Euler–Poisson–Darboux equation explained

In mathematics, the Euler–Poisson–Darboux^{[1] [2]} equation is the partial differential equation

u _{x,y} + (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

u _{rr} + (m)/(r) u _{r-u} _{tt} =0 ,

by

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

Notes and References

1. Book: Zwillinger, D. . 1997 . Handbook of Differential Equations 3rd edition . Academic Press, Boston, MA .
2. Book: Copson, E. T.. Partial differential equations. 1975. Cambridge University Press. 978-0521098939. Cambridge. 1499723.
3. Copson. E. T.. 1956-06-12. On a regular Cauchy problem for the Euler—Poisson—Darboux equation. Proc. R. Soc. Lond. A. en. 235. 1203. 560–572. 10.1098/rspa.1956.0106. 0080-4630. 1956RSPSA.235..560C. 2027/mdp.39015095254382. 122720337. free.
4. Shishkina. Elina L.. Sitnik. Sergei M.. 2017-07-15. The general form of the Euler--Poisson--Darboux equation and application of transmutation method. 1707.04733. math.CA.
5. Miles. E.P. Young. E.C. On a Cauchy problem for a generalized Euler-Poisson-Darboux equation with polyharmonic data. Journal of Differential Equations. en. 2. 4. 482–487. 10.1016/0022-0396(66)90056-8. 1966. 1966JDE.....2..482M. free.
6. Fusaro. B. A.. 1966. A Solution of a Singular, Mixed Problem for the Equation of Euler-Poisson- Darboux (EPD). The American Mathematical Monthly. 73. 6. 610–613. 10.2307/2314793. 2314793.