AEs for Example 5.

A numerical approach based on shifted Jacobi-Gauss collocation method for solving mixed Volterra-Fredholm integral equations is introduced. The novel technique with shifted Jacobi-Gauss nodes is applied to reduce the mixed Volterra-Fredholm integral equations to a system of algebraic equations that has an easy solved. The present algorithm is extended to solve the one and two-dimensional mixed Volterra-Fredholm integral equations. Convergence analysis for the present method is discussed and confirmed the exponential convergence of the spectral algorithm. Various numerical examples are approached to demonstrate the powerful and accuracy of the technique.

本文提出一种基于移位雅可比-高斯配置法（shifted Jacobi-Gauss collocation method）的数值方法，用于求解混合沃尔泰拉-弗雷德霍姆积分方程（mixed Volterra-Fredholm integral equations）。该结合移位雅可比-高斯节点的新颖技术，可将混合沃尔泰拉-弗雷德霍姆积分方程约化为易于求解的代数方程组。本文所提算法可推广至一维与二维混合沃尔泰拉-弗雷德霍姆积分方程的求解。文中对所提方法开展收敛性分析，验证了该谱算法（spectral algorithm）具备指数收敛特性。通过多组数值算例验证了该技术的高效性与准确性。