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.     

On well-posedness of the mixed problem for nonstrictly hyperbolic operators

By using a new concept of polynomial pencil, this paper obtains new sufficient conditions for the well-posedness of the Cauchy problem for a nonstrictly hyperbolic differential operator. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 24, Dynamical Systems and Optimization, 2005.     

On a boundary-value problem for a functional-differential hyperbolic equation of the third order

We consider a boundary-value problem for a functional-differential factorized hyperbolic equation of the third order not investigated earlier. We prove the existence and uniqueness of its solution.     

Boundary-value problem for a mixed parabolic-hyperbolic equation of the third order

The unique solvability of a certain boundary-value problem is proved for a mixed parabolic-hyperbolic type equation of the third order.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 1, pp. 20–29, January, 1995.     

The Cauchy problem for nonstrictly second order hyperbolic equations with nonregular coefficients

In this paper we consider the Cauchy problem for the equation , where the matrix {a _jk(x)} is non-negative, and the first derivatives of the coefficients have a singularity of orderq≥3 att=T>0; under these assumptions, the Cauchy problem is well-posed in all Gevrey classes of indexs<q/(q−1).     

The mixed problem for a degenerate second-order hyperbolic equation

We study conditions for uniqueness and existence of a solution for the mixed problem for a second-order hyperbolic equation that is degenerate at a finite instant of time.Translated fromMatematicheskie Melody i Fiziko-Mekhanicheskie Polya, Issue 34, 1991, pp. 39–42.     

Inverse coefficient problem for a wave equation in a bounded domain

The nonlinear inverse problem for a wave equation is investigated in a three-dimensional bounded domain subject to the Dirichlet boundary condition. Given a family of solutions to the equation defined on a closed surface within the original domain, it is required to reconstruct the coefficient determining the velocity of sound in the medium. The solutions used for this purpose correspond to the acoustic medium perturbations localized in the neighborhood of a certain closed surface. The inverse problem is reduced to a linear integral equation of the first kind, and the uniqueness of the solution to this equation is established. Numerical results are presented.     

First mixed problem for a parabolic difference-differential equation

The first mixed boundary value problem for a parabolic difference-differential equation with shifts with respect to the spatial variables is considered. The unique solvability of this problem and the smoothness of generalized solutions in some cylindrical subdomains are established. It is shown that the smoothness of generalized solutions can be violated on the interfaces of neighboring subdomains. Translated fromMatematicheskie Zametki, Vol. 66, No. 1, pp. 145–153, July, 1999.     

Solution of one boundary-value problem for a hyperbolic equation of the second order

We find conditions for the existence of the classical solution of the boundary-value problem u _tt -u _xx = f(x,t), u(0,t)=u(π, t)=0, u(x, 0)=u(x, 2π).     

The third mixed problem for the Sonin equation in a half space

We consider follwing mixed boundary-value problem:

On boundary-value problem for hyperbolic-type equation of the third order

We formulate some boundary-value problems for a linear third-order equation with hyperbolic operator in the main part and study the unique solvability. Under certain conditions to given functions, using the Riemann method, we obtain an integral representation of solutions. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 4, pp. 591–603, October–December, 2007.     

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...     

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.