The Euler-Poisson Darboux equations in the development of the theory of partial differential equations

The Euler-Poisson-Darboux (EPD) one-parameter family of equations was, for over a century, a major motivation and testbed for the theory of partial differential equations (PDEs). EPD equations came to be viewed as singular or degenerate when the Cauchy and Dirichlet problems became canonical, but they were studied for their own sake in the 1970s, leading to a theory of Fuchsian PDEs. This made it possible to show, in the early 1990s, that EPD-like equations are "universal" equations to which very general problems may be reduced ("Fuchsian reduction"). The main new points of the talk are the following:
(1) The first partial differential equation to appear in print is equivalent to an EPD equation (d'Alembert, 1743), before Euler's (1766) work, continued by Poisson and Darboux, on a two-parameter family of equations.
(2) The EPD equations waned in importance because the Cauchy and Dirichlet problems came to be viewed as prototypes of well-posed problems in the twentieth century.
(3) The theory of linear and nonlinear Fuchsian PDE, that contains the theory of the Cauchy and Dirichlet problems as a very special case, was developed by viewing the EPD equations as the PDE counterparts of ordinary differential equations of Fuchsian type.
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