Vol 28, No 142 (2023)

The following boundary value problem is considered:

${{D^{\alpha }}_0}_{+}x\left(t\right)+f(t,(Tx\left)\right(t\left)\right)=0,0<t<1, где \alpha \in (n-1,n], n\in Nф, n>2,$

$x\left(0\right)=x^{\text{'}}\left(0\right)=\cdots =x^{(n-2)}\left(0\right)=0,$

$x\left(1\right)=0$.

This problem reduces to an equivalent integral equation with a monotone operator in the space $C$ of functions continuous on $[0,1]$ (the space $C$ is assumed to be an ordered cone of nonnegative functions satisfying the boundary conditions of the problem under consideration). Using the well-known Krasnosel’sky theorem about fixed points of the operator of expansion (compression) of a cone, the existence of at least one positive solution of the problem under consideration is proved. An example is given that illustrates the fulfillment of sufficient conditions that ensure the solvability of the problem. The results obtained continue the author’s research (see [Russian Universities Reports. Mathematics, 27:138 (2022), 129–135]) devoted to the existence and uniqueness of positive solutions to boundary value problems for nonlinear functional-differential equations.

We present a review of the results devoted to describing families of operators obeying some inductive identities (e. g. Leibniz’s rule — the case of derivations, Fox derivation, and $(\sigma ,\tau )$-derivations) as characters on a suitable groupoid. We first give an implementation of this construction for derivations in group algebras and Fox derivations as characters on an action groupoid. It is also demonstrated how this construction can be realized for derivations on algebras generated by Maltsev semigroups, for the case of derivations with values in finite rings, and for $(\sigma ,\tau )$-derivations.

Let us consider the orthogonal Hermite system $\{{{\phi }_{2}n\left(x\right)\}}_{n\ge 0}$ of even index defined on $(-\infty ,\infty )$, where ${\phi }_{2}n\left(x\right)=\frac{e^{-{\displaystyle \frac{x^2}{2}}}}{\surd \left(2n\right)!{\pi }^{\frac{1}{4}}{2}^n)}{H}_{2}n\left(x\right),$

by $H_{2}n\left(x\right)$ we denote a Hermite polynomial of degree $2n$. In this paper, we consider a generalized system $\left\{{\psi }_{r,2n}\right(x\left)\right\}$ with $r>0$, $n\ge 0$ which is orthogonal with respect to the Sobolev type inner product on $(-\infty ,\infty )$, i.e. $⟨f,g⟩=\underset{(t\to -\infty )}{lim}\sum _{k=0}^{r-1}{f}^{\left(k\right)}\left(t\right)g^{\left(k\right)}\left(t\right)+\underset{-\infty }{\overset{{}^{\infty }}{\int }}{f}^{{}^{\left(r\right)}}\left(x\right)g^{\left(r\right)}\left(x\right)\rho \left(x\right)dx$

with $\rho \left(x\right)=e^{(}-x^2),$ and generated by ${\left\{{\phi }_{2}n\left(x\right)\right\}}_{n\ge 0}$. The main goal of this work is to study some properties related to the system ${\left\{{\psi }_{r,2n}\left(x\right)\right\}}_{n\ge 0}$, ${\psi }_{r,n}\left(x\right)=\frac{{(x-a)}^n}{n!}, n=0,1,2,\dots , r-1,$  ${\psi }_{r,r+n}\left(x\right)=\frac{1}{(r-1)!}{\int }_a^b(x-t{)}^{r-1}{\phi }_n(t)dt, n=0,1,2,\dots .$

We study the conditions on a function $f\left(x\right)$, given in a generalized Hermite orthogonal system, for it to be expandable into a generalized mixed Fourier series as well as the convergence of this Fourier series. The second result of the paper is the proof of a recurrent formula for the system ${\left\{{\psi }_{(}r,2n\left)\right(x)\right\}}_{n\ge 0}$. We also discuss the asymptotic properties of these functions, and this concludes our contribution.

In the paper, exact inequalities are found for the best approximation of an arbitrary analytic function $f$ in the unit circle by algebraic complex polynomials in terms of the modulus of continuity of the $m$ th order of the $r$ th order derivative $f^{\left(r\right)}$ in the weighted Bergman space $B_{2,\gamma }$. Also using the modulus of continuity of the $m$-th order of the derivative $f^{\left(r\right)}$, we introduce a class of functions $W_m^{\left(r\right)}(h,\Phi )$ analytic in the unit circle and defined by a given majorant $\Phi$, $h\in (0,\pi ⁄n]$, $n>r$, monotonically increasing on the positive semiaxis. Under certain conditions on the majorant , for the introduced class of functions, the exact values of some known $n$-widths are calculated. We use methods for solving extremal problems in normed spaces of functions analytic in a circle, as well as the method for estimating from below the -widths of functional classes in various Banach spaces developed by V.M. Tikhomirov. The results presented in this paper are a continuation and generalization of some earlier results on the best approximations and values of widths in the weighted Bergman space $B_{2,\gamma }$.

Algorithms for finding a symbolic-numerical solution of the initial- boundary value problem for a continuum transport equation are discussed. Analytical solution of such equations, as a rule, is impossible; therefore, approximate methods of solution that provide the condition of approximation, stability, and convergence are being actively developed. This article proposes a symbolic solution which is more convenient than a numerical one to be used, for example, in the synthesis of control systems. The algorithm is based on the approximation of partial derivatives with respect to one of the variables by a difference relation and the application of the Laplace transform to the resulting system of differential-difference equations. A block diagram of the algorithm is presented. The description of the structure of the software package based on the developed algorithm is carried out. The software package is developed in the Java programming language. To enter the initial data of the initial boundary value problem and output the solution, a web interface is used. The web interface of the software package is based on the Spring framework. An example of solving an initial boundary value problem with initial and boundary conditions using this software package is considered.

The results are of interest to researchers in applied areas related to heat transfer through a network coolant, transportation of viscous liquids through a network hydraulic carrier, and diffusion processes in biophysics. The developed algorithm can be used to solve some problems of automatic control.