  In this paper, we present the Partial Integral Equation (PIE) representation
of linear Partial Differential Equations (PDEs) in one spatial dimension, where
the PDE has spatial integral terms appearing in the dynamics and the boundary
conditions. The PIE representation is obtained by performing a change of
variable where every PDE state is replaced by its highest, well-defined
derivative using the Fundamental Theorem of Calculus to obtain a new equation
(a PIE). We show that this conversion from PDE representation to PIE
representation can be written in terms of explicit maps from the PDE parameters
to PIE parameters. Lastly, we present numerical examples to demonstrate the
application of the PIE representation by performing stability analysis of PDEs
via convex optimization methods.
