Abstract:
We consider the Euler–Darboux equation with parameters equal
to $\dfrac{1}{2}$ in absolute value. Since the Cauchy problem in
the classical formulation in 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},$$\beta=- \frac{1}{2},$ c)
$\alpha=\beta=- \frac{1}{2}.$ In the case а), the
modified Cauchy problem is solved by the Riemann method. We use
the obtained result to formulate the analog of the problem
$\Delta_1$ in the first quadrant with shifted boundary-value
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 limiting values of combination of the
solution and its normal derivatives. The problem is reduced to a
uniquely solvable system of integral equations.
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