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On a certain boundary value problem for the Euler-Darboux equation with positive parameter. (Russian) Zbl 0545.35072

For the equation \(Z_{xy}-\beta Z_ x/(x-y)+\alpha Z_ y/(x- y)=0\quad(\alpha>0,\quad \beta>0,\quad \alpha +\beta<1;\quad \alpha =\alpha_ 1,\quad \beta =\beta_ 1\quad if\quad y>x;\quad \alpha =\alpha_ 2,\quad \beta =\beta_ 2\quad if\quad y<x)\) in the domain \(D=D_ 1\cup D_ 2,\quad D_ 1=\{(x,y) | 0<x<y<1\},\quad D_ 2=\{(x,y) | 0<y<x<1\}.\) The authors study the Problem A: Find a function Z(x,y) such that \(1)\quad Z(x,y)\in C(\bar D);\quad 2)\quad Z(x,y)\in_ 0R^ 1\) in \(D_ i (i=1,2)\); \(3)\quad I_{0x}^{a_ 1,b,-a_ 1+\beta_ 1-1}Z(0,t)=\phi_ 1(x),\) 0\(\leq x\leq 1\), \(I_{0x}^{a_ 2,b,-a_ 2+\alpha_ 2-1}Z(t,0)=\phi_ 2(x),\) 0\(\leq x\leq 1\), where \[ \max \{-\alpha_ i,\beta_ i-1\}<a_ i<\min \{\beta_ i,1-\alpha_ i\},\quad b>-\alpha_ i-\beta_ i, \]
\[ \lim_{(y-x)\to +0}(y-x)^{\alpha_ 1+\beta_ 1}(Z_ x-Z_ y)=\lim_{(x-y)\to +0}(x-y)^{\alpha_ 2+\beta_ 2}(Z_ x-Z_ y)=\nu(x),\quad 0<x<1, \]
\[ I_{0x}^{a,b,\eta}f\equiv(x^{-a- b}/\Gamma(a))\int^{x}_{0}(x-t)^{a-1}F(a+b,-\eta,a;(x-t)/x)f(t)dt. \] It is proved that the problem A has a unique solution in some special cases and these solutions are obtained explicitly in the form of integral representations.
Reviewer: Ju.V.Kostarčuk

MSC:

35Q05 Euler-Poisson-Darboux equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35C15 Integral representations of solutions to PDEs
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