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 $n^{\rm th}$-order semi-discrete approximation of the PIDE. The $n^{\rm th}$-order semi-discrete approximation of the PIDE is an $n^{\rm th}$-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 $n\to\infty$. We illustrate our convergence results using numerical examples. The results in this work are useful for establishing the null controllability of the PIDE considered.