An efficient semi-implicit finite difference scheme for generalized nonlinear integro-differential equations with the Abel kernel

Abstract

In this work, we present a semi-implicit finite difference scheme for solving generalized nonlinear integro-differential equations involving the Abel kernel. In our approach, the temporal derivative is discretized using the backward Euler method, while the Abel–Liouville fractional integral term is approximated via a first-order convolution quadrature rule, resulting in a semi-discrete scheme in time. For completeness, this time-discrete scheme is coupled with central difference formulas for spatial discretization, thereby constructing a fully discrete numerical scheme. Additionally, the generalized nonlinear convection term is treated using a semi-implicit method to reduce computational costs. We establish the boundedness and convergence of the scheme in the L ² norm through an energy argument. Numerical experiments are conducted to validate the theoretical analysis.