**
**### Mathematics

1. Abasheeva N. L.

A linear inverse problem for a mixed type operator-differential equation with a parameter
**Abstract.** We study the inverse problem

Bu

_{t} + pLu = ϕ(t) + f(t, p), u(0, p) = u(T, p) = 0.

The operators

B, L are selfadjoint in the Hilbert space

E and the spectrum of the operator L is semibounded. The unique solvability of this problem is proved with using a series expansion in eigen and associated elements of the pencil L - λB.

**Keywords:** : inverse problem, mixed type equation.

2. Egorov I. E.

On fredholm solvability of Vragov boundary value problem for a mixed even-order equation
**Abstract.** We consider the boundary value problem of V. N. Vragov for mixed type equations of even order with elliptic operator in space variables. We prove the generalized solvability, dense solvability, uniqueness of generalized solutions and Fredholm solvability of the boundary value problem in the corresponding Sobolev spaces.

**Keywords:** mixed type equation, Fredholm solvability, boundary value problem, generalized solution, inequality, evaluation, operator equation.

3. Kozhanov A. I.

Inverse problems of recovering the right-hand side of a special type of parabolic equations
**Abstract.** We study the solvability of new inverse problems of finding a solution of some parabolic equation along with the unknown external source (the right-hand side) of a special type. The existence and uniqueness theorems for regular solutions are proved. The considered problems can be treated as generalizations known in the theory of parabolic equations inverse problems with final and integral overdetermination.

**Keywords:** parabolic equation, linear inverse problem, final or integral overdetermination, unknown coefficient of a special type, existence, uniqueness.

4. Pyatkov S. G. and Rotko V. V.

Recovering a source function in a one-dimensional parabolic equation with dead zones taking into account
**Abstract.** We examine the question of well-posedness in the Sobolev spaces of an inverse problem of determining a source function in a system comprising a parabolic equation and an ordinary differential equation. The overdetermination conditions are the values of concentration of an admixture at separate points. We prove existence and uniqueness of solutions to the problem.

**Keywords:** parabolic equation, inverse problem, heat-and-mass transfer, boundary value problem, source function.

5. Romanova E. A. and Fedorov V. E.

Resolving operators of a linear degenerate evolution equation with Caputo derivative. the sectorial case
**Abstract.** Unique solvability of the Cauchy problem for an equation in a Banach space with degenerate operator at the fractional Caputo derivative is studied. Previously found conditions of the existence of analytic in a sector resolving operators family for the equation with the degeneration on the kernel of the operator at the derivative is used. The form of the resolving operators is established. Under the satisfied conditions, the existence is shown for the unique solution of the Cauchy problem to the researched equation with initial data from the complement of the kernel of the operator at the derivative and the solution is presented using the resolving operators. The obtained results are applied to studying the linearized quasistationary time-fractional order system of the phase field equations.

**Keywords:** degenerate evolution equation, fractional Caputo derivative, analytic in a sector resolving operators family, Cauchy problem, initial boundary value problem, system of partial differential equations.

6. Tikhonova I. M.

Application of the stationary galerkin method to the first boundary value problem for a mixed high-order equation
**Abstract.** We consider the first boundary value problem for a mixed even-order equation and construct an approximate solution using the stationary Galerkin method. The existence of a regular solution for the boundary value problem is proved under certain conditions on coefficients of the equation. We obtain the error estimate of the Galerkin method.

**Keywords:** Galerkin method, mixed type equation, regular solution, a priori estimate.

7. Fedorov V. E. and Tikhonova I. M.

The stationary Galerkin method for a boundary value problem for a mixed second-order equation
**Abstract.** We prove the existence of the unique regular solution for the boundary value problem for the mixed type second-order equation in the Sobolev space. The stationary Galerkin method is applied, for which the error estimate is obtained using eigenvalues of the spectral problem for the Laplace equation in the variables

x ∈ R

^{n} and t.

**Keywords:** mixed type equation, boundary value problem, a priori estimate, stationary Galerkin method, error.

8. Khubiev K. U.

A problem with an integral condition in the hyperbolic part for a characteristically loaded hyperbolic-parabolic equation
**Abstract.** We prove the uniqueness and existence of solutions of a model characteristic ally loaded mixed hyperbolic-parabolic equation. The Tricomi method is applied for establishing the solution uniqueness and the existence is proved with the use of the integral equation method.

**Keywords:** : loaded equation, mixed type equation, hyperbolic-parabolic equation, nonlocal problem, integral condition.

### Mathematical modeling

9. Zhiltsov A. V.

Modified duality scheme for numerical simulation of the contact between elastic bodies
**Abstract.** We consider the problem of unilateral contact of two elastic bodies, a static problem in displacements. Bodies are influenced by volume and surface forces, while frictional forces are absent. Justification of use of the modified Lagrangian functionals method is given. We provide the results of numerical calculations.

**Keywords:** Lagrangian functional, finite element method, duality scheme, elastic contact.