In mathematics , the Euler–Poisson–Darboux(EPD) [ 1] [ 2] partial differential equation
u
x
,
y
+
N
(
u
x
+
u
y
)
x
+
y
=
0.
{\displaystyle 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
r
r
+
m
r
u
r
− − -->
u
t
t
=
0
,
{\displaystyle u_{rr}+{\frac {m}{r}}u_{r}-u_{tt}=0,}
by
x
=
r
+
t
{\displaystyle x=r+t}
y
=
r
− − -->
t
{\displaystyle y=r-t}
N
=
m
2
{\displaystyle N={\frac {m}{2}}}
[ 2] [ 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. (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 . ^ 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
