## Volume 25, # 2

### Mathematics

1. Olga A. Vikhreva On the ﬁrst boundary value problem for a strongly degenerate ordinary diﬀerential equation
Abstract. We consider a particular case of the earlier studied by the author second order degenerate diﬀerential operator with the same assumptions and designations. We focus on the study of the eﬀects associated with the “strong” degeneration. The problem is solved to be used in further researches of formally conjugated (coupled transposition operation) equation and also for obtaining some theorems of existence and uniqueness
for generalized solutions of formally conjugated equations fr om the proved theorem. The use of the following results is reduced to the operator equations in the simplest case. We study existence and uniqueness of the generalized solution of the ﬁrst boundary value problem for the given equation using the operator theory and obtain the generalized solution to the equation in the case connected with “strong” degeneration. The results
will be used in the future for research of equations with model operators which arise in mathematical modeling of various physical processes.

Keywords: Hilbert space, degenerate diﬀerential operator, generalized solution, “weak” degeneration, “strong” degeneration, nonhomogeneous equation, general solution, partial solution, model operator.

2.  Alexandra I. Grigorieva Initial-boundary problem with conjugation conditions for composite-type equations with two breakdown coeﬃcients
Abstract. In this paper we study the solvability of an initial-boundary value problem with conjugation conditions for two nonclassical diﬀerential equations of composite type. We describe the case when the coeﬃcients of each equation under consideration have a discontinuity of the ﬁrst kind at the point zero. The ﬁeld of research is given in the form of a band, due to the presence of a discontinuity point consisting of two subregions. Thus, the investigated equations are considered in two diﬀerent areas. To prove the existence and uniqueness of regular solutions (which have all the generalized derivatives entering into the equations) of the initial-boundary value problem, we use the method of continuation with respect to a parameter, which has a wide application in the theory of boundary value problems. Using the maximum principle, the presence of all necessary
a priori estimates for the solutions of the problem being studied is established.

Keywords: composite type equation, initial-boundary problem, breakdown coeﬃcients, conjugation problem, regular solution, existence and uniqueness, a priori estimate.

3. Natalia V. Zhukovskaya and Sergey M. Sitnik Euler type diﬀerential equations of fractional order
Abstract. Using the direct and inverse Mellin transforms, we ﬁnd a solution to the nonhomogeneous Euler-type diﬀerential equation with Riemann–Liouville fractional derivatives on the half-axis in the class of functions represented by the fractional integral in terms of the fractional analogue of the Green’s function. Fractional analogues of the Green’s function are constructed in the case when all roots of the characteristic polynomial are diﬀerent and in the case when there are multiple roots of the characteristic polynomial. Theorems of solvability of the nonhomogeneous fractional diﬀerential equations of Euler type on the half-axis are stated and proved. Special cases and examples are considered.

Keywords: fractional Riemann–Liouville integrals, Riemann–Liouville fractional derivatives, direct and inverse Mellin transforms, fractional analogue of Green’s function.

4. Zarina K. Fayazova Boundary control for a pseudo-parabolic equation
Abstract. Previously, a mathematical model for the following problem was considered. On a part of the border of the region $\Omega\subset R^3$ there is a heater with controlled temperature. It is required to ﬁnd such a mode of its operation that the average temperature in some subregion $D$ of $\Omega$ reaches some given value. In this paper, we consider a similar boundary control problem associated with a pseudo-parabolic equation on a segment. On the part of the border of the considered segment, the value of the solution with control parameter is given. Restrictions on the control are given in such a way that the average value of the solution in some part of the considered segment gets a given value. The auxiliary problem is solved by the method of separation of variables, while the problem in consideration is reduced to the Volterra integral equation of the second kind. By Laplace transform method, the existence and uniqueness theorems for admissible control are proved.

Keywords: pseudo-parabolic equation, boundary control, control parameter, Volterra integral equation, Laplace transform.

5. Valery B. Khokholov About the absolute value function with diﬀerent nodes of Lagrange interpolation
Abstract. Lagrange interpolation processes are considered for the following matrixes of interpolation nodes: the matrix of Chebyshev polynomial roots of the 1st kind, the matrix of Legendre polynomials roots, and the extended matrix of Legendre polynomials roots. For these matrixes the uniform convergence of Lagrange process of interpolation for the absolute value function proved. Also, we receive estimates on the order of convergence for each of these matrixes. To ensure the quality of convergence, the endpoints of the segment were added as nodes to the matrix of Legendre roots. However, for the absolute value function the order of convergence of the Legendre process does not change, but improves by approximately 8 times. For comparison, the negative result of equidistant nodes is taken.

Keywords: modulus of a number, interpolation, Lebesgue constant, Chebyshev and Legendre polynomials, extended matrix.

