Search

In this paper, we consider the problems on the solvability and constructing solutions of one nonlocal boundary-value problem for the multidimensional, fourth-order, integro-differential Benney–Luke equation with degenerate kernel and spectral parameters. For various values of spectral parameters, necessary and sufficient conditions of the existence of a solution are obtained. The Fourier series for solutions of the problem corresponding to various sets of spectral parameters are obtained. For regular values of spectral parameters, the absolute and uniform convergence of the series and the possibility of their termwise differentiation with respect to all variables are proved. The problem is also examined studied for cases of irregular values of spectral parameters. [ABSTRACT FROM AUTHOR]

In this paper, we study a boundary-value problem with discontinuous conjugation conditions on the line of type changing for a model equation of mixed hyperbolic-parabolic type with degeneration of order in the hyperbolicity domain. In the parabolic domain, the equation is the fractional diffusion equation, whereas in the hyperbolic domain it is the loaded one-speed transfer equation. We prove the uniqueness and existence theorem and propose an explicit solution of the problem in the parabolic and hyperbolic domains. [ABSTRACT FROM AUTHOR]

In this paper, a theorem on a priori estimates of solutions of the Tricomi problem for a second-order, mixed-type equation with the Gellerstedt operator in the hyperbolicity domain is proved. The a priori estimate obtained implies the uniqueness of the regular solution of the problem considered. [ABSTRACT FROM AUTHOR]

We examine the unique generalized solvability of the mixed problem for a higher-order nonlinear pseudoparabolic equation with two parameters in mixed derivatives. Using the Fourier variable separation method, we reduce the problem to a countable system of nonlinear integral equations whose unique solvability can be proved by the method of successive approximations. We prove the continuous dependence of a generalized solution to the mixed problem on the initial functions and the positive parameters. [ABSTRACT FROM AUTHOR]

Using the Fourier method of separation of variables, we examine the classical solvability and construct solutions of a nonlocal inverse boundary-value problem for the fourth-order Benney–Luke integro-differential equation with degenerate kernel. We prove the criterion of the unique solvability of the inverse boundary-value problem and examine the stability of solutions with respect to the recovery function. [ABSTRACT FROM AUTHOR]

In this paper, we study a boundary-value problem for a loaded hyperbolic-parabolic equation with degeneracy of order in the domain of its hyperbolicity. The existence and uniqueness theorems for solutions of this problem are proved and a representation of solutions is obtained. [ABSTRACT FROM AUTHOR]

We consider a nonlocal boundary-value problem for a third-order parabolic-hyperbolic equation with degeneracy of type and order in the domain of hyperbolicity, containing second-order derivatives in the boundary conditions. Sufficient conditions of the unique solvability of the problem are obtained. The Tricomi method is used to prove the uniqueness theorem for a solution. The solution of the problem is expressed in the explicit form. [ABSTRACT FROM AUTHOR]

In this paper, the classical solvability of a nonlocal boundary-value problem for a three dimensional, homogeneous, fourth-order, pseudoelliptic integro-differential equation with degenerate kernel is proved. The spectral Fourier method based on the separation of variables is used and a countable system of algebraic equations is obtained. A solution is constructed explicitly in the form of a Fourier series. The absolute and uniform convergence of the series obtained and the possibility of termwise differentiation of the solution with respect to all variables are justified. A criterion of the unique solvability of the problem considered is ascertained. [ABSTRACT FROM AUTHOR]

In this paper, we examine inverse problems for initial conditions for the wave and telegraph equations and state uniqueness criteria. Solutions of these problems are constructed in the series form. In the proof of uniform convergence of these series, the problem on small denominators appears. We prove estimates of small denominators separated from zero and obtain asymptotics that allow one to justify the convergence in the class of regular solutions. [ABSTRACT FROM AUTHOR]

We consider initial-boundary-value problems for three classes of inhomogeneous degenerate equations of mixed parabolic-hyperbolic type: mixed-type equations with degenerate hyperbolic part, mixed-type equations with degenerate parabolic part, and mixed-type equations with power degeneration. In each case, we state a criterion of uniqueness of a solution to the problem. We construct solutions as series with respect to the system of eigenfunctions of the corresponding one-dimensional spectral problem. We prove that the uniqueness of the solution and the convergence of the series depend on the ratio of sides of the rectangular from the hyperbolic part of the mixed domain. In the proof of the existence of solutions to the problem, small denominators appear that impair the convergence of series constructed. In this connection, we obtain estimates of small denominators separated from zero and the corresponding asymptotics, which allows us, under certain conditions, to prove that the solution constructed belongs to the class of regular solutions. [ABSTRACT FROM AUTHOR]