A problem with an analog of Frankl condition on the characteristics for gellerstedt equation with singular coefficient

We investigate the problem with an analog of Frankl condition on boundary characteristics for generalized Tricomi equation. We prove that the formulated problem is correct.     

Problem with Frankl and Bitsadze-Samarskii conditions on the degeneration line and on parallel characteristics for the gellerstedt equation with a singular coefficient

We study the well-posedness of a problem for the Gellerstedt equation with a singular coefficient and with the Frankl and Bitsadze-Samarskii conditions on the degeneration line and on parallel characteristics.The uniqueness of the solution of the considered problem is proved with the use of the extremum principle, and the existence of the solution of the problem is justified with the use of the theories of singular integral equations, Wiener-Hopf equations, and Fredholm integral equations.     

The Bitsadze-Samarskii problem with the Frankl condition on the degeneration line for a mixed-type equation with singular coefficient

We consider the Bitsadze-Samarskii problem with the Frankl condition for the Gellerstedt equation with a singular coefficient and prove its well-posedness.     

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.     

Keldysh problem for mixed type equation with strong characteristic degeneration and singular coefficient

We study a mixed-type equation of the second kind with a singular coefficient. With the help of the spectral analysis method we establish a uniqueness criterion for a solution of the problem with incomplete boundary data. The solution represents the sum of the Fourier–Bessel series. The substantiation of its uniform convergence is based on an estimate of the separation from zero of the small denominator with the corresponding asymptotic behavior. This allows us to prove the convergence of the series in the class of regular solutions.     

A nonlocal boundary-value problem for the gellerstedt equation

We consider the boundary-value problem for the Gellerstedt equation

Tricomi problem for a mixed parabolic-hyperbolic equation in a rectangular domain

Mathematical Notes - For a class of equations of mixed type, we study an analog of the Dirichlet problem satisfying the conjugation conditions on the line of change of the type. We establish a...     

Tricomi problem for a nonlinear equation of mixed type with functional delay and advance

We study a boundary value problem for a nonlinear equation of mixed type with the Lavrent’ev–Bitsadze operator in the principal part and with functional delay and advance in lower-order terms. The general solution of the equation is constructed. The problem is uniquely solvable.     

On the solvability of the Tricomi problem with a generalized Frankl transmission condition

We study the solvability of the Tricomi problem for the Lavrent’ev-Bitsadze equation with mixed boundary conditions in the elliptic part of the domain. On the type change line of the equation, the solution gradient is subjected to a condition that is usually referred to as the generalized Frankl transmission condition. We show that the inhomogeneous Tricomi problem either has a unique solution or is conditionally solvable and the homogeneous problem has only the trivial solution. We write out an integral representation of the solution of this problem.     

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.     

On the conditional solvability of the Tricomi problem with mixed boundary conditions

In the present paper, we consider the Tricomi problem with mixed boundary conditions. One of these conditions specifies a directional derivative with constant inclination angle. We show that the problem is either conditionally solvable or has a unique solution depending on the inclination angle.     

Cauchy problem for a second-order hyperbolic operator-differential equation with a singular coefficient

In a Hilbert space, we study the well-posedness of the Cauchy problem for a second-order operator-differential equation with a singular coefficient.     

On a problem with deviation from a characteristic for the gellerstedt equation

We prove the unique regular solvability of a problem with deviation from a characteristic for the Gellerstedt equation.     

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.     

Uniqueness of solution of Tricomi problem for equation of mixed type

Mathematical Notes -     

A nonlocal problem for a mixed-type equation with a singular coefficient in an unbounded domain

In this paper we study a nonlocal problem for a mixed-type equation in a domain whose elliptic part is the first quadrant of the plane and the hyperbolic part is the characteristic triangle. With the help of the method of integral equations and the principle of extremum we prove the unique solvability of the considered problem.     

Interior-boundary value problem with Saigo operators for the gellerstedt equation

For an equation of mixed elliptic-parabolic type, we consider an interior-boundary value problem in which the Dirichlet condition is posed on the elliptic part of the boundary and a point condition relating generalized derivatives and fractional integrals with the Gauss hypergeometric function of the values of the solution on the characteristics to the values of the solution and its derivative on the parabolic degeneration line is posed on the hyperbolic part.     

A dirichlet problem for an equation of mixed type with a discontinuous coefficient

Summary A Dirichlet problem for an equation of mixed type with a discontinuous coefficient is considered for a rectangle. In one part of the rectangle, Laplace's equation is specified while the wave equation is specified in the other part. The questions of existence, non-existence, uniqueness and non-uniqueness are discussed.     

A three-dimensional analog of the Tricomi problem for a parabolic-hyperbolic equation

For a parabolic-hyperbolic equation, we study the three-dimensional analog of the Tricomi problem with a noncharacteritic plane on which the type of the equation changes. The uniqueness of the solution to the problem is proved by the method of a priori estimates, and the existence of a solution is reduced to the existence of a solution to a Volterra integral equation of the second kind.