## Vol 67, No 3 (2021): Dedicated to 70th anniversary of the President of the RUDN University V. M. Filippov

**Year:**2021**Articles:**13**URL:**http://journals.rudn.ru/CMFD/issue/view/1482**DOI:**https://doi.org/10.22363/2413-3639-2021-67-3

###### Abstract

**Contemporary Mathematics. Fundamental Directions**. 2021;67(3):423-426

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We consider sequences of orthogonal polynomials with varying weights, i.e., depending on the number of the polynomial. We obtain extensions of applicability classes of well-known asymptotic formulas for large numbers.

**Contemporary Mathematics. Fundamental Directions**. 2021;67(3):427-441

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In this paper, we present the numerical solution of the Stefan problem to calculate the temperature of the tungsten sample heated by the laser pulse. Mathematical modeling is carried out to analyze field experiments, where an instantaneous heating of the plate to 9000 K is observed due to the effect of a heat flow on its surface and subsequent cooling. The problem is characterized by nonlinear coefficients and boundary conditions. An important role is played by the evaporation of the metal from the heated surface. Basing on Samarskii’s approach, we choose to implement the method of continuous counting considering the heat conductivity equation in a uniform form in the entire domain using the Dirac delta function. The numerical method has the second order of approximation with respect to space, the interval of smoothing of the coefficients is 5 K. As a result, we obtain the temperature distributions on the surface and in the cross section of the sample during cooling.

**Contemporary Mathematics. Fundamental Directions**. 2021;67(3):442-454

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This paper contains the proof of general results on the calculation of the norms of monotone operators acting from one ideal space to another under matching convexity and concavity properties of the operator and the norms in ideal spaces.

**Contemporary Mathematics. Fundamental Directions**. 2021;67(3):455-471

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In this paper, we introduce a new version of the definition of a quasi-norm (in particular, a norm) in Lebesgue spaces with variable order of summability. Using it, we prove an analogue of Holder’s inequality for such spaces, which is more general and more precise than those known earlier.

**Contemporary Mathematics. Fundamental Directions**. 2021;67(3):472-482

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For autonomous delay differential equations $x\text{'}\left(t\right)=f\left({x}_{t}\right)$ we construct a continuous semiflow of continuously differentiable solution operators ${x}_{0}\to {x}_{t}$, $t\le 0$, on open subsets of the Fre´chet space $C\left(\right(-\infty ,0],{R}^{n})$. For nonautonomous equations this yields a continuous process of differentiable solution operators. As an application, we obtain processes which incorporate all solutions of Volterra integro-differential equations $x\text{'}\left(t\right)={{\int}_{0}}^{t}k(t,s)h\left(x\right(s\left)\right)ds$.

**Contemporary Mathematics. Fundamental Directions**. 2021;67(3):483-506

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Abstract Volterra integro-differential equations with kernels of integral operators representable by Stieltjes integrals are investigated. The presented results are based on the approach related to the study of one-parameter semigroups for linear evolution equations. We present the method of reduction of the original initial-value problem for a model integro-differential equation with operator coefficients in a Hilbert space to the Cauchy problem for a first-order differential equation in an extended function space. The existence of the contractive C_{0}-semigroup is proved. An estimate for the exponential decay of the semigroup is obtained.

**Contemporary Mathematics. Fundamental Directions**. 2021;67(3):507-525

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In this work, we consider the magnetohydrodynamics system with the Hall and ion-slip effects in R^{3}. The main result is a sufficient condition for regularity on a time interval [0,*T*] expressed in terms ∞,∞ of the norm of the homogeneous Besov space ${\dot{B}}_{\infty ,\infty}^{0}$ with respect to the pressure and the BMO-norm with respect to the gradient of the magnetic field, respectively

${\int}_{0}^{T}({\Vert \Delta \pi \left(t\right)\Vert}_{{\dot{B}}_{\infty ,\infty}^{0}}^{2/3}+{\Vert \Delta B\left(t\right)\Vert}_{BMO}^{2})dt<\infty $,

which can be regarded as improvement of the result in [3].

**Contemporary Mathematics. Fundamental Directions**. 2021;67(3):526-534

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In this paper, we investigate the movement of an inverted pendulum, the suspension point of which performs high-frequency oscillations along a line making a small angle with the vertical. We prove that under certain conditions on the function describing the oscillations of the suspension point of the pendulum, a periodic motion of the pendulum arises, and it is asymptotically stable.

**Contemporary Mathematics. Fundamental Directions**. 2021;67(3):535-548

###### Abstract

Explicit formulas for the first-order partial differential equations system solving were obtained. Solution found for the system with initial conditions. Calculation examples establishing statements truth mentioned. Searching for partial differential equations system solution mathematical expectation became more difficult issue as partial differential equations system with random processes coefficients were covered. Gaussian coefficients and uniformly distributed random process cases examples has been reviewed.

**Contemporary Mathematics. Fundamental Directions**. 2021;67(3):549-563

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We consider the generalized Neumann problem for a *2l*th-order elliptic equation with constant real higher-order coefficients in an infinite domain containing the exterior of some circle and bounded by a sufficiently smooth contour. It consists in specifying of the $({k}_{j}-1)$th-order normal derivatives where $1\le {k}_{1}<...<{k}_{l}\le 2l$; for ${k}_{j}=j$ it turns into the Dirichlet problem, and for ${k}_{j}=j+1$ into the Neumann problem. Under certain assumptions about the coefficients of the equation at infinity, a necessary and sufficient condition for the Fredholm property of this problem is obtained and a formula for its index in Holder spaces is given.

**Contemporary Mathematics. Fundamental Directions**. 2021;67(3):564-575

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We consider the second boundary-value problem for a second-order differential-difference equation with variable coefficients on the interval (0,*d*). We investigate the existence of a generalized solution and obtain conditions on the right-hand side of the equation which ensure the smoothness of generalized solutions on the entire interval (0,*d*).

**Contemporary Mathematics. Fundamental Directions**. 2021;67(3):576-595

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By a bi-variational system we mean any system of equations generated by two different Hamiltonian actions. A connection between their variational symmetries is established. The effective use of the nonclassical Hamiltonian actions for the construction of approximate solutions with the high accuracy for the given dissipative problem is demonstrated. We also investigate the potentiality of the given operator equation with the second-order time derivative, construct the corresponding functional and find necessary and sufficient conditions for the operator S to be a generator of symmetry of the constructed functional. Theoretical results are illustrated by some examples.

**Contemporary Mathematics. Fundamental Directions**. 2021;67(3):596-608