Euler–Poisson–Darboux equation
From HandWiki
In mathematics, the Euler–Poisson–Darboux[1][2] equation is the partial differential equation
- [math]\displaystyle{ u_{x,y}+\frac{N(u_x+u_y)}{x+y}=0. }[/math]
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
- [math]\displaystyle{ u_{rr}+\frac{m}{r}u_r-u_{tt}=0, }[/math]
by [math]\displaystyle{ x=r+t }[/math], [math]\displaystyle{ y=r-t }[/math], where [math]\displaystyle{ N=\frac{m}{2} }[/math] [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.
- ↑ 2.0 2.1 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" (in en). Proc. R. Soc. Lond. A 235 (1203): 560–572. doi:10.1098/rspa.1956.0106. ISSN 0080-4630. Bibcode: 1956RSPSA.235..560C.
- ↑ 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" (in en). Journal of Differential Equations 2 (4): 482–487. doi:10.1016/0022-0396(66)90056-8. Bibcode: 1966JDE.....2..482M.
- ↑ 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.
External links
- Hazewinkel, Michiel, ed. (2001), "Euler–Poisson–Darboux equation", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=E/e130050
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