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.

翻译：微分消元算法通过对偏微分方程系统施加有限次微分和消元操作。对于具有有理数系数的多项式非线性系统，这些算法能确保包含缺失的可积性条件，并陈述此类系统局部解析解的存在唯一性定理。此外，它们有助于将系统转化为更便于精确解法和近似解法处理的形式。Maple求解常微分方程和偏微分方程的\maple{dsolve}与\maple{pdsolve}算法在应用过程中常自动调用此类程序。事实上，即使是Maple中dsolve和pdsolve命令的普通用户，也可能在不知不觉中使用了Maple的微分消元算法。假设某偏微分方程系统经微分消元约简后，其自动存在唯一性算法被判定为有限维系统。本文提出一种算法，可将输出结果重写为参数化常微分方程系统。针对常微分方程和微分代数方程的精确解法与数值解法均可应用于该形式。