Euler–Poisson–Darboux equation

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In mathematics, the Euler–Poisson–Darboux(EPD)^[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]

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]

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. (2025-08-06). "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. (2025-08-06). "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.
^ Stewart, J.M. (2009). "The Euler–Poisson–Darboux equation for relativists,". Gen. Rel. Grav. 41: 2045–2071. doi:10.1007/s10714-009-0829-3.
^ Barna, I.F.; Kersner, R. (2010). "Heat conduction: a telegraph-type model with self-similar behavior of solutions". Journal of Physics A: Mathematical and General. 43: 375210. arXiv:1204.4386. doi:10.1088/1751-8113/43/37/375210.
^ Garra, R.; Orsingher, E.; Shishkina, Shishkina (2019). "Solutions to Non-linear Euler-Poisson-Darboux Equations by Means of Generalized Separation of Variables". Lobachevskii Journal of Mathematics. 40 (640–647). doi:10.1134/S1995080219050093.

External links

Moro?anu, C. (2001) [1994], "Euler–Poisson–Darboux equation", Encyclopedia of Mathematics, EMS Press

This mathematical analysis–related article is a stub. You can help Wikipedia by expanding it.

[Zwillinger1997-1] Zwillinger, D. (1997). Handbook of Differential Equations 3rd edition. Academic Press, Boston, MA.

[:0-2] Copson, E. T. (1975). Partial differential equations. Cambridge: Cambridge University Press. ISBN 978-0521098939. OCLC 1499723.

[3] Copson, E. T. (2025-08-06). "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.

[4] Shishkina, Elina L.; Sitnik, Sergei M. (2025-08-06). "The general form of the Euler--Poisson--Darboux equation and application of transmutation method". arXiv:1707.04733 [math.CA].

[5] 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.

[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. doi:10.2307/2314793. JSTOR 2314793.

[7] Stewart, J.M. (2009). "The Euler–Poisson–Darboux equation for relativists,". Gen. Rel. Grav. 41: 2045–2071. doi:10.1007/s10714-009-0829-3.

[8] Barna, I.F.; Kersner, R. (2010). "Heat conduction: a telegraph-type model with self-similar behavior of solutions". Journal of Physics A: Mathematical and General. 43: 375210. arXiv:1204.4386. doi:10.1088/1751-8113/43/37/375210.

[9] Garra, R.; Orsingher, E.; Shishkina, Shishkina (2019). "Solutions to Non-linear Euler-Poisson-Darboux Equations by Means of Generalized Separation of Variables". Lobachevskii Journal of Mathematics. 40 (640–647). doi:10.1134/S1995080219050093.

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