Three finite difference schemes for generalized nonlinear integro-differential equations with tempered singular kernel

Abstract

This paper presents and investigates three different finite difference schemes for solving generalized nonlinear integro-differential equations with tempered singular kernel. For the temporal derivative, the backward Euler (BE), Crank–Nicolson (CN), and second-order backward differentiation formula (BDF2) schemes are employed. The corresponding convolution quadrature rules are utilized for the integral term involving tempered fractional kernel. In order to ensure second-order accuracy in the spatial direction, the standard central difference formula is applied, leading to fully discrete difference schemes. The convergence and stability of these three schemes are proved, by using energy method and cut-off method. Furthermore, we apply a fixed point iterative algorithm to calculate the proposed schemes, and the numerical results are consistent with the theoretical analysis.