A Symbolic Approach to Boundary Problems for Linear Partial Differential Equations: Applications to the Completely Reducible Case of the Cauchy Problem with Constant Coefficients

TL;DR

This paper develops an algebraic symbolic framework for boundary problems in linear PDEs, focusing on the Cauchy problem with constant coefficients, and demonstrates its theoretical foundation and implementation.

Contribution

It introduces a novel algebraic approach to boundary problems for linear PDEs, specifically addressing the completely reducible case with constant coefficients.

Findings

Algebraic framework successfully models boundary problems in PDEs.

Implementation supports the theoretical methods developed.

Framework applicable to the Cauchy problem for reducible PDEs.

Abstract

We introduce a general algebraic setting for describing linear boundary problems in a symbolic computation context, with emphasis on the case of partial differential equations. The general setting is then applied to the Cauchy problem for completely reducible partial differential equations with constant coefficients. While we concentrate on the theoretical features in this paper, the underlying operator ring is implemented and provides a sufficient basis for all methods presented here.