Towards identification of explicit solutions to overdetermined systems of differential equations

TL;DR

This paper introduces an algorithm for explicitly solving overdetermined systems of PDEs by transforming them into algebraic polynomial equations, enabling solutions for complex systems like hydrodynamics.

Contribution

The paper presents a novel algorithm that finds explicit solutions to overdetermined PDE systems using algebraic polynomial methods, addressing computational challenges.

Findings

Explicit solutions for some overdetermined PDE systems are obtainable.

The algorithm can handle large systems with millions of polynomial equations.

Potential to derive general solutions for complex physical models like hydrodynamics.

Abstract

The authors proposed a general way to find particular solutions for overdetermined systems of PDEs previously, where the number of equations is greater than the number of unknown functions. In this paper, we propose an algorithm for finding solutions for overdetermined PDE systems, where we use a method for finding an explicit solution for overdetermined algebraic (polynomial) equations. Using this algorithm, the solution of some overdetermined PDE systems can be obtained in explicit form. The main difficulty of this algorithm is the huge number of polynomial equations that arise, which need to be investigated and solved numerically or explicitly. For example, the overdetermined hydrodynamic equations obtained earlier by the authors give a minimum of 10 million such equations. However, if they are solved explicitly, then we can write out the solution of the hydrodynamic equations in a general form, which is of great scientific interest.