Euler–Poisson–Darboux equation




In mathematics, the Euler–Poisson–Darboux[1][2] equation is the partial differential equation


ux,y+N(ux+uy)x+y=0.{displaystyle u_{x,y}+{frac {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


urr+mrur−utt=0,{displaystyle u_{rr}+{frac {m}{r}}u_{r}-u_{tt}=0,}{displaystyle u_{rr}+{frac {m}{r}}u_{r}-u_{tt}=0,}

by x=r+t{displaystyle x=r+t}{displaystyle x=r+t}, y=r−t{displaystyle y=r-t}{displaystyle y=r-t}, where N=m2{displaystyle N={frac {m}{2}}}{displaystyle N={frac {m}{2}}} [2] and some sources quote this equation when referring to the Euler–Poisson–Darboux equation.[3][4][5][6]



References





  1. ^ Zwillinger, D. (1997). Handbook of Differential Equations 3rd edition. Academic Press, Boston, MA..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}


  2. ^ ab 1901-1980., Copson, E. T. (Edward Thomas), (1975). Partial differential equations. Cambridge: Cambridge University Press. ISBN 0521098939. OCLC 1499723.


  3. ^ 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. ISSN 0080-4630.


  4. ^ 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].


  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.




External links



  • Moroşanu, C. (2001) [1994], "Euler–Poisson–Darboux equation", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4









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