Maurice Janet's algorithms on systems of linear partial differential equations
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by
Kenji Iohara, Philippe Malbos
2020
Abstract
This article presents 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, M. 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 unicity 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. M. 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 20th century in various algebraic contexts.
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