A spectral solution method for a boundary-value problem for a mixed-type equation with two singularity lines

We consider a boundary-value problem for a mixed-type equation with two perpendicular singularity lines given in a domain whose elliptic part is a rectangle, while the hyperbolic one is a vertical half-strip. This problem differs from the Dirichlet one by the fact that at the left boundary of the rectangle and the half-strip we specify the vanishing order of the desired function rather than its value. We find a solution to the problem by a spectral method with the use of the Fourier–Bessel series and prove the uniqueness of the solution. We substantiate the uniform convergence of the corresponding series under certain requirements to the problem statement.     

Classical solution of the first mixed problem for the Klein-Gordon-Fock equation in a half-strip

We consider a classical solution of the first boundary value problem for the Klein-Gordon-Fock equation in a half-strip in the one-dimensional case. We prove the existence and uniqueness of the classical solution under certain smoothness conditions and matching conditions for the given functions. To solve the problem, one should solve Volterra integral equations of the second kind.     

Initial-boundary value problems in a half-strip for two-dimensional Zakharov–Kuznetsov equation

Initial-boundary value problems in a half-strip with different types of boundary conditions for two-dimensional Zakharov–Kuznetsov equation are considered. Results on global existence, uniqueness and long-time decay of weak and regular solutions are established.     

A Mixed Problem for a Hyperbolic System on the Plane with Delay in the Boundary Conditions

We examine well-posedness of the boundary value problem in a half-strip for a first-order linear hyperbolic system with delay (lumped and distributed) in the boundary conditions. In the case of the negative real parts of the eigenvalues of the corresponding spectral problem we prove a time uniform estimate for a solution to the homogeneous problem which enables us to justify the linearization principle for analysis of stability of stationary solutions to the nonlinear problem.     

Classical solution of the first mixed problem for a third-order hyperbolic equation with the wave operator

We study the classical solution of the boundary value problem for a third-order hyperbolic equation defined on the plane in a half-strip with the Cauchy conditions on the lower base of the domain and with the Dirichlet conditions on the lateral boundary. By the method of characteristics, in the case of two independent variables, we find the solution of the problem in closed form and prove its uniqueness.     

Asymptotic behavior of solutions of the heat equation in domains with curvilinear boundaries

The first boundary value problem with null boundary conditions is studied for the one-dimensional heat equation in a domain with curvilinear lateral boundaries. It is proved that for domains sufficiently close to a half-strip, solutions with a sign change for any time value, for unbounded time increase, tend to zero faster than positive solutions.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 14, pp. 57–65, 1989.     

Boundary value problems for a nonstrictly hyperbolic equation of the third order

We study classical solutions of boundary value problems for a nonstrictly hyperbolic third-order equation. The equation is posed in a half-strip and a quadrant of the plane of two independent variables. The Cauchy conditions are posed on the lower boundary of the domain, and the Dirichlet conditions are posed on the lateral boundaries. By using the method of characteristics, we find the analytic form of the solution of considered problems. The uniqueness of the solutions is proved.     

Solvability of Nonlocal Boundary Value Problems for an Equation of Mixed Type with Various Boundary Conditions

Two boundary value problems in which one of the conditions is nonlocal and contains a real parameter are studied for an equation of mixed type in a half-strip. Sufficient conditions for the unique solvability of these problems are obtained under some restrictions on the parameter.     

Mixed problem for the Kawahara equation in a half-strip

We prove the global well-posedness of the mixed problem for the Kawahara equation in a half-strip under natural conditions on the boundary data.     

An Initial Boundary-Value Problem in a Half-Strip for the Korteweg–De Vries Equation in Fractional-Order Sobolev Spaces

Abstract

An initial boundary-value problem in a half-strip with one boundary condition for the Korteweg–de Vries equation is considered and results on global well-posedness of this problem are established in Sobolev spaces of various orders, including fractional. Initial and boundary data satisfy natural (or close to natural) conditions, originating from properties of solutions of a corresponding initial-value problem for a linearized KdV equation. An essential part of the study is the investigation of special solutions of a “boundary potential” type for this linearized KdV equation.     

Increasing smoothness of solutions to a hyperbolic system on the plane with delay in the boundary conditions

Under consideration is a mixed problem in the half-strip Π = {(x, t): 0 < x < 1, t > 0} for a first order homogeneous linear hyperbolic system with delay in t in the boundary conditions. We study the behavior of the Laplace transform of a solution to this problem for the large values of the complex parameter. The boundary conditions are found under which the smoothness of a solution to the corresponding mixed problem increases with t.     