6. Pavel V. Chernikov Absolute σ-retracts and Luzin’s theorem
Abstract. We prove some properties of absolute $\sigma$-retracts. The generalization of the classical Luzin theorem about approximation of a measurable mapping by continuous mappings is given. Namely, we prove the following statement:
Theorem. Let $Y$ be a complete separable metric space in $AR_{\sigma}(\mathfrak{M})$, where $AR_{\sigma}(\mathfrak{M})$ is the whole complex of all absolute $\sigma$-retracts. Suppose that $X$ is a normal space, $A$ is a closed subset in $X$, $\mu\geq0$ is the Radon measure on $A$, and $f:A\rightarrow Y$ is a $\mu$-measurable mapping. Given $\varepsilon>0$, there exist a closed subset $A_{\varepsilon}$ of $A$ such that
$\mu(A\backslash A_{\varepsilon})\leq\varepsilon$ and a continuous mapping $f_{\varepsilon}:X\rightarrow Y$ such that $f_{\varepsilon}(x)=f(x)$ for all
$x\in A_{\varepsilon}$.
Note that a connected separable $ANR(\mathfrak{M})$-space belongs to $AR_{\sigma}(\mathfrak{M})$.

Keywords: absolute σ-retract, Luzin’s theorem.

7. Marco Kostić Weyl-almost periodic and asymptotically Weyl-almost periodic properties of solutions to linear and semilinear abstract Volterra integro-diﬀerential equations
Abstract. The main purpose of paper is to considerWeyl-almost periodic and asymptotically Weyl-almost periodic solutions of linear and semilinear abstract Volterra integro-differential equations. We focus our attention to the investigations of Weyl-almost periodic and asymptotically Weyl-almost periodic properties of both, ﬁnite and inﬁnite convolution product, working in the setting of complex Banach spaces. We introduce the class of asymptotically (equi)-Weyl-p-almost periodic functions depending on two parametres and prove a composition principle for the class of asymptotically equi-Weyl-p-almost periodic functions. Basically, our results are applicable in any situations where the variation of parameters formula takes a role. We provide several new contributions to abstract linear and semilinear Cauchy problems, including equations with the Weyl-Liouville fractional derivatives and the Caputo fractional derivatives. We provide some applications of our abstract theoretical results at the end of paper, considering primarily abstract degenerate diﬀerential equations, including the famous Poisson heat equation and its fractional analogues.

Keywords: Weyl-p-almost periodic functions, asymptotically Weyl-p-almost periodic functions, abstract Volterra integro-diﬀerential equations.

8. Elina L. Shishkina, Mesut Karabacak Singular Cauchy problem for generalized homogeneous Euler–Poisson–Darboux equation
Abstract. In this paper, we solve singular Cauchy problem for a generalised form of an homogeneous Euler–Poisson–Darboux equation with constant potential, wh ere Bessel operator acts instead of the each second derivative. In the classical formulation, the Cauchy problem for this equation is not correct. However, S. A. Tersenov observed that, considering the form of a general solution of the classical Euler–Poisson–Darboux equation, the derivative in the second initial condition must be multiplied by a power function whose degree is equal to the index of the Bessel operator acting on the time variable. The ﬁrst initial condition remains in the usual formulation. With the chosen form of the initial conditions, the considering equation has a solution. Obtained solution is represented as the sum of two terms. The ﬁrst tern is an integral containing the normalized Bessel function and the weighted spherical mean. The second term is expressed in terms of the derivative of the square of the time variable from the integral, which is similar in structure to the ﬁrst term.

Keywords: Bessel operator, Euler–Poisson–Darboux equation, singular Cauchy problem.

### Mathematical modeling

9. Germogen F. Krymsky, Yury A. Romashchenko and Egor P. Sharin Plasma conﬁnement in the ﬁeld of a magnetic dipole
Abstract. When constructing a model of the outer shell of a pulsar, or considering processes during solar ﬂares in studying the processes during solar ﬂares or studying the dynamics of the magnetosphere, one must encounter problems of equilibrium and nonequilibrium conﬁgurations of magnetized plasma. These problems are very complex, because they often reduce to solving second-order nonlinear equations, the analytical method of solving which at the present time being not suﬃciently developed. These problems are very complex. The analytical method for solving such problems is still not suﬃciently developed. In this paper we consider a model problem on the equilibrium of a plasma bundle in a magnetic dipole ﬁeld. The problem is solved in the two-dimensional case, which made it possible to use the mathematical apparatus of analytic functions. One assumption is made: a second plasma symmetric with respect to the dipole was replaced by an equivalent linear current, which led to a single-connected problem. The results of the problem can be used in astrophysics and in the study of magnetospheric processes.

Keywords: plasma, magnetic dipole, Grad–Shafranov equation.