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.
翻译:在本文中,我们用一个空间层面的线性部分分布式(PEE)表示线性部分分布式(PDE),PDE在动态和边界条件中都有空间整体术语。PIE的表示式是通过改变变量获得的,每个PDE州都用其最高和定义明确的衍生物替换,使用Calculus基本理论来获得一个新的方程(a PIE)。我们表明,从PDE代表式转换为PIE代表式可以写成从PDE参数到PIE参数的清晰地图。最后,我们提出数字例子,通过配置优化方法对PDEs进行稳定性分析,以证明PIE代表式的应用。