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The second-order [[Hyperbolic partial differential equation|hyperbolic partial differential equation]]
 
The second-order [[Hyperbolic partial differential equation|hyperbolic partial differential equation]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130050/e1300501.png" /></td> </tr></table>
+
\begin{equation*} 0 = L ( \alpha , \beta ) u = \left\{ \partial _ { x } \partial _ { y } - \frac { \alpha - \beta } { x - y } \partial _ { x } + \frac { \alpha ( \beta - 1 ) } { ( x - y ) ^ { 2 } } \right\} u = 0, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130050/e1300502.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130050/e1300503.png" /> are real positive parameters such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130050/e1300504.png" /> (see [[#References|[a8]]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130050/e1300505.png" /> denotes the partial derivative of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130050/e1300506.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130050/e1300507.png" />.
+
where $\alpha$ and $\beta$ are real positive parameters such that $\alpha + \beta &lt; 1$ (see [[#References|[a8]]]) and $\partial _ { x } u$ denotes the partial derivative of the function $u$ with respect to $x$.
  
 
This equation appears in various areas of mathematics and physics, such as the theory of surfaces [[#References|[a4]]], the propagation of sound [[#References|[a3]]], the colliding of gravitational waves [[#References|[a6]]], etc.. The Euler–Poisson–Darboux equation has rather interesting properties, e.g. in relation to Miller symmetry and the Laplace sequence, and has a relation to, e.g., the Toda molecule equation (see [[#References|[a4]]]).
 
This equation appears in various areas of mathematics and physics, such as the theory of surfaces [[#References|[a4]]], the propagation of sound [[#References|[a3]]], the colliding of gravitational waves [[#References|[a6]]], etc.. The Euler–Poisson–Darboux equation has rather interesting properties, e.g. in relation to Miller symmetry and the Laplace sequence, and has a relation to, e.g., the Toda molecule equation (see [[#References|[a4]]]).
Line 9: Line 17:
 
A formal solution to the Euler–Poisson–Darboux equation has the form [[#References|[a8]]]
 
A formal solution to the Euler–Poisson–Darboux equation has the form [[#References|[a8]]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130050/e1300508.png" /></td> </tr></table>
+
\begin{equation*} \phi ( \lambda , \mu ; \alpha , \beta ; x , y ) = \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130050/e1300509.png" /></td> </tr></table>
+
\begin{equation*} \sum _ { n \in Z } \frac { [ \lambda + \alpha ; n ] [ \mu - n + 1 ; n ] } { [ \mu - n + \beta ; n ] [ \lambda + 1 ; n ] } x ^ { \lambda + n } y ^ { \mu - n }, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130050/e13005010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130050/e13005011.png" /> is the [[Gamma-function|gamma-function]].
+
where $[ \lambda ; n ] = \Gamma ( \lambda + n ) / \Gamma ( \lambda )$ and $\Gamma ( \lambda )$ is the [[Gamma-function|gamma-function]].
  
By conjugate transformation of the differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130050/e13005012.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130050/e13005013.png" /> one obtains the operator
+
By conjugate transformation of the differential operator $L ( \alpha , \beta )$ with $( x - y ) ^ { - a }$ one obtains the operator
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130050/e13005014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
\begin{equation} \tag{a1} \overline{E} ( \alpha , \beta ) = \partial _ { x } \partial _ { y } - \frac { \beta } { x - y } \partial _ { x } + \frac { \alpha } { x - y } \partial y. \end{equation}
  
 
Many papers deal with the equation
 
Many papers deal with the equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130050/e13005015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
\begin{equation} \tag{a2} \overline{E} ( \alpha , \beta ) = 0 \end{equation}
  