On possibilities of the method of homogeneous solutions in the mixed problem of elasticity theory for a half-strip

The mixed plane boundary value problem for an elastic half-strip under kinematic loading of its endface and lateral sides free of force loads is examined. An asymptotic is obtained for the coefficients of the series expansion of the displacement vector in homogeneous solutions. Regularization of the series in homogeneous solutions is proposed which would permit computation of the stress field on the half-strip enface.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 18, pp. 3–8, 1987.     

Decay mode solution of nonlinear boundary–initial value problems for the cylindrical (spherical) Boussinesq–Burgers equations

The major target of this paper is to construct new nonlinear boundary–initial value problems for Boussinesq–Burgers Equations, and derive the solutions of these nonlinear boundary–initial value problems by the simplified homogeneous balance method. The nonlinear transformation and its inversion between the Boussinesq–Burgers Equations and the linear heat conduction equation are firstly derived; then a new nonlinear boundary–initial value problem for the Boussinesq–Burgers equations with variable damping on the half infinite straight line is put forward for the first time, and the solution of this nonlinear boundary–initial value problem is obtained, especially, the decay mode solution of nonlinear boundary–initial value problem for the cylindrical (spherical) Boussinesq–Burgers equations is obtained.     

Integral representation of the solution of a Gellerstedt problem

The solution of the Gellerstedt problem for the Lavrent’ev-Bitsadze equation was earlier obtained in the form of a series for the case of a half-strip in the elliptic part of the domain and of nonzero boundary data on the characteristics in the hyperbolic part of the domain. In the present paper, we give an integral representation of the solutions of this problem as well as some related problems.     

Boundary control of a hyperbolic system with one space variable

We consider a mixed problem in a half-strip for a hyperbolic system with one space variable and with constant coefficients. The control problem is to find boundary conditions ensuring that the system has a given state vector at a given instant of time. We study whether the problem is asymptotically solvable, i.e., whether there exists a sequence of boundary conditions such that the corresponding sequence of final state vectors uniformly converges to the given vector. We reduce the construction of a family of such sequences of boundary conditions with a function parameter to the solution of a Fredholm integral equation of the second kind and prove a sufficient condition for its unique solvability in terms of the problem data.     

Neumann initial–boundary value problem for Benjamin–Ono equation

We consider the Neumann initial–boundary value problem for Benjamin–Ono equation on a half-line. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial–boundary value problem and the asymptotic behavior of solutions for large time.     

Mixed initial–boundary value problem for the Benjamin–Ono equation

We consider the mixed initial–boundary value problem for the Benjamin–Ono equation on a half-line. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial–boundary value problem and the asymptotic behavior of solutions for large time.     

On the Uniqueness of the Solution of the Inverse Sturm–Liouville Problem with Nonseparated Boundary Conditions on a Geometric Graph

For the first time, the inverse Sturm–Liouville problem with nonseparated boundary conditions is studied on a star-shaped geometric graph with three edges. It is shown that the Sturm–Liouville problem with general boundary conditions cannot be uniquely reconstructed from four spectra. Nonseparated boundary conditions are found for which a uniqueness theorem for the solution of the inverse Sturm–Liouville problem is proved. The spectrum of the boundary value problem itself and the spectra of three auxiliary problems are used as reconstruction data. It is also shown that the Sturm–Liouville problem with these nonseparated boundary conditions can be uniquely recovered if three spectra of auxiliary problems are used as reconstruction data and only five of its eigenvalues are used instead of the entire spectrum of the problem.     

Asymptotic expansions for hyperbolic–parabolic systems with a small parameter

In this paper we consider initial-boundary value problems for systems with a small parameter ?. The problems are mixed hyperbolic–parabolic when ? > 0 and hyperbolic when ? = 0. Often the solution can be expanded asymptotically in ? and to first approximation it consists of the solution of the corresponding hyperbolic problem and a boundary layer part. We prove sufficient conditions for the expansion to exist and give estimates of the remainder. We also examine how the boundary conditions should be choosen to avoid O(1) boundary layers.     

Solution of a Gellerstedt problem for the Lavrent’ev-Bitsadze equation

We write out the solution of the Gellerstedt problem for the Lavrent’ev-Bitsadze equation in the form of a series for the case in which the elliptic part of the domain is a half-strip and the boundary data are nonzero only on the characteristics in the hyperbolic parts of the domain. We obtain new results on the basis property, completeness, and minimality of the system of sines with discontinuous phase used in the series representation of the solution of the Gellerstedt problem. We prove the uniform convergence and justify the possibility of term-by-term differentiation of the series.