The advantages of the barycentric rational interpolation (BRI) introduced by Floater and Hormann include the stability of interpolation, no poles, and high accuracy for any sufficiently smooth function. In this paper we design a transformed BRI scheme to solve two dimensional fractional Volterra integral equation (2D-FVIE), whose solution may be non-smooth since its derivatives may be unbounded near the integral domain boundary. The transformed BRI method is constructed based on bivariate BRI and some smoothing transformations, hence inherits the advantages of the BRI even for a singular function. First, the smoothing transformations are employed to change the original 2D-FVIE into a new form, so that the solution of the new transformed 2D-FVIE has better regularity. Then the transformed equation can be solved efficiently by using the bivariate BRI together with composite Gauss–Jacobi quadrature formula. Last, some inverse transformations are used to obtain the solution of the original equation. The whole algorithm is easy to be implemented and does not require any integral computation. Besides, we analyze the convergence behavior via the transformed equation. Several numerical experiments are provided to illustrate the features of the proposed method.

Floater和Hormann引入的重心有理插值（BRI）的优点包括插值的稳定性，无极点以及对任何足够平滑函数的高精度。在本文中，我们设计了一种变换的BRI方案，用于求解二维分数阶Volterra积分方程（2D-FVIE），该方程的解可能是非光滑的，因为其导数可能在积分域边界附近无界。变换的BRI方法是基于双变量BRI和一些平滑变换构造的，因此即使对于奇异函数也继承了BRI的优点。首先，采用平滑变换将原始的2D-FVIE更改为新形式，以使新的变换后的2D-FVIE的解具有更好的规则性。然后，通过使用二元BRI和复合高斯-雅各比正交公式，可以有效地求解变换后的方程。最后，使用一些逆变换来获得原始方程的解。整个算法易于实现，不需要任何积分计算。此外，我们通过变换方程分析了收敛行为。提供了几个数值实验，以说明该方法的特征。