(see, e.g., [[#References|[a11]]], [[#References|[a8]]], [[#References|[a7]]], [[#References|[a10]]], [[#References|[a12]]]). In the characteristic triangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130050/e13005016.png" /> and under the conditions
+
(see, e.g., [[#References|[a11]]], [[#References|[a8]]], [[#References|[a7]]], [[#References|[a10]]], [[#References|[a12]]]). In the characteristic triangle $\Omega = \{ ( x , y ) \in \mathbf{R} ^ { 2 } : 0 &lt; x &lt; y &lt; 1 \}$ and under the conditions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130050/e13005017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
+
\begin{equation} \tag{a3} u | _ { x  = y} = \tau ( x ), \end{equation}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130050/e13005018.png" /></td> </tr></table>
+
\begin{equation*} ( y - x ) ^ { \alpha + \beta } \left( \frac { \partial u } { \partial y } - \frac { \partial u } { \partial x } \right) | _ { x = y } = \nu ( x ), \end{equation*}
  
 
the solution of (a2) can be expressed as (see [[#References|[a12]]]):
 
the solution of (a2) can be expressed as (see [[#References|[a12]]]):
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130050/e13005019.png" /></td> </tr></table>
+
\begin{equation*} u ( x , y ) = \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130050/e13005020.png" /></td> </tr></table>
+
\begin{equation*} = \frac { \Gamma ( \alpha + \beta ) } { \Gamma ( \alpha ) \Gamma ( \beta ) } \int _ { 0 } ^ { 1 } \tau ( x + ( y - x ) t ) t ^ { \beta - 1 } ( 1 - t ) ^ { \alpha - 1 } d t + \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130050/e13005021.png" /></td> </tr></table>
+
\begin{equation*} + \frac { \Gamma ( 1 - \alpha - \beta ) } { 2 \Gamma ( 1 - \alpha ) \Gamma ( 1 - \beta ) } ( y - x ) ^ { t - \alpha - \beta }. \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130050/e13005022.png" /></td> </tr></table>
+
\begin{equation*} .\int _ { 0 } ^ { 1 } \nu ( x + ( y - x ) t ) t ^ { - \alpha } ( 1 - t ) ^ { - \beta } d t. \end{equation*}
  
Formulas for the general solution of (a2) are known for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130050/e13005023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130050/e13005024.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130050/e13005025.png" />; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130050/e13005026.png" />. For other values of the parameters, an explicit representation of the solution can be given using a regularization method for the divergent integral (see [[#References|[a7]]]). The unique solvability of a boundary value problem for (a2) with a non-local boundary condition, containing the Szegö fractional integration and differentiation (cf. [[Fractional integration and differentiation|Fractional integration and differentiation]]) operators, is proved in [[#References|[a11]]]. For (a2) local solutions, propagation of singularities, and holonomic solutions of hypergeometric type are studied in [[#References|[a14]]]. For hypergeometric functions of several variables occurring as solutions of boundary value problems for (a2), see also [[#References|[a14]]].
+
Formulas for the general solution of (a2) are known for $| \alpha | &lt; 1$, $| \beta | &lt; 1$; $\alpha = \beta$; and $\alpha + \beta = 1$. For other values of the parameters, an explicit representation of the solution can be given using a regularization method for the divergent integral (see [[#References|[a7]]]). The unique solvability of a boundary value problem for (a2) with a non-local boundary condition, containing the Szegö fractional integration and differentiation (cf. [[Fractional integration and differentiation|Fractional integration and differentiation]]) operators, is proved in [[#References|[a11]]]. For (a2) local solutions, propagation of singularities, and holonomic solutions of hypergeometric type are studied in [[#References|[a14]]]. For hypergeometric functions of several variables occurring as solutions of boundary value problems for (a2), see also [[#References|[a14]]].
  
A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130050/e13005028.png" />-difference analogue of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130050/e13005029.png" /> is considered in [[#References|[a8]]]; it has been proved that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130050/e13005030.png" />-deformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130050/e13005031.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130050/e13005032.png" />-difference operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130050/e13005033.png" />.
+
A $q$-difference analogue of the operator $E ( \alpha , \beta ) = ( x - y ) \bar{E} ( \alpha , \beta )$ is considered in [[#References|[a8]]]; it has been proved that the $q$-deformation of $E ( \alpha , \beta )$ is the $q$-difference operator $E _ { q } ( \alpha , \beta ) = [ \theta _ { x } + \alpha ] _ { q } [ \partial _ { y } ] _ { q } - [ \theta _ { y } + \beta ] [ \partial _ { x } ] _ { q }$.
  
