This manuscript displays the following coupled differential-integral equations, including the Caputo fractional operator of variable orders. A numerical method based on the shifted Jacobi–Gauss collocation scheme is used to solve these equations numerically. While using this numerical method, a system of algebraic equations is constructed. This system can be solved with a recursive method in the nonlinear case and with algebraic formulas in the linear case. Finally, in order to show the high performance of the suggested method, three Examples are illustrated.

Keywords: Coupled differential-integral equation; Caputo fractional operator; Shifted fractional Jacobi collocation method; Variable-order.