Robust numerical scheme for 2D fractional integro-differential equations of Volterra type

This article provides a numerical study of two-dimensional Volterra integro-differential equations involving fractional derivatives in the Caputo sense of order \( \alpha ,\gamma \) \( (0< \alpha ,\gamma <1). \) First, we establish a sufficient condition for the existence and uniqueness of the solution using the Banach fixed point theorem. Due to the limitation of finding the exact analytical solution, we derive and analyze an efficient numerical scheme to approximate the solution. The proposed scheme uses the L1 technique to discretize the differential components, whereas a composite trapezoidal rule is used to approximate the double integral. The convergence analysis and error estimation are carried out. It is shown that the proposed scheme converges with an optimal convergence rate of \( \min \{2-\alpha ,2-\gamma \} \) for sufficiently smooth initial data. In addition, we apply the proposed difference scheme to solve the semilinear problem. The well-known Newton’s linearization technique is used to deal with semilinearity. Finally, a couple of numerical experiments are conducted to support our theoretical findings and validate the proposed scheme.