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

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.

本报告展示了 Maple 软件包 TDDS（Thomas 分解微分系统）。对于包含方程以及不等式的多项式非线性微分系统，本包能将其分解为有限个微分三角且代数上简单的子系统，其方程子集具有可逆性。通常，分解后的系统在解析和数值上均易于研究及求解。Thomas 分解的独特属性在于输出子系统的解集相互独立。因此，对于一个良定初值问题，其解属于且仅属于一个输出子系统。Thomas 分解完全算法化。它允许执行对输入微分系统进行代数分析的重要元素，例如：验证一致性，即解的存在性；检测一般解析解中的任意性；给定一个额外的方程，检查该方程是否满足输入系统的所有公共解；在可能的情况下，从系统中消除部分依赖变量；揭示对依赖变量的隐藏约束等。本包的使用示例亦被提供。