We present a new analytical and numerical framework for solution of Partial Differential Equations (PDEs) that is based on an exact transformation that moves the boundary constraints into the dynamics of the corresponding governing equation. The framework is based on a Partial Integral Equation (PIE) representation of PDEs, where a PDE equation is transformed into an equivalent PIE formulation that does not require boundary conditions on its solution state. The PDE-PIE framework allows for a development of a generalized PIE-Galerkin approximation methodology for a broad class of linear PDEs with non-constant coefficients governed by non-periodic boundary conditions, including, e.g., Dirichlet, Neumann and Robin boundaries. The significance of this result is that solution to almost any linear PDE can now be constructed in a form of an analytical approximation based on a series expansion using a suitable set of basis functions, such as, e.g., Chebyshev polynomials of the first kind, irrespective of the boundary conditions. In many cases involving homogeneous or simple time-dependent boundary inputs, an analytical integration in time is also possible. We present several PDE solution examples in one spatial variable implemented with the developed PIE-Galerkin methodology using both analytical and numerical integration in time. The developed framework can be naturally extended to multiple spatial dimensions and, potentially, to nonlinear problems.

翻译：本文提出一种新的解析与数值框架用于求解偏微分方程（PDE），该框架基于将边界约束转化为相应控制方程动力学过程的精确变换。该框架基于偏微分方程的偏积分方程（PIE）表示，将PDE方程转化为等价的PIE公式，其中解状态不再需要边界条件。PDE-PIE框架为宽泛的满足非周期边界条件（例如狄利克雷、诺伊曼和罗宾边界）的非定系数线性PDE类提供了通用PIE-伽辽金逼近方法的发展。这一结果的重要意义在于：现在几乎任意线性PDE的解都可以通过使用合适的基函数集（例如第一类切比雪夫多项式）进行级数展开，以解析逼近的形式构建，且与边界条件无关。在涉及齐次或简单时变边界输入的诸多情形中，时间上的解析积分同样可行。我们通过开发的PIE-伽辽金方法，采用时间解析积分和数值积分两种方式，展示了若干一维空间变量PDE求解实例。该框架可自然扩展到多维空间，并具有拓展至非线性问题的潜力。