Euler–Poisson–Darboux equation

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

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

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

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

References

^ Zwillinger, D. (1997). Handbook of Differential Equations 3rd edition. Academic Press, Boston, MA.
^ ^a ^b Copson, E. T. (1975). Partial differential equations. Cambridge: Cambridge University Press. ISBN 978-0521098939. OCLC 1499723.
^ Copson, E. T. (1956-06-12). "On a regular Cauchy problem for the Euler—Poisson—Darboux equation". Proc. R. Soc. Lond. A. 235 (1203): 560–572. Bibcode:1956RSPSA.235..560C. doi:10.1098/rspa.1956.0106. hdl:2027/mdp.39015095254382. ISSN 0080-4630. S2CID 122720337.
^ Shishkina, Elina L.; Sitnik, Sergei M. (2017-07-15). "The general form of the Euler--Poisson--Darboux equation and application of transmutation method". arXiv:1707.04733 [math.CA].
^ Miles, E.P; Young, E.C (1966). "On a Cauchy problem for a generalized Euler-Poisson-Darboux equation with polyharmonic data". Journal of Differential Equations. 2 (4): 482–487. Bibcode:1966JDE.....2..482M. doi:10.1016/0022-0396(66)90056-8.
^ 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. doi:10.2307/2314793. JSTOR 2314793.