On the convergence of a numerical scheme for a boundary controlled 1D linear parabolic PIDE

We consider an 1D partial integro-differential equation (PIDE) comprising of an 1D parabolic partial differential equation (PDE) and a nonlocal integral term. The control input is applied on one of the boundaries of the PIDE. Partitioning the spatial interval into n+1 subintervals and approximating the spatial derivatives and the integral term with their finite-difference approximations and Riemann sum, respectively, we derive an nth-order semi-discrete approximation of the PIDE. The nth-order semi-discrete approximation of the PIDE is an nth-order ordinary differential equation (ODE) in time. We establish some of its salient properties and using them prove that the solution of the semi-discrete approximation converges to the solution of the PIDE as n → ∞. We illustrate our convergence results using numerical examples. The results in this work are useful for establishing the null controllability of the PIDE considered.