 
The existence and uniqueness of global generalized solutions of mixed problems for the generalized Euler–Poisson–Darboux equation
 
The existence and uniqueness of global generalized solutions of mixed problems for the generalized Euler–Poisson–Darboux equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130050/e13005034.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>
+
<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130050/e13005034.png"/></td> <td style="width:5%;text-align:right;" valign="top">(a4)</td></tr></table>
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130050/e13005035.png" /></td> </tr></table>
+
\begin{equation*} = f ( t , x , u , u _ { t } , \nabla u ) \end{equation*}
  
 
are studied in [[#References|[a15]]], using Galerkin approximation. Moreover, the classical solution of (a4) has been obtained by using properties of Sobolev spaces and imbedding theorems (cf. also [[Imbedding theorems|Imbedding theorems]]). See [[#References|[a2]]], [[#References|[a11]]], [[#References|[a1]]], [[#References|[a9]]] for various aspects of (a4).
 
are studied in [[#References|[a15]]], using Galerkin approximation. Moreover, the classical solution of (a4) has been obtained by using properties of Sobolev spaces and imbedding theorems (cf. also [[Imbedding theorems|Imbedding theorems]]). See [[#References|[a2]]], [[#References|[a11]]], [[#References|[a1]]], [[#References|[a9]]] for various aspects of (a4).
Line 54: Line 62:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C.Y. Chan,  K.K. Nip,  "Quenching for semilinear Euler–Poisson–Darboux equations"  J. Wiener (ed.) , ''Partial Differential Equations. Proc. Internat. Conf. Theory Appl. Differential Equations (Univ. Texas-Pan American, Edinburg, Texas, May 15-18, 1991)'' , ''Pitman Res. Notes'' , '''273''' , Longman  (1992)  pp. 39–43</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C.Y. Chan,  K.K. Nip,  "On the blow-up of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130050/e13005036.png" /> at quenching for semilinear Euler–Poisson–Darboux equations"  ''Comput. Appl. Math.'' , '''14''' :  2  (1995)  pp. 185–190</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E.T. Copson,  "Partial differential equations" , Cambridge Univ. Press  (1975)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Darboux,  "Sur la théeorie générale de surfaces" , '''II''' , Chelsea, reprint  (1972)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  V.N. Denisov,  "On the stabilization of means of the solution of the Cauchy problem for hyperbolic equations in symmetric spaces"  ''Soviet Math. Dokl.'' , '''42''' :  3  (1991)  pp. 738–742  ''Dokl. Akad. Nauk. SSSR'' , '''315''' :  2  (1990)  pp. 266–271</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  I. Hauser,  F.J. Ernst,  "Initial value problem for colliding gravitational plane wave"  ''J. Math. Phys.'' , '''30''' :  4  (1989)  pp. 872–887</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  R.S. Khairullin,  "On the theory of the Euler–Poisson–Darboux equation"  ''Russian Math.'' , '''37''' :  11  (1993)  pp. 67–74  ''Izv. Vyssh. Uchebn. Zaved. Mat.'' :  11  (1993)  pp. 69–76</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  K. Nagamoto,  Y. Koga,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130050/e13005037.png" />-difference analogue of the Euler–Poisson–Darboux equation and its Laplace sequence"  ''Osaka J. Math.'' , '''32''' :  2  (1995)  pp. 451–465</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  S.V. Pan'ko,  "On a representation of the solution of a generalized Euler–Poisson–Darboux equation"  ''Diff. Uravnen.'' , '''28''' :  2  (1992)  pp. 278–281  (In Russian)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  O.A. Repin,  "Boundary value problems with shift for equations of hyperbolic and mixed type" , Samara: Izd. Sartovsk. Univ.  (1992)  (In Russian)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  O.A. Repin,  "A nonlocal boundary value problem for the Euler–Poisson–Darboux equation"  ''Diff. Eqs.'' , '''31''' :  1  (1995)  pp. 160–162  ''Diff. Uravn.'' , '''31''' :  1  (1995)  pp. 171–172</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  M. Saigo,  "A certain boundary value problem for the Euler–Poisson–Darboux equation"  ''Math. Japon.'' , '''24''' :  4  (1979)  pp. 377–385</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  M.M. Smirnov,  "Degenerate hyperbolic equations" , Izd. Vysh. Shkola, Minsk  (1977)  (In Russian)</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  N. Takayama,  "Propagation of singularities of solutions of the Euler–Poisson–Darboux equation and a global structure of the space of holonomic solutions I"  ''Funkc. Ekvacioj, Ser. Internat.'' , '''35'''  (1992)  pp. 343–403</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  J. Wang,  "Mixed problems for nonlinear hyperbolic equations with singular dissipative terms"  ''Acta Math. Appl. Sin.'' , '''16'''  (1993)  pp. 23–30  (In Chinese)  (English summary)</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  C.Y. Chan,  K.K. Nip,  "Quenching for semilinear Euler–Poisson–Darboux equations"  J. Wiener (ed.) , ''Partial Differential Equations. Proc. Internat. Conf. Theory Appl. Differential Equations (Univ. Texas-Pan American, Edinburg, Texas, May 15-18, 1991)'' , ''Pitman Res. Notes'' , '''273''' , Longman  (1992)  pp. 39–43</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  C.Y. Chan,  K.K. Nip,  "On the blow-up of $| u_{ tt } |$ at quenching for semilinear Euler–Poisson–Darboux equations"  ''Comput. Appl. Math.'' , '''14''' :  2  (1995)  pp. 185–190</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  E.T. Copson,  "Partial differential equations" , Cambridge Univ. Press  (1975)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  G. Darboux,  "Sur la théeorie générale de surfaces" , '''II''' , Chelsea, reprint  (1972)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  V.N. Denisov,  "On the stabilization of means of the solution of the Cauchy problem for hyperbolic equations in symmetric spaces"  ''Soviet Math. Dokl.'' , '''42''' :  3  (1991)  pp. 738–742  ''Dokl. Akad. Nauk. SSSR'' , '''315''' :  2  (1990)  pp. 266–271</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  I. Hauser,  F.J. Ernst,  "Initial value problem for colliding gravitational plane wave"  ''J. Math. Phys.'' , '''30''' :  4  (1989)  pp. 872–887</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  R.S. Khairullin,  "On the theory of the Euler–Poisson–Darboux equation"  ''Russian Math.'' , '''37''' :  11  (1993)  pp. 67–74  ''Izv. Vyssh. Uchebn. Zaved. Mat.'' :  11  (1993)  pp. 69–76</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  K. Nagamoto,  Y. Koga,  "$q$-difference analogue of the Euler–Poisson–Darboux equation and its Laplace sequence"  ''Osaka J. Math.'' , '''32''' :  2  (1995)  pp. 451–465</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  S.V. Pan'ko,  "On a representation of the solution of a generalized Euler–Poisson–Darboux equation"  ''Diff. Uravnen.'' , '''28''' :  2  (1992)  pp. 278–281  (In Russian)</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  O.A. Repin,  "Boundary value problems with shift for equations of hyperbolic and mixed type" , Samara: Izd. Sartovsk. Univ.  (1992)  (In Russian)</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  O.A. Repin,  "A nonlocal boundary value problem for the Euler–Poisson–Darboux equation"  ''Diff. Eqs.'' , '''31''' :  1  (1995)  pp. 160–162  ''Diff. Uravn.'' , '''31''' :  1  (1995)  pp. 171–172</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  M. Saigo,  "A certain boundary value problem for the Euler–Poisson–Darboux equation"  ''Math. Japon.'' , '''24''' :  4  (1979)  pp. 377–385</td></tr><tr><td valign="top">[a13]</td> <td valign="top">  M.M. Smirnov,  "Degenerate hyperbolic equations" , Izd. Vysh. Shkola, Minsk  (1977)  (In Russian)</td></tr><tr><td valign="top">[a14]</td> <td valign="top">  N. Takayama,  "Propagation of singularities of solutions of the Euler–Poisson–Darboux equation and a global structure of the space of holonomic solutions I"  ''Funkc. Ekvacioj, Ser. Internat.'' , '''35'''  (1992)  pp. 343–403</td></tr><tr><td valign="top">[a15]</td> <td valign="top">  J. Wang,  "Mixed problems for nonlinear hyperbolic equations with singular dissipative terms"  ''Acta Math. Appl. Sin.'' , '''16'''  (1993)  pp. 23–30  (In Chinese)  (English summary)</td></tr></table>

Latest revision as of 17:44, 1 July 2020

The second-order hyperbolic partial differential equation

\begin{equation*} 0 = L ( \alpha , \beta ) u = \left\{ \partial _ { x } \partial _ { y } - \frac { \alpha - \beta } { x - y } \partial _ { x } + \frac { \alpha ( \beta - 1 ) } { ( x - y ) ^ { 2 } } \right\} u = 0, \end{equation*}

where $\alpha$ and $\beta$ are real positive parameters such that $\alpha + \beta < 1$ (see [a8]) and $\partial _ { x } u$ denotes the partial derivative of the function $u$ with respect to $x$.

