On Formulation of Modified Problems for the Euler-Darboux Equation with Parameters Equalto 1/2 in Absolute Value

We consider the Euler-Darboux equation with parameters equal to 1/2 in absolute value. Since the Cauchy problem in the classical formulation in ill-posed for such values of parameters, we proposeformulations and solutions of modified Cauchy-type problems with the following values of parameters: a)α = β = 1 , b) α = - 1 , β = - 1 , c) α = β = - 1 . In the case а), the modified Cauchy problem is solved2 2 2 2by the Riemann method. We use the obtained result to formulate the analog of the problem Δ1 in the first quadrant with shifted boundary-value conditions on axes and nonstandard conjunction conditions on thesingularity 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.

ON CHARACTERISTIC PROBLEMS FOR ONE HYPERBOLIC SYSTEM IN THREE-DIMENSIONAL SPACE

We consider characteristic problems for a hyperbolic system with three independent variables. Using the Riemann method and theory of integral equations we obtain conditions of one-valued solvability for this problems.

The variant of the Goursat problem for the sixth order equation

Our study is considered as a continuation of results in [1-8]. More precisely we investigate the Goursat problem in class $ C^{2+2+2}(D)$ for the equation (1) below by Riemann method variant from the works [8] and [9].