Bivariate modified Bernstein-Kantorovich operators for the numerical solution of two-dimensional fractional Volterra integral equations

A class of two-dimensional fractional Volterra integral equations (2D-FVIEs) of the second kind is considered. The solution may have unbounded derivatives near the integral domain boundary. Therefore, smoothing transformations are employed to change the original 2D-FVIEs into new transformed 2D-FVIEs with better regularity. The novelty in this research concerns both the theoretical investigation of the bivariate modified Bernstein-Kantorovich (B-MBK) operators and the numerical application of these operators for approximating the unknown solution of 2D-FVIEs. In this regard, an algorithm is given utilizing the B-MBK operators and discretization that approximates the solution of the transformed discretized equation. Further, an inverse transformation is applied to obtain the solution of the original equation. Additionally, we illustrate the applicability of the proposed method on examples from the literature.