Some boundary value problems for the Euler-Darboux equation with positive parameters (Q2768772)

This paper deals with the problem NEWLINE\[NEWLINE\begin{aligned} U_{xy}+\beta(y-x)^{-1}U_{x}-\alpha(y-x)^{-1}U_{y}&=0,\\ U_{y}(0,y)+\beta y^{-1}U(0,y)&=\phi(y),\quad y\in[0,h],\\ U_{x}(x,h)+\alpha (h-x)^{-1}U(x,h)&=f(x),\quad x\in[0,h]\end{aligned}NEWLINE\]NEWLINE in the domain \(G=\{(x,y)\mid 0<x<y<h\}\). In particular, the authors prove that if \(f(x)\in C[0,h)\cap R[0,h]\cap C^1[0,h)\), \(\int_{0}^{h}f(t)(h-t)^{\alpha} dt=0\), \(f(x)=O((h-x)^{-(\alpha+\beta)})\), \(f'(x)=O((h-x)^{-(1+\alpha+\beta)})\), \(\phi(y)\in C(0,h]\cap R[0,h]\cap C^1(0,h]\), \(\int_{0}^{h}\phi(t)t^{\beta} dt=0\), \(\phi(y)=O(y^{-(\alpha+\beta)})\), \(\phi'(y)=O(y^{-(1+\alpha+\beta)})\), then the unique solution \(U(x,y)\in C(\bar G)\) of the considered problem has the form NEWLINE\[NEWLINEU(x,y)= \int_{0}^{x}f(t)R(t,h,x,y) dt-\int_{y}^{h}\phi(t)R(0,t,x,y) dt,NEWLINE\]NEWLINE where \(R(x,y,x_0,y_0)\) is the Riemann function of the given equation.