On a boundary value problem with shift for an equation of mixed type in an unbounded domain

We study the unique solvability of a problem with shift for an equation of mixed type in an unbounded domain. We prove the uniqueness theorem under inequality-type constraints for known functions for various orders of the fractional differentiation operators in the boundary condition. The existence of a solution is proved by reduction to a Fredholm equation of the second kind, whose unconditional solvability follows from the uniqueness of the solution of the problem.     

Solvability of a nonlocal problem for a loaded parabolic-hyperbolic equation

For a mixed-type equation we study a problem with generalized fractional integrodifferentiation operators in the boundary condition. We prove its unique solvability under inequality-type conditions imposed on the known functions for various orders of fractional integrodifferentiation operators. We prove the existence of a solution to the problem by reducing the latter to a fractional differential equation.     

Integrability of Solutions to Mixed Stochastic Differential Equations

We prove that the standard conditions for the unique solvability of a mixed stochastic differential equation guarantee that its solution possesses finite moments. We also give conditions supplying the existence of exponential moments. For a special equation whose coefficients do not satisfy the linear growth condition, we prove the integrability of its solution.     

A problem with a nonlocal,with respect to time,condition for multidimensional hyperbolic equations

We study a boundary-value problem for a hyperbolic equation with a nonlocal with respect to time-variable integral condition. We obtain sufficient conditions for unique solvability of the nonlocal problem. The proof is based on reduction of the nonlocal first-type condition to the second-type one. This allows to reduce the nonlocal problem to an operator equation. We show that unique solvability of the operator equation implies the existence of a unique solution to the problem.     

The Impedance Boundary Value Problem for the Helmholtz Equation in a Half-Plane

We prove unique existence of solution for the impedance (or third) boundary value problem for the Helmholtz equation in a half-plane with arbitrary L_∞ boundary data. This problem is of interest as a model of outdoor sound propagation over inhomogeneous flat terrain and as a model of rough surface scattering. To formulate the problem and prove uniqueness of solution we introduce a novel radiation condition, a generalization of that used in plane wave scattering by one-dimensional diffraction gratings. To prove existence of solution and a limiting absorption principle we first reformulate the problem as an equivalent second kind boundary integral equation to which we apply a form of Fredholm alternative, utilizing recent results on the solvability of integral equations on the real line in [5]. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

A problem with operators of fractional differentiation in boundary condition for mixed-type equation

We consider a question on unique solvability of a boundary-value problem with fractional derivatives for a mixed-type equation of second order. We prove first a uniqueness theorem. The existence theorem is proved by means of reduction to Fredholm equation of the second kind, and its unconditional solvability follows from the uniqueness of solution.     

Nonlocal problem for an equation of mixed type in a domain whose elliptic part is a half-strip

For the Gellerstedt equation, we study a problem with shift in a domain whose elliptic part is an infinite half-strip. By using generalized fractional differentiation operators, we specify a linear combination that relates the value of the solution on characteristics of the equation with the value of the solution and its derivative on the parabolic degeneration line. We prove the unique solvability of this problem.     

Many-dimensional Dirichlet and Tricomi problems for one class of hyperbolic-elliptic equations

For the generalized many-dimensional Lavrent’ev-Bitsadze equation, we prove the unique solvability of the Dirichlet and Tricomi problems. We also establish the existence and uniqueness of a solution of the Dirichlet problem in the hyperbolic part of a mixed domain.     

On the Unique Solvability of an Inverse Problem for Parabolic Equations under a Final Overdetermination Condition

We consider the unique solvability of the inverse problem of determining the right-hand side of a parabolic equation with the leading coefficient depending on time and space variables under a final overdetermination condition. We obtain two types of conditions that are sufficient for the local solvability of the inverse problem and also prove the so-called Fredholm solvability of the inverse problem under study.     

An analog of the Tricomi problem with a nonlocal integral conjugate condition

We prove the unique solvability of an analog of the Tricomi problem for an elliptic-hyperbolic equation with a nonlocal integral conjugate condition on the characteristic line.     

Proof of the solvability of a system of partial differential equations in the nonlinear theory of shallow shells of Timoshenko type

We study the solvability of a system of second-order partial differential equations under given boundary conditions. To prove the existence of a solution of the system, we reduce it to a single nonlinear partial differential equation whose solvability is proved with the use of the contraction mapping principle.     

