A new treatment of boundary conditions in PDE solution with Galerkin methods via Partial Integral Equation framework

We present a new mathematical framework for solution of Partial Differential Equations (PDEs), which is based on an exact transformation of the underlying PDE that removes the boundary constraints from the solution state and moves them into the dynamics of the equivalent transformed equation. The framework is based on a Partial Integral Equation (PIE) representation of a PDE or a system of PDEs, where Partial Integral Equation does not require boundary conditions on its solution state. The PDE-PIE framework allows for a development of a generalized and consistent treatment of boundary conditions in constructing spectrally convergent solution approximations to a broad class of linear PDEs with non-constant coefficients governed by non-periodic boundary conditions, including, e.g., Dirichlet, Neumann and Robin boundaries, among others. The significance of this result is that a solution to almost any linear PDE in a form of a function series approximation can now be systematically constructed, irrespective of the boundary conditions. Furthermore, in many cases, the resulting ODE system for the expansion coefficients can now be integrated analytically in time, which allows us to obtain solution approximations to a broad class of unsteady PDEs with unprecedented accuracy. 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. We also present comparison of the PIE methods with some classical direct PDE solution methods, further demonstrating advantages and potential limitations of the PIE approach. The developed framework can be naturally extended to multiple spatial dimensions and, potentially, to nonlinear problems.