Modified Problems for Euler–Darboux Equations with Parameters with Absolute Values Equal to 12

We consider the Euler–Darboux equation with parameters such that their absolute values are equal to 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) α = β = 1 2 ; (b) α = − 1 2 and β = 1 2 ; (c) α = β = − 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 Δ 1 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.