Ravshan Ashurov

Ravshan Ashurov
National University of Uzbekistan · Department of differential equations

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153
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Publications

Publications (153)
Preprint
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Actual problems of applied mathematics and information technologies - Al-Khwarizmi 2024: abstracts of the international scientific conference (22-23 October 2024, Tashkent, Uzbekistan). – Tashkent. 2024.
Article
In recent years, the fractional partial differential equation of the Boussinesq type has attracted much attention from researchers due to its practical importance. In this paper, we study a non-local problem for the Boussinesq type equation Dtαu(t)+A Dtαu(t)+ν2Au(t) =0, 0<t<T, 1<α<3∕2, where Dtα is the Caputo fractional derivative, and A is an abst...
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We consider the Cauchy problem for a system of partial differential equations of fractional order DtB\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${D}_{t}^{\mathcal{B...
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In the present paper, we investigate the initial-boundary value problem for fractional order parabolic equation on a metric star graph in Sobolev spaces. First, we prove the existence and uniqueness results of strong solutions which are proved with the classical functional method based on a priori estimates. Moreover, the inverse source problem wit...
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The paper considers the initial‐boundary value problem for equation Dtρu(x,t)+(−Δ)σu(x,t)=0,ρ∈(0,1),σ>0$$ {D}_t^{\rho }u\left(x,t\right)+{\left(-\Delta \right)}^{\sigma }u\left(x,t\right)=0,\rho \in \left(0,1\right),\sigma >0 $$, in an N‐dimensional domain Ω$$ \Omega $$ with a homogeneous Dirichlet condition. The fractional derivative is taken in t...
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An inverse problem of determining the right-hand side of the abstract subdiffusion equation with a fractional Caputo derivative is considered in a Hilbert space H. For the forward problem, instead of the Cauchy condition, the non-local in time condition u(0)=u(T) is taken. The right-hand side of the equation has the form g(t)f with a given function...
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The article studies direct and inverse problems for subdiffusion equations involving a Hilfer fractional derivative. An arbitrary positive self-adjoint operator $A$ is taken as the elliptic part of the equation. In particular, as the operator $A$ we can take the Laplace operator with the Dirichlet condition. First, the existence and uniqueness of a...
Preprint
The paper considers the Cauchy problem for the system of partial differential equations of fractional order $D_t^{\mathcal{B}} {U}(t,x) + \mathbb{A}(D) {U} (t,x)=H(t,x) $. Here $U$ and $H$ are vector-functions, the $m\times m$ matrix of differential operators $\mathbb{A}(D)$ is triangular (elements above or below the diagonal are zero). Operators l...
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This study investigates the inverse problem of determining the right-hand side of a telegraph equation given in a Hilbert space. The main equation under consideration has the form \((D_{t}^{\rho })^{2}u(t)+2\alpha D_{t}^{\rho }u(t)+Au(t)=p( t)q+f(t)\), where \(0<t\le T\), \(0<\rho <1\) and \(D_{t}^{\rho }\) is the Caputo derivative. The equation co...
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It is well known that Luzin’s conjecture has a positive solution for one-dimensional trigonometric Fourier series, but in the multidimensional case it has not yet found its confirmation for spherical partial sums of multiple Fourier series. Historically, progress in solving Luzin’s conjecture has been achieved by considering simpler problems. In th...
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In this paper, for a fractional order equation with a fractional differentiation operator in the sense of Gerasimov–Caputo, we study a nonlocal problem of the Bitsadze–Samarskii type. To prove uniqueness and existence theorems for a regular solution to this problem, the spectral method is used. Spectral questions for the corresponding ordinary diff...
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The paper addresses the issues of well-posedness of the linear inverse problem for the three-dimensional Chaplygin equation in a prismatic unbounded domain with semi-local boundary conditions. Using ε-regularization methods, a priori estimates, and a sequence of approximations with application of the Fourier transform, we establish the existence an...
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We consider a nonlocal inverse problem of Bitsadze–Samarskii type for a degenerate fractional order parabolic equation with the Gerasimov–Caputo operator in two spatial variables. The problem is reduced to the study of a spectral boundary value problem for a second order ordinary differential equation with respect to the spatial variable. We study...
Preprint
In the Hilbert space $H$, the inverse problem of determining the right-hand side of the abstract subdiffusion equation with the fractional Caputo derivative is considered. For the forward problem, a non-local in time condition $u(0)=u(T)$ is taken. The right-hand side of the equation has the form $fg(t)$, and the unknown element is $f\in H$. If fun...
Preprint
This study investigates the inverse problem of determining the right-hand side of a telegraph equation given in a Hilbert space. The main equation under consideration has the form $(D_{t}^{\rho})^{2}u(t)+2\alpha D_{t}^{\rho}u(t)+Au(t)=p( t)q+f(t)$, where $0<t\leq T$, $0<\rho<1$ and $D_{t}^{\rho}$ is the Caputo derivative. The equation contains a se...