This equation appears in various areas of mathematics and physics, such as the theory of surfaces [a4], the propagation of sound [a3], the colliding of gravitational waves [a6], etc.. The Euler–Poisson–Darboux equation has rather interesting properties, e.g. in relation to Miller symmetry and the Laplace sequence, and has a relation to, e.g., the Toda molecule equation (see [a4]).

A formal solution to the Euler–Poisson–Darboux equation has the form [a8]

\begin{equation*} \phi ( \lambda , \mu ; \alpha , \beta ; x , y ) = \end{equation*}

\begin{equation*} \sum _ { n \in Z } \frac { [ \lambda + \alpha ; n ] [ \mu - n + 1 ; n ] } { [ \mu - n + \beta ; n ] [ \lambda + 1 ; n ] } x ^ { \lambda + n } y ^ { \mu - n }, \end{equation*}

where $[ \lambda ; n ] = \Gamma ( \lambda + n ) / \Gamma ( \lambda )$ and $\Gamma ( \lambda )$ is the gamma-function.

By conjugate transformation of the differential operator $L ( \alpha , \beta )$ with $( x - y ) ^ { - a }$ one obtains the operator

\begin{equation} \tag{a1} \overline{E} ( \alpha , \beta ) = \partial _ { x } \partial _ { y } - \frac { \beta } { x - y } \partial _ { x } + \frac { \alpha } { x - y } \partial y. \end{equation}

Many papers deal with the equation

\begin{equation} \tag{a2} \overline{E} ( \alpha , \beta ) = 0 \end{equation}

(see, e.g., [a11], [a8], [a7], [a10], [a12]). In the characteristic triangle $\Omega = \{ ( x , y ) \in \mathbf{R} ^ { 2 } : 0 < x < y < 1 \}$ and under the conditions

\begin{equation} \tag{a3} u | _ { x = y} = \tau ( x ), \end{equation}

\begin{equation*} ( y - x ) ^ { \alpha + \beta } \left( \frac { \partial u } { \partial y } - \frac { \partial u } { \partial x } \right) | _ { x = y } = \nu ( x ), \end{equation*}

the solution of (a2) can be expressed as (see [a12]):

\begin{equation*} u ( x , y ) = \end{equation*}

\begin{equation*} = \frac { \Gamma ( \alpha + \beta ) } { \Gamma ( \alpha ) \Gamma ( \beta ) } \int _ { 0 } ^ { 1 } \tau ( x + ( y - x ) t ) t ^ { \beta - 1 } ( 1 - t ) ^ { \alpha - 1 } d t + \end{equation*}

\begin{equation*} + \frac { \Gamma ( 1 - \alpha - \beta ) } { 2 \Gamma ( 1 - \alpha ) \Gamma ( 1 - \beta ) } ( y - x ) ^ { t - \alpha - \beta }. \end{equation*}

\begin{equation*} .\int _ { 0 } ^ { 1 } \nu ( x + ( y - x ) t ) t ^ { - \alpha } ( 1 - t ) ^ { - \beta } d t. \end{equation*}

Formulas for the general solution of (a2) are known for $| \alpha | < 1$, $| \beta | < 1$; $\alpha = \beta$; and $\alpha + \beta = 1$. For other values of the parameters, an explicit representation of the solution can be given using a regularization method for the divergent integral (see [a7]). The unique solvability of a boundary value problem for (a2) with a non-local boundary condition, containing the Szegö fractional integration and differentiation (cf. Fractional integration and differentiation) operators, is proved in [a11]. For (a2) local solutions, propagation of singularities, and holonomic solutions of hypergeometric type are studied in [a14]. For hypergeometric functions of several variables occurring as solutions of boundary value problems for (a2), see also [a14].

A $q$-difference analogue of the operator $E ( \alpha , \beta ) = ( x - y ) \bar{E} ( \alpha , \beta )$ is considered in [a8]; it has been proved that the $q$-deformation of $E ( \alpha , \beta )$ is the $q$-difference operator $E _ { q } ( \alpha , \beta ) = [ \theta _ { x } + \alpha ] _ { q } [ \partial _ { y } ] _ { q } - [ \theta _ { y } + \beta ] [ \partial _ { x } ] _ { q }$.

