A fast difference scheme for the nonlinear differential equation with two Caputo fractional derivatives

In this paper, we study a nonlinear differential equation with two Caputo fractional derivatives. The existence and uniqueness of solutions to the above equation are proved. In addition, based on the nonsmooth regularity assumption, a finite difference scheme on the graded mesh is constructed by applying the sum-of-exponentials (SOE) technique to approximate the kernel function, which significantly improves computational efficiency. Meanwhile, the stability and convergence of the proposed scheme are proved. Finally, some numerical examples demonstrate the effectiveness of numerical schemes and confirm the theoretical analysis.