Bitsadze-Samarskii problem with a lacking shift condition for the Gellerstedt equation with a singular coefficient

For the Gellerstedt equation with a singular coefficient, we study the well-posedness of the problem with the Bitsadze-Samarskii conditions on the ellipticity boundary and on a segment of the degeneration line and with a shift condition on parts of boundary characteristics. We use the maximum principle to prove the uniqueness of the solution of the problem in the class of Hölder functions and the method of integral equations to prove its existence.     

A mixed problem with Tricomi and Frankl conditions for the gellerstedt equation with a singular coefficient

We consider a generalized Tricomi equation with a singular coefficient. For this equation in a mixed domain we study the corresponding problem in the case, when a part of the boundary characteristic is free of boundary conditions; the deficient Tricomi condition is equivalently substituted by a nonlocal Frankl condition on a segment of the degeneration line. We prove that the stated problem is well-posed.     

A combination of the Tricomi problem and a boundary-value problem with a shift for the Gellerstedt equation

We study a problem whose statement combines the Tricomi problem and the problem with a shift considered by V. I. Zhegalov and A. M. Nakhushev for the Gellerstedt equation with a singular coefficient. We prove its solvability by the method of integral equations, and we do the uniqueness of the solution with the help of the extremum principle.     

A problem with nonlocal conditions for mixed-type equations

We study a problem with the Bitsadze-Samarskii conditions on the ellipticity boundary and on a segment of degeneration line and with the displacement condition on parts of boundary characteristics of the Gellerstedt equation with singular coefficient. With the help of the maximum principle we prove the uniqueness of a solution to the problem, and with the help of the method of integral equations we prove the existence of a solution to the problem.     

A problem with conditions analogous to Frankl’ and Bitsadze-Samarskii ones for the Gellerstedt equation

In this paper we study a problem with conditions analogous to Frankl’ and Bitsadze-Samarskii ones for the Gellerstedt equation. We prove that the stated problem is well-posed.     

A problem with a Frankl condition on the boundary of the ellipticity domain for the Gellerstedt equation with a singular coefficient

We study a problem with the Frankl and Bitsadze-Samarskii conditions on the elliptic boundary and on the degeneration line for the Gellerstedt equation with a singular coefficient. We prove the correctness of the stated problem.     

Problem with an Analog of the Frankl Condition on an Internal Characteristic for an Equation of Mixed Type

Differential Equations - For the equation $$(\mathrm {sign}\thinspace y)|y|^{m}u_{{xx}}+u_{{yy}}-(m/2y)u_{y}=0$$ considered in a mixed domain, we prove theorems on the uniqueness and existence of a...     

On a problem with a shift and with the Frankl condition on a segment of the degeneration line for a class of equations of mixed type

We consider the Frankl-Nakhushev problem. By using the maximum principle, we prove the uniqueness of the solution of the problem in the class of Hölder functions, and by using the method of integral equations, in particular, the recently developed method of Wiener-Hopf equations, we prove its existence.     

Tricomi-Nakhushev problem

In the Tricomi problem, the values of the unknown function are given at all points of a characteristic. We study the well-posedness of a problem in which part of the characteristic is free of the boundary condition and the lacking Tricomi condition is equivalently replaced by A.M. Nakhushev’s nonlocal condition with shift.     

Problem with Nonlocal Conditions,Specified on Parts of the Boundary Characteristics and on the Degeneracy Segment,for the Gellerstedt Equation with Singular Coefficient

Russian Mathematics - For the Gellerstedt equation with a singular coefficient, we investigate a boundary value problem with nonlocal conditions, given on parts of the boundary characteristics, and...