A unified approach to solving parabolic Volterra partial integro-differential equations for a broad category of kernels: Numerical analysis and computing

This work is concerned with solving parabolic Volterra partial integro-differential equations (PIDE) considering differentiable and singular kernels. The implicit finite difference scheme is implemented to approximate the differential operator, and the nonlocal term is discretized based on an open-type formula with two distinct time step sizes related to the nature of the time level to guarantee to avoid the singular terms at the endpoints and denominators. The properties of the plied scheme are investigated, more precisely, its stability and consistency. Four detailed examples are implemented to demonstrate the efficiency and reliability of the applied finite difference scheme.