrature weight function and eliminate the influence of the weakly singular kernel on the method. Convergence analysis in the -norm is carried out and the exponential convergence rate is obtained. Finally, we demonstrate the efficiency of the proposed method by numerical examples.

中文翻译：

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二维弱奇异 Volterra-Hammerstein 积分方程的隐式线性 Jacobi 谱配置方法

第二类弱奇异Volterra积分方程在积分区间初始点附近通常存在非光滑解，严重影响谱方法的精度。提出基于平滑变换和隐式线性方法求解二维弱奇异Volterra-Hammerstein积分方程的雅可比谱搭配法。经过平滑变换后的平滑方程的解比原解平滑得多，可以使用谱法。对于非线性Hammerstein项，采用隐式线性方法来简化计算，提高精度。采用雅可比高斯求积公式对弱奇异积分项进行离散化，将弱奇异核函数吸收到求积权函数中，消除了弱奇异核对方法的影响。在-范数下进行收敛分析，得到指数收敛率。最后，我们通过数值例子证明了所提出方法的效率。