Algorithmic reduction of polynomially nonlinear PDE systems to parametric ODE systems

TL;DR

This paper introduces an algorithm that reduces polynomially nonlinear PDE systems to parametric ODE systems, facilitating the application of existing solution methods and improving the analysis of such systems.

Contribution

The paper presents a novel algorithm for converting polynomial nonlinear PDE systems into parametric ODE systems, enabling easier solution and analysis.

Findings

The reduction allows the use of ODE solvers on PDE systems.

It guarantees inclusion of missing integrability conditions.

The method improves the form of PDE systems for solution techniques.

Abstract

Differential-elimination algorithms apply a finite number of differentiations and eliminations to systems of partial differential equations. For systems that are polynomially nonlinear with rational number coefficients, they guarantee the inclusion of missing integrability conditions and the statement of of existence and uniqueness theorems for local analytic solutions of such systems. Further, they are useful in obtaining systems in a form more amenable to exact and approximate solution methods. Maple's \maple{dsolve} and \maple{pdsolve} algorithms for solving PDE and ODE often automatically call such routines during applications. Indeed, even casual users of Maple's dsolve and pdsolve commands have probably unknowingly used Maple's differential-elimination algorithms. Suppose that a system of PDE has been reduced by differential-elimination to a system whose automatic existence and uniqueness algorithm has been determined to be finite-dimensional. We present an algorithm for rewriting the output as a system of parameterized ODE. Exact methods and numerical methods for solving ODE and DAE can be applied to this form.