Modified Problems for Euler–Darboux Equations with Parameters with Absolute Values Equal to $$ \frac{1}{2} $$

Abstract

We consider the Euler–Darboux equation with parameters such that their absolute values are equal to \( \frac{1}{2} \). Since the Cauchy problem in the classical formulation is ill-posed for such values of parameters, we propose formulations and solutions of modified Cauchy-type problems with the following values of parameters:(a) \( \alpha =\beta =\frac{1}{2}; \)(b) \( \alpha =-\frac{1}{2}\ \mathrm{and}\ \beta =\frac{1}{2}; \)(c) \( \alpha =\beta =-\frac{1}{2}. \)In case (a), the modified Cauchy problem is solved by the Riemann method. We use the obtained result to formulate the analog of the problem Δ1 in the first quadrant with translated boundary conditions on axes and nonstandard conjunction conditions on the singularity line of the coefficients of the equation y = x. The first condition is gluing normal derivatives of the solution and the second one contains limit values of a combination of the solution and its normal derivatives. The problem is reduced to a uniquely solvable system of integral equations.