In this paper, we study the inverse problem of determining the time-dependent coefficient in a one-dimensional fractional order equation with initial-boundary conditions and over-determination conditions. Using the Fourier method, this problem is reduced to equivalent integral equations. Then, using estimates of the Mittag--Leffler function and the method of successive approximations, an estimate of the solution to the direct problem is obtained through the norm of the unknown coefficient, which will be used in the study of the inverse problem. The inverse problem is reduced to an equivalent integral equation. To solve this equation, the contraction mapping principle is used. The results of local existence and uniqueness are proved.

обратная задача; адвекция-дисперсия; дробная производная; функция Мит\-таг-Леффлера; $H$-функция.

When solving many problems in the theory of approximate integration and differential equations, it is the correct choice of spaces that is the key to success. A very clearly chosen approach was demonstrated in the famous works of S.L. Sobolev on the polyharmonic equation. S.L. Sobolev posed and solved by the variational method the first boundary value problem for the equation ${{\Delta }^{\ell }}u=f$ with boundary conditions on surfaces of various dimensions.

Problems of optimization of approximate integration formulas consist in minimizing the norm of the error functional of the formula on selected normalized spaces, and most of them are considered in the Sobolev space.

Until now, we have considered cubature formulas, with the help of which a definite integral of a function is approximately calculated when the values of this function at individual points of the nodes of the cubature formula are unknown. But more general cubature formulas are possible, which include both the values of the function and the values of its derivatives of one order or another.

If we know not only the values of a function at some points of the $n-$ dimensional unit sphere, but also the values of its derivatives of one order or another, then it is natural that if all this data is used correctly, we can expect a more accurate result than if we use only function values.

This paper examines cubature formulas, which require special attention to the construction of the most economical formulas; according to N.S. Bakhvalov, such formulas are called practical.

\selectlanguage{english} A homogeneous linear conjugate problem on a closed contour for a two-dimensional piecewise analytic vector is considered. Each solution of the problem is given a pair of functions, which are relations of limit values on the contour of the corresponding components of this solution. The relations connecting the elements of the $H$-continuous matrix-functions of the problem, providing the existence of its two solutions, for which the corresponding components of the pair differ by rational multipliers, and the problem itself admits a solution in closed form, are specified.

The paper discusses two approaches to defining the computability of numberings of families of total functions. We consider both the classical definition of computable numbering of a family of computable functions, according to which the number of a function in this numbering effectively provides its G\"odel number, and, expanding the previous one, a definition based on the uniform application of the concept of the left-c.e. element of Baire space. The main question studied in the paper is the possibility of generating all computable numberings of a family by the closure with respect to the reducibility of infinite direct sums of uniform sequences of its single-valued, positive, and minimal numberings.

Curved pipe systems are widely used in mechanical engineering, the nuclear industry, offshore oil production, and aerospace engineering. The purpose of the work is to study small vibrations of a viscoelastic helical spring. Small vibrations of a thin curved rod, the elastic line of which is a flat curve and one of the main directions of the cross-section of which lies in the plane of the curve, break down into two types: vibrations with displacements in the plane of the curve and with displacements perpendicular to the plane of the curve. The viscoelastic properties of materials are taken into account using complex elastic moduli. Asymptotic expansions are constructed for the eigenfunctions and eigenfrequencies corresponding to both types of oscillations of a repeatedly twisted flat spiral spring with fixed ends. A technique has been developed for obtaining resolving equations corresponding to the boundary conditions.

Let $\mathfrak{P}$ be a~non-empty set of~primes. We prove that any $\mathfrak{P}$\nobreakdash-bounded nilpotent group is~$\mathfrak{P}$\nobreakdash-potent, and~the~tree product~$T$ of~a~finite number of~$\mathfrak{P}$\nobreakdash-bounded nilpotent groups with~proper locally cyclic edge subgroups is~residually a~finite $\mathfrak{P}$\nobreakdash-group if and~only if any vertex group of~$T$ has no~$\mathfrak{P}^{\prime}$\nobreakdash-torsion and~any edge subgroup of~$T$ is~$\mathfrak{P}^{\prime}$\nobreakdash-isolated in~the~vertex group containing~it. We prove also that the~tree product of~a~finite number of~groups with~locally cyclic edge subgroups is~residually a~finite $p$\nobreakdash-group if~all its vertex groups have this property and~any edge subgroup is~separable in~the~corresponding vertex group by~the~class of~finite $p$\nobreakdash-groups.

In this work, we study a family of multi-parameter polynomial differential systems of degree eleven. We prove that the considered family has an invariant algebraic curve, which is given explicitly. Subsequently, we demonstrate the integrability of these systems and derive an explicit expression for a first integral. Moreover, we provide sufficient conditions for the systems to possess two explicitly given limit cycles. The applicability of our results is illustrated by a concrete example.

A typical approximation problem is the interpolation problem. The classical method for solving it is to construct an interpolation polynomial. However, polynomials have a number of disadvantages, such as being a tool for approximating functions with singularities and functions with not very high smoothness. In practice, in order to approximate functions well, instead of constructing a high-degree interpolation polynomial, splines are used, which are very convenient to use.

This paper examines the construction of interpolation splines using the Sobolev method, minimizing the norm in a certain Hilbert space.

For the first time, S.L. Sobolev \cite{6} posed the problem of finding the extremal function for the interpolation formula and calculating the norm of the error functional in the Sobolev space.

In this work, the extremal function of the interpolation formula is found in explicit form in the Sobolev space $W_{2}^{\left( m \right)}\left( {{R}^{n}} \right)$; a function whose generalized derivatives of order $m$ are square integrable. Basically, the problem of constructing optimal interpolation formulas in the space of S.L. Sobolev $\tilde{W}_{2}^{\left( m \right)}\left( {{T}_{1}} \right )$ for $m=4$ is considered.

In this paper a number of extreme problems related to the best polynomial approximation of analytical in a circle $U:=\{z\in\mathbb{C}:|z|<1\}$ functions belonging to the Bergman's space $B_2$ are being solved. The bilateral inequality is proved, which is a generalization of the result of periodic functions $f\in L_{2}$, by M.Sh.Shabozov--G.A.Yusupov obtained for the class $L_{2}^{(r)}[0,2\pi]$-in which $(r-1)$ the derivative of $f^{(r-1)}$ is absolutely continuous, and the derivative of $r $ is order of $f^{(r)}\ in L_{2}$ in the case of a polynomial approximation of $f\in \mathcal{A}(U)$ belonging to $B_{2}^{(r)}(U)$.

A number of cases are given when the bilateral inequality turns into equality. For some classes of functions belonging to $B_2$, the exact values of the known $n$-diameters are found, and the problem of joint approximation of functions and their intermediate derivatives is solved.