A New Treatment of Boundary Conditions in PDE Solution with Galerkin Methods via Partial Integral Equation Framework

TL;DR

This paper introduces a novel PDE solution framework using Partial Integral Equations that transforms boundary conditions into the dynamics, enabling analytical and numerical solutions for a broad class of linear PDEs with various boundary conditions.

Contribution

The paper develops a PDE-PIE framework that eliminates boundary conditions in PDE solutions, allowing for generalized Galerkin approximations applicable to diverse linear PDEs.

Findings

Successfully applied to several 1D PDE examples

Allows analytical series solutions regardless of boundary conditions

Framework extendable to multiple dimensions and nonlinear problems

Abstract

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.