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:我们提出了一个新的分析和数字框架,用于解决部分差别(PDE),这一框架的基础是精确转换,将边界限制转化为相应的治理方程的动态;这个框架基于PDE的局部整体方程(PIE)代表PDE,其中PDE的方程转换成等效的PIE配方,不要求其解决方案状态的边界条件。PDE-PIE框架允许为具有非定期边界条件(例如Dirichlet、Neumann和Robin边界)的非一致系数的广类线性PDE制定通用PIE-G-Galerkin近似方法。这一结果的重要性是,现在几乎任何线性PDE的解决方案都可以以分析性近似形式形成,其依据是一系列适当的基础功能,例如,无论边界条件如何,先行的Chebyshev 聚氨基(Chebyshev),许多类型的线性PIE-G。在涉及单一或简单时间独立的边界输入的许多情况下,在时间上进行分析的整合,在时间上进行的分析整合,在不具有一定的空间上,我们目前采用多种空间方法,可能采用多种方法。