A problem with tricomi and frankl conditions on the characteristic for a class of mixed-type equations

In this paper we prove the correctness of a problem with Tricomi and Frankl conditions on the characteristic for a certain class of mixed-type equations.     

Problem with analogs of the Frankl’ condition on a characteristic and the degeneration segment for an equation of mixed type with a singular coefficient

For the equation (sign y)|y| ^m u _xx +u _yy ?m(2y)^?1 u _y = 0, where m > 0, considered in some mixed domain, we prove existence and uniqueness theorems for the solution of the boundary value problem with an analog of the Frankl’ condition on a characteristic and on the degeneration segment of the equation.     

A generalization of Bitsadze–Samarskii problem for mixed type equation

We study a boundary-value problem with Bitsadze–Samarskii conditions on boundary characteristic on a special inner curve and on a segment of degeneration of mixed type equation. Its solvability is proved by method of integral equations, and uniqueness of solution is established by means of the maximum principle.     

Problem with lacking Tricomi condition on a characteristic and an analog of the Frankl condition on a segment of the degeneration line for a class of mixed type equations

For the Gellerstedt equation with a singular coefficient, we consider a boundary value problem that differs from the Tricomi problem in that the boundary characteristic AC is arbitrarily divided into two parts AC ₀ and C ₀ C and the Tricomi condition is posed on the first of them, while the second part C ₀ C is free of boundary conditions. The lacking Tricomi condition is equivalently replaced by an analog of the Frankl condition on a segment of the degeneration line. The well-posedness of this problem is proved.     

Problem with displacement conditions on internal characteristics and the Frankl condition on a segment of the degeneration line for the Gellerstedt equation with a singular coefficient

We study the well-posedness of a problem with displacement conditions on internal characteristics and an analog of the Frankl condition on a segment of the degeneration line for the Gellerstedt equation with a singular coefficient. The uniqueness of a solution is proved with the use of an extremum principle. The proof of the existence uses the method of integral equations.     

The Problem With Missing Shift Condition for the Gellerstedt Equation With a Singular Coefficient

For the Gellerstedt equation with singular coefficient we prove theorems of uniqueness and existence of solution to the problemwith the missing shift condition on the boundary characteristics and the Frankl type condition on the degeneration segment of the equation.     

On a problem with three variants of shift conditions for mixed type equation

For mixed type equation we study a problem where shift conditions are given in inner characteristics, on degeneration line, and on the boundary of elliptic domain. 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 a nonlocal boundary condition on the characteristic for a class of equations of mixed type

We study the boundary-value problem with a nonlocal boundary condition on the characteristic for a class of equations of mixed type. The unique solvability of the problem is proved.