Персона: Костин, Андрей Борисович
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Recovery of Multifactor Source in Parabolic Equation with Integral Type Observation
© 2019, Springer Science+Business Media, LLC, part of Springer Nature.We consider the inverse multifactor source problem for a uniformly parabolic equation with integral overdetermination conditions and study the existence, uniqueness, and stability of a solution. We obtain two forms of sufficient conditions for the unique solvability and present examples of inverse problems satisfying the assumptions of the proved theorems.
Basis Property of the System of Root Functions of the Oblique Derivative Problem for the Laplace Operator in a Disk
© 2019, Pleiades Publishing, Ltd.We study the spectral oblique derivative problem for the Laplace operator in a disk D. The asymptotic properties of the eigenvalues are established, and the basis property with parentheses in the space L2(D) is proved for the system of root functions of the above problem.
Enveloping of Riemann’s Zeta Function Values and Curious Approximation
In this note, by the example of approximate calculation of $ pi
Линейные и нелинейные обратные задачи с нелокальным наблюдением для параболических уравнений в пространствах Соболева
Computation of Sums of Natural Powers of the Inverses of Roots of an Equation Connected with a Spectral Problem
© 2019, Springer Science+Business Media, LLC, part of Springer Nature.We consider equations arising in the oblique derivative spectral problem of the formazJv′(z)+bJv(z)=0,z∈ℂ, where ν, a, b ∈ ℂ are parameters such that |a|+|b| > 0 and Jν(z) is the Bessel function. For roots of the equation we prove summation relations. The results obtained agree with the theory of Rayleigh sums which are calculated in terms of zeros of the Bessel functions.
Inverse source problem for the abstract fractional differential equation
© 2020 Walter de Gruyter GmbH, Berlin/Boston 2020.In a Banach space, the inverse source problem for a fractional differential equation with Caputo-Dzhrbashyan derivative is considered. The initial and observation conditions are given by elements from D (A) D(A), and the operator function on the right side is sufficiently smooth. Two types of the observation operator are considered: integral and at the final point. Under the assumptions that operator A is a generator of positive and compact semigroup the uniqueness, existence and stability of the solution are proved.
ON TAYLOR COEFFICIENTS OF ANALYTIC FUNCTION RELATED WITH EULER NUMBER
We consider a classical construction of second remarkable limit. We pose a question on asymptotically sharp description of the character of such approximation of the number e. In view of this we need the information on behavior of the coefficients in the power expansion for the function (Formula Presented) converging in the interval (Formula Presented). We obtain a recurrent rule regulating the forming of the mentioned coefficients. We show that the coefficients form a sign-alternating sequence of rational numbers (−1)nan, where n ∈ N ∪ {0} and a0 = 1, the absolute values of which strictly decay. On the base of the Faá di Bruno formula for the derivatives of a composed function we propose a combinatorial way of calculating the numbers an as n ∈ N. The original function f(x) is the restriction of the function f(z) on the real ray x andgt; −1 having the same Taylor coefficients and being analytic in the complex plane C with the cut along (−∞, −1]. By the methods of the complex analysis we obtain an integral representation for an for each value of the parameter n ∈ N. We prove that an → 1/e as n → ∞ and find the convergence rate of the difference an − 1/e to zero. We also discuss the issue on choosing the contour in the integral Cauchy formula for calculating the Taylor coefficients (−1)nan of the function f(z). We find the exact values of arising in calculations special improper integrals. The results of the made study allows us to give a series of general two-sided estimates for the deviation e−(1+x) 1/x consistent with the asymptotics s of f(x) as x → 0. We discuss the possibilities of applying the obtained statements © Kostin A.B., Sherstyukov V.B. 2022
Asymptotic Behavior of Remainders of Special Number Series
© 2020, Springer Science+Business Media, LLC, part of Springer Nature.We consider a one-parameter family of number series involving the generalized harmonic series and study asymptotic properties of the remainders. Using R(Np)≡∑n=N∞1/np as an example, we describe the typical obtained results: we obtain the integral representation, find the complete asymptotic expansion with respect to the parameter 2N − 1 as N →∞, and prove that R(N, p) is enveloped by its asymptotic series. The possibilities of the proposed approach are demonstrated by the problem of exact two-sided estimates for the central binomial coefficient.
THE INVERSE PROBLEM OF DETERMINING THE LOWEST COEFFICIENT IN A HIGHER-ORDER PARABOLIC EQUATION WITH WEAK DEGENERACY
We consider the inverse problem of finding the coefficient before u(t, x) in a higher-order parabolic equation, which is not assumed to be uniformly parabolic and can admit weak degeneracy. The required coefficient is considered to depend only on the spatial variable x is an element of [0, l]. Additional information is taken in the form of an integral over the variable t is an element of [0, T] of the solution with a given weight function (integral observation). The initial condition and m boundary conditions of the first kind (2m order of the equation) are specified in a standard way. It is assumed that the leading coefficient rho before u(t) in the equation is non-negative, and its reciprocal 1/rho belongs to the space L-q(Q) for some q > 1. For the considered inverse problem, existence and uniqueness theorems for the generalized solution are proved. In the course of its research, the corresponding theorems on the solvability of the direct problem were formulated and proved. In this case, the approaches and results of the well-known work of S. N. Kruzhkov(1979) were used. In the conclusion, we give an example of the inverse problem for which the conditions of the theorems proved are satisfied. It is shown that for all sufficiently large values of T > 0 its solution exists, is unique, and an estimate for the required coefficient is written out.
Inverse Source and Coefficient Problems for Elliptic and Parabolic Equations in Holder and Sobolev Spaces
© 2019, Springer Science+Business Media, LLC, part of Springer Nature. We review some results obtained by the authors during the last 15 years. In particular, we present the existence and uniqueness theorems for linear and nonlinear inverse problems of reconstructing unknown coefficients in elliptic and parabolic equations.