Symbolic Solutions of Simultaneous First-order PDEs in One Unknown

TL;DR

This paper introduces a novel algorithm for solving overdetermined first-order PDE systems in one unknown, using the Bour-Mayer method to determine compatibility and recursively solving the system more efficiently than traditional differential Gr"obner bases.

Contribution

The paper presents a new recursive algorithm for solving compatible PDE systems that relies on Bour-Mayer brackets, offering a more straightforward alternative to differential Gr"obner bases.

Findings

Algorithm successfully solves compatible PDE systems

More efficient than differential Gr"obner bases

Implementation demonstrates practical viability

Abstract

We propose and implement an algorithm for solving an overdetermined system of partial differential equations in one unknown. Our approach relies on Bour-Mayer method to determine compatibility conditions via Jacobi-Mayer brackets. We solve compatible systems recursively by imitating what one would do with pen and paper: Solve one equation, substitute the solution into the remaining equations and iterate the process until the equations of the system are exhausted. The method we employ for assessing the consistency of the underlying system differs from the traditional use of differential Gr\"obner bases yet seems more efficient and straightforward to implement. We are not aware of a computer algebra system that adopts the procedure we advocate in this work.