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

Abstract

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(hx4−α+hy4−β) for 0<α,β<1. Finally, experiments with four numerical examples are shown, to support the theoretical findings and to illustrate the efficiency of our proposed method.

A High-Order Approximate Solution for the Nonlinear 3D Volterra Integral Equations with Uniform Accuracy

“…In the future, we hope to construct a high-order approximate solution for a peridynamics plate model based on the idea of [29]. In addition, we intend to construct an efficient high-order numerical solution with uniform accuracy for 3D-VIEs with a generally weak nonlinear singular kernel function based on the ideas of [30,31]. Finally, we will apply a fast algorithm to implement the high-order numerical scheme for large-scale practical engineering problems based on the idea of [32].…”

Hybridization of Block-Pulse and Taylor Polynomials for Approximating 2D Fractional Volterra Integral Equations

“…Asgari et al [31] adopted the Bernstein polynomials to approximate the 2DFVIEs. Abdollahi et al [32] presented an operational matrix scheme based 2D Haar wavelets, whereas Wang et al [33] used 2D Euler polynomials combined with Gauss-Jacobi quadrature technique to simulate the 2DFVIEs. Liu et al [34] employed the Bivariate barycentric rational interpolation for the 2DFVIEs.…”

“…Khan et al [35] implemented 2D Bernstein's approximation to approximate the 2DFVIEs. Wang et al [33] applied the modified block-by-block technique, while Mohammad et al [36] proposed an efficient approach based on Framelets for solving the 2DFVIEs. Ahsan et al [37] used optimal Homotopy asymptotic scheme and Fazeli et al [38] considered the Chebyshev polynomials for approximating the 2DFVIEs.…”