An improved high-order uniform accuracy scheme for nonlinear system of 2D general fractional Volterra integral equations

Abstract This paper is dedicated to investigating an improved high-order uniform accuracy numerical scheme to solve the system of 2D general fractional nonlinear Volterra integral equations. Using classical fixed point theory, we establish a sufficient condition for the existence and uniqueness of solutions to the system. Based on the idea of the modified block-by-block method, we establish an improved high-order uniform accuracy numerical scheme by using piecewise biquadratic interpolation to approximate the integrand function other than the general fractional kernel. Furthermore, the convergence order O h 1 3 + ϑ 1 + h 2 3 + ϑ 2 $O\left({h}_{1}^{3+{\vartheta }_{1}}+{h}_{2}^{3+{\vartheta }_{2}}\right)$ is proposed by using an appropriate dimensionality reduction technique and discrete Grönwall inequality, where 0 < ϑ 1 , ϑ 2 < 1 are order of the fractional kernel and h 1 , h 2 are the step sizes of 2D variables. Finally, the effectiveness and reliability of the proposed method are validated by means of several numerical experiments which include cases with smooth and non-smooth solutions.