This article describes the emergence of formal methods in theory of partial differential equations (PDE) in the French school of mathematics through Janet’s work in the period 1913–1930. In his thesis and in a series of articles published during this period, Janet introduced an original formal approach to deal with the solvability of the problem of initial conditions for finite linear PDE systems. His constructions implicitly used an interpretation of a monomial PDE system as a generating family of a multiplicative set of monomials. He introduced an algorithmic method on multiplicative sets to compute compatibility conditions, and to study the problem of the existence and the uniqueness of a solution to a linear PDE system with given initial conditions. The compatibility conditions are formulated using a refinement of the division operation on monomials defined with respect to a partition of the set of variables into multiplicative and non-multiplicative variables. Janet was a pioneer in the development of these algorithmic methods, and the completion procedure that he introduced on polynomials was the first one in a long and rich series of works on completion methods which appeared independently throughout the twentieth-century in various algebraic contexts.

中文翻译：

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Maurice Janet 关于线性偏微分方程组的算法

本文通过珍妮特在 1913 年至 1930 年期间的工作，描述了法国数学学派偏微分方程 (PDE) 理论中形式化方法的出现。在他的论文和在此期间发表的一系列文章中，Janet 引入了一种原始的形式化方法来处理有限线性 PDE 系统的初始条件问题的可解性。他的构造隐含地将单项式偏微分方程系统解释为单项式乘法集的生成族。他引入了乘法集的算法方法来计算兼容性条件，并研究具有给定初始条件的线性 PDE 系统的解的存在性和唯一性问题。兼容性条件是使用对单项式的除法运算的细化来制定的，这些单项式是关于将变量集划分为乘法和非乘法变量定义的。Janet 是这些算法方法发展的先驱，他在多项式上引入的补全过程是关于补全方法的长而丰富的系列作品中的第一个，这些作品在整个 20 世纪独立出现在各种代数环境中。