On the Euler-Poisson-Darboux equation theory (Q1902794)

In the studies of differential equations with partial derivatives, where coefficients contain singularities, a great attention is paid to the Euler-Poisson-Darboux (EPD) equation: \[ E(\beta, \alpha): u_{xy}- {\alpha\over x- y} u_x+ {\beta\over x- y} u_y= 0,\tag{1} \] where \(\alpha\) and \(\beta\) are real parameters. A number of model degenerated hyperbolic equations have form (1) in characteristic coordinates. The formulas for general solution to the EPD equation are known for the cases \(|\alpha|< 1\), \(|\beta|< 1\); \(\alpha= \beta\) and \(\alpha+ \beta= 1/2\). For other values of parameters we give an explicit representation of solutions. For these purposes we use the method of regularization of a divergent integral.