Euler–Poisson–Darboux equation

In mathematics, the Euler–Poisson–Darboux(EPD){{cite book

| author = Zwillinger, D.

| year = 1997

| title = Handbook of Differential Equations 3rd edition

| publisher = Academic Press, Boston, MA

| isbn =

}}{{Cite book|title=Partial differential equations|last=Copson|first=E. T.|date=1975|publisher=Cambridge University Press|isbn=978-0521098939|location=Cambridge|oclc=1499723}} equation is the partial differential equation

: $u\_\{x,y\}+\backslash frac\{N(u\_x+u\_y)\}\{x+y\}=0.$

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

: $u\_\{rr\}+\backslash frac\{m\}\{r\}u\_r-u\_\{tt\}=0,$

by $x=r+t$, $y=r-t$, where $N=\backslash frac\{m\}\{2\}$ and some sources quote this equation when referring to the Euler–Poisson–Darboux equation.{{Cite journal|last=Copson|first=E. T.|date=1956-06-12|title=On a regular Cauchy problem for the Euler—Poisson—Darboux equation|journal=Proc. R. Soc. Lond. A|language=en|volume=235|issue=1203|pages=560–572|doi=10.1098/rspa.1956.0106|issn=0080-4630|bibcode=1956RSPSA.235..560C|hdl=2027/mdp.39015095254382|s2cid=122720337|hdl-access=free}}{{cite arXiv|last1=Shishkina|first1=Elina L.|last2=Sitnik|first2=Sergei M.|date=2017-07-15|title=The general form of the Euler--Poisson--Darboux equation and application of transmutation method|eprint=1707.04733|class=math.CA}}{{Cite journal|last1=Miles|first1=E.P|last2=Young|first2=E.C|title=On a Cauchy problem for a generalized Euler-Poisson-Darboux equation with polyharmonic data|journal=Journal of Differential Equations|language=en|volume=2|issue=4|pages=482–487|doi=10.1016/0022-0396(66)90056-8|year=1966|bibcode=1966JDE.....2..482M|doi-access=free}}{{Cite journal|last=Fusaro|first=B. A.|date=1966|title=A Solution of a Singular, Mixed Problem for the Equation of Euler-Poisson- Darboux (EPD)|journal=The American Mathematical Monthly|volume=73|issue=6|pages=610–613|doi=10.2307/2314793|jstor=2314793}}

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.{{Cite journal |last=Stewart |first=J.M. |date=2009 |title=The Euler–Poisson–Darboux equation for relativists, |url=https://link.springer.com/article/10.1007/s10714-009-0829-3#citeas |journal=Gen. Rel. Grav. |volume=41 |pages=2045–2071 |doi=10.1007/s10714-009-0829-3}}

Compact support self-similar solution of the EPD equation for thermal conduction was derived starting from the modified Fourier-Cattaneo law.{{Cite journal |last=Barna |first=I.F. |last2=Kersner |first2=R. |date=2010 |title=Heat conduction: a telegraph-type model with self-similar behavior of solutions |url=https://iopscience.iop.org/journal/0305-4470 |journal=Journal of Physics A: Mathematical and General |volume=43 |pages=375210 |arxiv=1204.4386 |doi=10.1088/1751-8113/43/37/375210}}

It is also possible to solve the non-linear EPD equations with the method of generalized separation of variables.{{Cite journal |last=Garra |first=R. |last2=Orsingher |first2=E. |last3=Shishkina |first3=Shishkina |date=2019 |title=Solutions to Non-linear Euler-Poisson-Darboux Equations by Means of Generalized Separation of Variables |url=https://link.springer.com/article/10.1134/S1995080219050093 |journal=Lobachevskii Journal of Mathematics |volume=40 |issue=640–647 |doi=10.1134/S1995080219050093}}