A nonlocal boundary-value problem for a fourth-order mixed-type equation

The criterion of uniqueness of a solution of the problem with periodicity and nonlocal and boundary conditions is established by the spectral analysis for a fourth-order mixed-type equation in a rectangular region. When constructing a solution in the form of the sum of a series, we use the completeness in the space L_2, the system of eigenfunctions of the corresponding problem orthogonally conjugate. When proving the convergence of a series, the problem of small denominators arises. Under some conditions imposed on the parameters of the data of the problem and given functions, the stability of the solution is proved.

ON SOME PROBLEMS FOR A LOADED PARABOLIC-HYPERBOLIC EQUATION

Some problems with various boundary conditions for the loaded mixed type equation in rectangular area are studied. The criterion of uniqueness is established and theorems of an existence of solutions to the problems are proved. The solutions are constructed as Fourier series with respect to eigenfunctions of a corresponding one-dimensional problem.

DIRICHLET PROBLEM FOR PULKIN’S EQUATION IN A RECTANGULAR DOMAIN

In the given article for the mixed-type equation with a singular coefficient the first boundary value problem is studied. On the basis of property of completeness of the system of own functions of one-dimensional spectral problem the criterion of uniqueness is established. The solution the problem is constructed as the sum of series of Fourier - Bessel. At justification of convergence of a row there is a problem of small denominators. In connection with that the assessment about apartness of small denominator from zero with the corresponding asymptotic which allows to prove the convergence of the series constructed in a class of regular solutions under some restrictions is given.

Keldysh problem for Pulkin’s equation in a rectangular domain

In this article for the mixed type equation with a singular coefficient Keldysh problem of incomplete boundary conditions is studied. On the basis of property of completeness of the system of own functions of one-dimensional spectral prob- lem the criterion of uniqueness is established. The solution is constructed as the summary of Fourier-Bessel row. At the foundation of the uniform convergence of a row there is a problem of small denominators.Under some restrictions on these tasks evaluation of separation from zero of a small denominator with the corresponding asymptotics was found, which helped to prove the uniform con- vergence and its derivatives up to the second order inclusive, and the existence theorem in the class of regular solutions.

Inverse problems for the heat equation

The inverse problem of finding initial conditions and the right-hand side had been studied for the inhomogeneous heat equation on the basis of formulas for the solution of the first initial-boundary value problem. A criterion of uniqueness of solution of the inverse problem for finding the initial condition was found with Spectral analysis. The right side of the heat equation is represented as a product of two functions, one of which depends on the spatial coordinates and the other from time. In one task, along with an unknown solution is sought factor on the right side, depending on the time, and in another - a factor that depends on the spatial coordinates. For these tasks, we prove uniqueness theorems, the existence and stability of solution.

CORRECTNESS OF THE LOCAL BOUNDARY VALUE PROBLEM IN A CYLINDRICAL DOMAIN FOR ONE CLASS OF MULTIDIMENSIONAL ELLIPTIC EQUATIONS

Correctness of boundary value problems in a plane for elliptical equations has been studied properly using the method of the theory of analytic functions. At investigation of analogous problems, when the number of independent variables is more than two, there arise principle difficulties. Quite good and convenient method of singular integral equations has to be abandoned because there is no complete theory of multidimensional singular integral equations. Boundary value problems for second-order elliptical equations in domains with edges have been studied properly earlier. Explicit classical solutions to Dirichlet and Poincare problems in cylindrical domains for one class of multidimensional elliptical equations can be found in the author’s works. In this article,the author proved that the local boundary value problem, which is the generalization of Dirichet and Poincare problem, has only solution. Besides, the criterion of uniqueness of regular solution is obtained.

The Uniqueness of Solution for a Class of Fractional Order Nonlinear Systems withp-Laplacian Operator

We are concerned with the uniqueness of solutions for a class ofp-Laplacian fractional order nonlinear systems with nonlocal boundary conditions. Based on some properties of thep-Laplacian operator, the criterion of uniqueness for solutions is established.