Numerical analysis of finite difference schemes arising from time-memory partial integro-differential equations - Fakharany, M.; El-Borai, Mahmoud M.; Abu Ibrahim, M. A. (Frontiers in Applied Mathematics and Statistics, 2022)
This paper investigates the partial integro-differential equation of memory type numerically. The differential operator is discretized based on θ-finite difference schemes, while the integral operator is approximated using Simpson's rule. The mesh points of the integral part are adapted to coincide with the nodes of the computational domain using the Heaviside function. The stability of the proposed numerical methods is established based on Gerschgoren's theorems. Also, its consistency is investigated to guarantee the numerical solutions' convergence. Several examples are provided to discuss the efficiency of the used finite difference schemes and compare them with previous works.