A Higher-Order Numerical Scheme for Two-Dimensional Nonlinear Fractional Volterra Integral Equations with Uniform Accuracy

In this paper, based on the modified block-by-block method, we propose a higher-order numerical scheme for two-dimensional nonlinear fractional Volterra integral equations with uniform accuracy. This approach involves discretizing the domain into a large number of subdomains and using biquadratic Lagrangian interpolation on each subdomain. The convergence of the high-order numerical scheme is rigorously established. We prove that the numerical solution converges to the exact solution with the optimal convergence order O(h(x)(4-alpha) + h(y)(4-beta)) for 0 < alpha,beta < 1. Finally, experiments with four numerical examples are shown, to support the theoretical findings and to illustrate the efficiency of our proposed method.

Automated summary

This paper proposes a higher-order numerical scheme for solving two-dimensional nonlinear fractional Volterra integral equations with uniform accuracy. The scheme is based on the modified block-by-block method and involves discretizing the domain into subdomains and using biquadratic Lagrangian interpolation. The convergence of the scheme is rigorously established, and it is proven that the numerical solution converges to the exact solution with the optimal convergence order of O(h(x)(4-alpha) + h(y)(4-beta)) for 0 < alpha,beta < 1. Experimental results with four numerical examples are presented to support the theory and illustrate the efficiency of the proposed method.