The solvability of a coupled thermoviscoelastic contact problem with small Coulomb friction and linearized growth of frictional heat

A coupled thermoviscoelastic frictional contact problem is investigated. The contact is modelled by the Signorini condition for the displacement velocities and the friction by the Coulomb law. The heat generated by friction is described by a non‐linear boundary condition with at most linear growth. The weak formulation of the problem consists of a variational inequality for the elasticity part and a variational equation for the heat conduction part. In order to prove the existence of a solution to this problem we first use an approximation of the Signorini condition by the penalty method. The existence of a solution for the approximate problem is shown using the fixed‐point theorem of Schauder. This theorem is applied to the composition of the solution operator for the contact problem with given temperature field and the solution operator for the heat equation problem with known displacement field. To obtain this proof, the unique solvability of both problems is necessary. Due to this reason it is necessary to introduce the penalty method. While the penalized contact problem has a unique solution, this is not clear for the original contact problem. The solvability of the original frictional contact problem is verified by an investigation of the limit for vanishing penalty parameter. Copyright © 1999 John Wiley & Sons, Ltd.     

Boundary-value problem with saigo operators for mixed type equation of the third order with multiple characteristics

We study a boundary-value problem with shift formixed-type equation of the third order. In the hyperbolic field boundary condition contains a linear combination of generalized operators of fractional integro-differentiation. We prove a unique solvability of the problem.     

Nonlocal problem with integral conditions for a system of hyperbolic equations in characteristic rectangle

We consider a nonlocal problem with integral conditions for a system of hyperbolic equations in rectangular domain. We investigate the questions of existence of unique classical solution to the problemunder consideration and approaches of its construction. Sufficient conditions of unique solvability to the investigated problem are established in the terms of initial data. The nonlocal problem with integral conditions is reduced to an equivalent problem consisting of the Goursat problem for the system of hyperbolic equations with functional parameters and functional relations. We propose algorithms for finding a solution to the equivalent problem with functional parameters on the characteristics and prove their convergence. We also obtain the conditions of unique solvability to the auxiliary boundary-value problem with an integral condition for the system of ordinary differential equations. As an example, we consider the nonlocal boundary-value problem with integral conditions for a two-dimensional system of hyperbolic equations.     

Boundary-value problems for a hyperbolic equation with nonlocal conditions of the I and II kind

In this paper we consider two initial-boundary value problems with nonlocal conditions. The main goal is to propose a method for proving the solvability of nonlocal problems with integral conditions of the first kind. The proposed method is based on the equivalence of a nonlocal problem with an integral condition of the first kind and a nonlocal problem with an integral condition of the second kind in a special form. We prove the unique existence of generalized solutions to both problems.     

Problem on small motions of ideal rotating relaxing fluid

We study an evolution problem on small motions of the ideal rotating relaxing fluid in bounded domains. We begin from the problem posing. Then we reduce the problem to a second-order integrodifferential equation in a Hilbert space. Using this equation, we prove a strong unique solvability problem for the corresponding initial-boundary value problem.     

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.     

On the inverse problem of determining the right-hand side of a parabolic equation under an integral overdetermination condition

We study the unique solvability of the inverse problem of determining the righthand side of a parabolic equation whose leading coefficient depends on both the time and the spatial variable under an integral overdetermination condition with respect to time. We obtain two types of condition sufficient for the local solvability of the inverse problem as well as study the so-called Fredholm solvability of the inverse problem under consideration.Translated from Matematicheskie Zametki, vol. 77, no. 4, 2005, pp. 522–534.Original Russian Text Copyright © 2005 by V. L. Kamynin.This revised version was published online in April 2005 with a corrected issue number.     

On perturbations of abstract fractional differential equations by nonlinear operators

We prove the unique solvability of a Cauchy-type problem for an abstract parabolic equation containing fractional derivatives and a nonlinear perturbation term. The result is applied to establish the solvability of the inverse coefficient problem for a fractional-order equation.     

Existence of a Global Classical Solution of One Problem Arising in Combustion Theory

We consider a multidimensional free-boundary problem for a parabolic equation that arises in combustion theory. We prove the existence of a global classical solution. The idea of the method is as follows: first, we perform the differential–difference approximation of the problem and establish its solvability; then we prove uniform estimates and perform a limit transition.