The modified problems for the equation of Euler--Darboux in the case of parameters on the module equal to 1/2

Abstract

We consider the Euler--Darboux equation with parameters modulo 1/2 and generalization to the space 3D analogue. Due to the fact that the Cauchy problem in its classical formulation is incorrect for such parameter values, the authors propose formulations and solutions of modified Cauchy-type problems with parameter values: a) $\alpha=\beta=\displaystyle\frac{1}{2}$, b) $\alpha=-\,\displaystyle\frac{1}{2}$, $\beta=+\,\displaystyle\frac{1}{2}$, c) $\alpha=\beta=-\,\displaystyle\frac{1}{2}$. The obtained result is used to formulate an analogue of the $\Delta_1$ problem in the first quadrant with the setting of boundary conditions with displacement on the coordinate axes and non-standard conjugation conditions on the singularity line of the coefficients of the equation $y=x$. The first of these conditions glues the normal derivatives of the desired solution, the~second contains the limit values of the combination of the solution and its normal derivatives. The problem was reduced to a uniquely solvable system of integral equations.