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UDC 517.9 We consider a Schrödinger equation i ∂ t ρ u ( x , t ) - u x x ( x , t ) = p ( t ) q ( x ) + f ( x , t ) , 0 < t ≤ T , 0 < ρ < 1 , with the Riemann–Liouville derivative. An inverse problem is investigated in which, parallel with u ( x , t ) , a time-dependent factor p ( t ) of the source function is also unknown. To solve this inverse pro...
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In recent years, much attention has been paid to the study of forward and inverse problems for the Rayleigh-Stokes equation in connection with the importance of this equation for applications. This equation plays an important role, in particular, in the study of the behavior of certain non-Newtonian fluids. The equation includes a fractional deriva...
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The inverse problems of determining the right-hand side of the Schrödinger and the sub-diffusion equations with the fractional derivative is considered. In the problem 1, the time-dependent source identification problem for the Schrödinger equation , in a Hilbert space is investigated. To solve this inverse problem, we take the additional condition...
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A nonlocal boundary value problem for the fractional version of the Rayleigh–Stokes equation, well-known in fluid dynamics, is studied. Namely, the condition u(x,T)=βu(x,0)+φ(x), where β is an arbitrary real number, is proposed instead of the initial condition. If β=0, then we have the inverse problem in time, called the backward problem. It is wel...
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The Cauchy problem for the telegraph equation (Dtρ)2u(t)+2αDtρu(t)+Au(t)=f(t) (0<t≤T,0<ρ<1, α>0), with the Caputo derivative is considered. Here, A is a selfadjoint positive operator, acting in a Hilbert space, H; Dt is the Caputo fractional derivative. Conditions are found for the initial functions and the right side of the equation that guarantee...
Preprint
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The Cauchy problem for the telegraph equation $(D_{t}^{\rho })^{2}u(t)+2\alpha D_{t}^{\rho }u(t)+Au(t)=f(t)$ ($0<t\leq T, \, 0<\rho<1$), with the Caputo derivative is considered. Here $A$ is a selfadjoint positive operator, acting in a Hilbert space $H$, $D_t$ is the Caputo fractional derivative. Existence and uniqueness theorems for the solution t...
Preprint
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In this paper, we consider forward and inverse problems for subdiffusion equations with time-dependent coefficients. The fractional derivative is taken in the sense of Riemann-Liouville. Using the classical Fourier method, the theorem of the uniqueness and existence of forward and inverse problems for determining the right-hand side of the equation...
Preprint
Full-text available
In recent years, much attention has been paid to the study of forward and inverse problems for the Rayleigh-Stokes equation in connection with the importance of this equation for applications. This equation plays an important role, in particular, in the study of the behavior of certain non-Newtonian fluids. The equation includes a fractional deriva...
Preprint
Full-text available
A nonlocal boundary value problem for the fractional version of the well known in fluid dynamics Rayleigh-Stokes equation is studied. Namely, the condition $u(x,T)=\beta u(x,0)+\varphi(x)$, where $\beta $ is an arbitrary real number, is proposed instead of the initial condition. If $\beta=0$, then we get the inverse problem in time, called the back...
Article
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A fractional wave equation with a fractional Riemann–Liouville derivative is considered. An arbitrary self-adjoint operator A with a discrete spectrum was taken as the elliptic part. We studied the inverse problem of determining the order of the fractional time derivative. By setting the value of the projection of the solution onto the first eigenf...
Preprint
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The inverse problem of determining the right-hand side of the subdiffusion equation with the fractional Caputo derivative is considered. The right-hand side of the equation has the form $f(x)g(t)$ and the unknown is function $f(x)$. The condition $ u (x,t_0)= \psi (x) $ is taken as the over-determination condition, where $t_0$ is some interior poin...
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This paper presents the method of separation of variables to find conditions on the right-hand side and on the initial data in the Rayleigh-Stokes problem, which ensure the existence and uniqueness of the solution. Further, in the Rayleigh-Stokes problem, instead of the initial condition, the non-local condition is considered: u(x,T)=βu(x,0)+φ(x),...
Preprint
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The Fourier method is used to find conditions on the right-hand side and on the initial data in the Rayleigh-Stokes problem, which ensure the existence and uniqueness of the solution. Then, in the Rayleigh-Stokes problem, instead of the initial condition, consider the non-local condition: $u(x,T)=\beta u(x,0)+\varphi(x)$, where $\beta$ is either ze...
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Initial boundary value problems with a time-nonlocal condition for a subdiffusion equation with the Riemann-Liouville time-fractional derivatives are considered. The elliptical part of the equation is the Laplace operator, defined in an arbitrary N−dimensional domain Ω with a sufficiently smooth boundary ∂Ω. The existence and uniqueness of the solu...
