Euler–Poisson–Darboux equation

In mathematics, the Euler–Poisson–Darboux(EPD)^[1][2] equation is the partial differential equation

${\displaystyle u_{x,y}+{\frac {N(u_{x}+u_{y})}{x+y}}=0.}$

This equation is named for Siméon Poisson, Leonhard Euler, and Gaston Darboux. It plays an important role in solving the classical wave equation.

This equation is related to

${\displaystyle u_{rr}+{\frac {m}{r}}u_{r}-u_{tt}=0,}$

by ${\displaystyle x=r+t}$, ${\displaystyle y=r-t}$, where ${\displaystyle N={\frac {m}{2}}}$ ^[2] and some sources quote this equation when referring to the Euler–Poisson–Darboux equation.^[3][4][5][6]

The EPD equation equation is the simplest linear hyperbolic equation in two independent variables whose coefficients exhibit singularities, therefore it has an interest as a paradigm to relativity theory.^[7]

Compact support self-similar solution of the EPD equation for thermal conduction was derived starting from the modified Fourier-Cattaneo law.^[8]

It is also possible to solve the non-linear EPD equations with the method of generalized separation of variables.^[9]