The existence and uniqueness of global generalized solutions of mixed problems for the generalized Euler–Poisson–Darboux equation

(a4)

\begin{equation*} = f ( t , x , u , u _ { t } , \nabla u ) \end{equation*}

are studied in [a15], using Galerkin approximation. Moreover, the classical solution of (a4) has been obtained by using properties of Sobolev spaces and imbedding theorems (cf. also Imbedding theorems). See [a2], [a11], [a1], [a9] for various aspects of (a4).

See [a5] for necessary and sufficient conditions for stabilization of the solution of the Cauchy problem for the Euler–Poisson–Darboux equation in a homogeneous symmetric space.

References

[a1] C.Y. Chan, K.K. Nip, "Quenching for semilinear Euler–Poisson–Darboux equations" J. Wiener (ed.) , Partial Differential Equations. Proc. Internat. Conf. Theory Appl. Differential Equations (Univ. Texas-Pan American, Edinburg, Texas, May 15-18, 1991) , Pitman Res. Notes , 273 , Longman (1992) pp. 39–43
[a2] C.Y. Chan, K.K. Nip, "On the blow-up of $| u_{ tt } |$ at quenching for semilinear Euler–Poisson–Darboux equations" Comput. Appl. Math. , 14 : 2 (1995) pp. 185–190
[a3] E.T. Copson, "Partial differential equations" , Cambridge Univ. Press (1975)
[a4] G. Darboux, "Sur la théeorie générale de surfaces" , II , Chelsea, reprint (1972)
[a5] V.N. Denisov, "On the stabilization of means of the solution of the Cauchy problem for hyperbolic equations in symmetric spaces" Soviet Math. Dokl. , 42 : 3 (1991) pp. 738–742 Dokl. Akad. Nauk. SSSR , 315 : 2 (1990) pp. 266–271
[a6] I. Hauser, F.J. Ernst, "Initial value problem for colliding gravitational plane wave" J. Math. Phys. , 30 : 4 (1989) pp. 872–887
[a7] R.S. Khairullin, "On the theory of the Euler–Poisson–Darboux equation" Russian Math. , 37 : 11 (1993) pp. 67–74 Izv. Vyssh. Uchebn. Zaved. Mat. : 11 (1993) pp. 69–76
[a8] K. Nagamoto, Y. Koga, "$q$-difference analogue of the Euler–Poisson–Darboux equation and its Laplace sequence" Osaka J. Math. , 32 : 2 (1995) pp. 451–465
[a9] S.V. Pan'ko, "On a representation of the solution of a generalized Euler–Poisson–Darboux equation" Diff. Uravnen. , 28 : 2 (1992) pp. 278–281 (In Russian)
[a10] O.A. Repin, "Boundary value problems with shift for equations of hyperbolic and mixed type" , Samara: Izd. Sartovsk. Univ. (1992) (In Russian)
[a11] O.A. Repin, "A nonlocal boundary value problem for the Euler–Poisson–Darboux equation" Diff. Eqs. , 31 : 1 (1995) pp. 160–162 Diff. Uravn. , 31 : 1 (1995) pp. 171–172
[a12] M. Saigo, "A certain boundary value problem for the Euler–Poisson–Darboux equation" Math. Japon. , 24 : 4 (1979) pp. 377–385
[a13] M.M. Smirnov, "Degenerate hyperbolic equations" , Izd. Vysh. Shkola, Minsk (1977) (In Russian)
[a14] N. Takayama, "Propagation of singularities of solutions of the Euler–Poisson–Darboux equation and a global structure of the space of holonomic solutions I" Funkc. Ekvacioj, Ser. Internat. , 35 (1992) pp. 343–403
[a15] J. Wang, "Mixed problems for nonlinear hyperbolic equations with singular dissipative terms" Acta Math. Appl. Sin. , 16 (1993) pp. 23–30 (In Chinese) (English summary)
How to Cite This Entry:
Euler-Poisson-Darboux equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler-Poisson-Darboux_equation&oldid=12800
This article was adapted from an original article by C. MoroÅŸanu (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article