Euler–Poisson–Darboux equation
In mathematics, the Euler–Poisson–Darboux(EPD)[1][2] equation is the partial differential equation
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
by , , where [2] and some sources quote this equation when referring to the Euler–Poisson–Darboux equation.[3][4][5][6]
The EPD 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.
- ↑ 2.0 2.1 Copson, E. T. (1975). Partial differential equations. Cambridge: Cambridge University Press. ISBN 978-0-521-09893-9. 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.
- ↑ Stewart, J.M. (2009). "The Euler–Poisson–Darboux equation for relativists". Gen. Rel. Grav. 41 (9): 2045–2071. doi:10.1007/s10714-009-0829-3. Bibcode: 2009GReGr..41.2045S. https://link.springer.com/article/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 (37). doi:10.1088/1751-8113/43/37/375210. Bibcode: 2010JPhA...43K5210B. https://iopscience.iop.org/journal/0305-4470.
- ↑ 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): 640–647. doi:10.1134/S1995080219050093. https://link.springer.com/article/10.1134/S1995080219050093.
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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