The MAPLE package TDDS for computing Thomas decompositions of systems of nonlinear PDEs

TL;DR

The paper introduces the Maple package TDDS, which computes Thomas decompositions of nonlinear PDE systems, simplifying analysis and solution by partitioning the system into disjoint, involutive subsystems.

Contribution

It presents a fully algorithmic Maple package for Thomas decomposition of differential systems, enabling easier analysis and solution of nonlinear PDEs.

Findings

Decomposes complex PDE systems into simpler, involutive subsystems.

Facilitates verification of solution existence and consistency.

Enables variable elimination and detection of hidden constraints.

Abstract

We present the Maple package TDDS (Thomas Decomposition of Differential Systems). Given a polynomially nonlinear differential system, which in addition to equations may contain inequations, this package computes a decomposition of it into a finite set of differentially triangular and algebraically simple subsystems whose subsets of equations are involutive. Usually the decomposed system is substantially easier to investigate and solve both analytically and numerically. The distinctive property of a Thomas decomposition is disjointness of the solution sets of the output subsystems. Thereby, a solution of a well-posed initial problem belongs to one and only one output subsystem. The Thomas decomposition is fully algorithmic. It allows to perform important elements of algebraic analysis of an input differential system such as: verifying consistency, i.e., the existence of solutions; detecting the arbitrariness in the general analytic solution; given an additional equation, checking whether this equation is satisfied by all common solutions of the input system; eliminating a part of dependent variables from the system if such elimination is possible; revealing hidden constraints on dependent variables, etc. Examples illustrating the use of the package are given.