A higher-order numerical scheme for system of two-dimensional nonlinear fractional Volterra integral equations with uniform accuracy

We give a modified block-by-block method for the nonlinear fractional order Volterra integral equation system by using quadratic Lagrangian interpolation based on the classical block-by-block method. The core of the method is that we divide its domain into a series of subdomains, that is, block it, and use piecewise quadratic Lagrangian interpolation on each subdomain to approximate $\mathit{\kappa }(x,y,s,r,u(s,r))$. Our proposed method has uniform accuracy and its convergence order is $O(h_x^{4-\alpha }+h_y^{4-\beta })$. We give a strict proof for the error analysis of the method, and give several numerical examples to verify the correctness of the theoretical analysis.