Euler-Poisson-Darboux equation

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)