Numerical analysis of fractional integro-differential equations with Caputo derivatives of different orders

This study investigates a class of boundary value problems governed by fractional integro-differential equations involving fractional derivatives of different orders defined in the Caputo sense. A robust finite difference scheme is developed in which the Caputo fractional derivative of order 0<α<1 is approximated using the $L1$ discretization, while the fractional derivative of order 1<β<2 is approximated using an explicit finite difference approximation. The integral terms are approximated via the composite trapezoidal rule. The resulting numerical method is simple to implement. A rigorous error analysis is performed, establishing first-order convergence of the proposed scheme. To further enhance the accuracy, Richardson extrapolation is incorporated. The applicability of the method to semilinear fractional integro-differential equations is also demonstrated. Numerical experiments are presented to validate the theoretical results, and CPU times are reported to document the method's practical computational cost.