  The Partial Integral Equation (PIE) framework was developed to
computationally analyze linear Partial Differential Equations (PDEs) where the
PDE is first converted to a PIE and then the analysis problem is solved by
solving operator-valued optimization problems. Previous works on the PIE
framework focused on the analysis of PDEs with spatial derivatives up to
$2^{nd}$-order. In this paper, we extend the class of PDEs by including
integral terms and performing stability analysis using the PIE framework. More
specifically, we show that PDEs with the integral terms where the integration
is with respect to the spatial variable and the kernel of the integral operator
is matrix-valued polynomials can be converted to PIEs if the boundary
conditions satisfy certain criteria. The conversion is performed by using a
change of variable where every PDE state is substituted in terms of its highest
derivative and boundary values to obtain a new equation (a PIE) in a variable
that does not have any continuity requirements. Later, we show that this change
of variable can be represented using explicit maps from the parameters of the
PDE to the parameters of the PIE and the stability test can be posed as an
optimization problem involving these parameters. Lastly, we present numerical
examples to demonstrate the simplicity and application of this method.