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Let $A$ be an arbitrary positive selfadjoint operator, defined in a separable Hilbert space $H$. The inverse problems of determining the right-hand side of the equation and the function $\phi$ in the non-local boundary value problem $D_t^{\rho} u(t) + Au(t) = f(t)$ ($0 < \rho < 1, 0 < t \leq T$), $u(\xi) = \alpha u(0) + \phi$, ($\alpha$ is a consta...
Preprint
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The Schr\"odinger equation $i \partial_t^\rho u(x,t)-u_{xx}(x,t) = p(t)q(x) + f(x,t)$ ( $0<t\leq T, \, 0<\rho<1$), with the Riemann-Liouville derivative is considered. An inverse problem is investigated in which, along with $u(x,t)$, also a time-dependent factor $p(t)$ of the source function is unknown. To solve this inverse problem, we take the ad...
Article
As it is known various dynamical processes can be modeled through systems of time-fractional order pseudo-differential equations. In the modeling process one frequently faces with the problem of determination of adequate orders of time-fractional derivatives in the sense of Riemann–Liouville or Caputo. This problem is qualified as an inverse proble...
Article
Full-text available
The nonlocal boundary value problem, dtρu(t)+Au(t)=f(t) (0<ρ<1, 0<t≤T), u(ξ)=αu(0)+φ (α is a constant and 0<ξ≤T), in an arbitrary separable Hilbert space H with the strongly positive selfadjoint operator A, is considered. The operator dt on the left hand side of the equation expresses either the Caputo derivative or the Riemann–Liouville derivative...
Article
Full-text available
It is well known that Luzins conjecture has a positive solution for one-dimensional trigonometric Fourier series, but in the multidimensional case it has not yet found its confirmation for spherical partial sums of multiple Fourier series. Historically, progress in solving Luzins hypothesis has been achieved by considering simpler problems. In this...
Article
В работе исследована обратная задача по определению порядка дробной производной в смысле Герасимова-Капуто в волновом уравнении с произвольным положительным самосопряженным оператором $A$, имеющий дискретный спектр. Классическим методом Фурье доказано, что значение проекции решения на некоторую собственную функцию в фиксированный момент времени одн...
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The nonlocal boundary value problem, dtρu(t)+Au(t)=f(t) (0<ρ<1, 0<t≤T), u(ξ)=αu(0)+φ (α is a constant and 0<ξ≤T), in an arbitrary separable Hilbert space H with the strongly positive selfadjoint operator A, is considered. The operator dt on the left hand side of the equation expresses either the Caputo derivative or the Riemann-Liouville derivative...
Article
Full-text available
The nonlocal boundary value problem, d ρ t u(t) + Au(t) = f (t) (0 < ρ < 1, 0 < t ≤ T), u(ξ) = αu(0) + φ (α is a constant and 0 < ξ ≤ T), in an arbitrary separable Hilbert space H with the strongly positive selfadjoint operator A, is considered. The operator dt on the left hand side of the equation expresses either the Caputo derivative or the Riem...
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Full-text available
In this paper we consider an inverse problem for simultaneously determining the order of the Riemann-Liouville time fractional derivative and a source function in subdiffusion equations. Using the classical Fourier method, we prove the uniqueness and the existence theorem for this inverse problem.
Article
An initial-boundary value problem for a time-fractional subdiffusion equation with the Riemann-Liouville derivatives on N-dimensional torus is considered. Uniqueness and existence of the classical solution of the posed problem are proved by the classical Fourier method. Sufficient conditions for the initial function and for the right-hand side of the...
Preprint
Full-text available
It is well known that Luzin's conjecture has a positive solution for one-dimensional trigonometric Fourier series, but in the multidimensional case it has not yet found its confirmation for spherical partial sums of multiple Fourier series. Historically, progress in solving Luzin's hypothesis has been achieved by considering simpler problems. In th...
Preprint
Full-text available
The inverse problem of determining the order of the fractional Riemann- Liouville derivative with respect to time in the subdi_usion equation with an arbitrary positive self-adjoint operator having a discrete spectrum is considered. Using the classical Fourier method it is proved, that the value of the norm jju(t)jj of the solution at a_xed time in...
Preprint
Full-text available
As it is known various dynamical processes can be modeled through the systems of time-fractional order pseudo-differential equations. In the modeling process one frequently faces with determining the adequate orders of time-fractional derivatives in the sense of Riemann-Liouville or Caputo. This problem is qualified as an inverse problem. The right...
Preprint
Full-text available
An initial-boundary value problem for a time-fractional subdiffusion equation with the Riemann-Liouville derivatives on N-dimensional torus is considered. The uniqueness and existence of the classical solution of the posed problem are proved by the classical Fourier method. Sufficient conditions for the initial function and for the right-hand side...
Preprint
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The backward problem for subdiffusion equation with the fractional Riemann-Liouville time-derivative of order ? 2 (0; 1) and an arbitrary positive self-adjoint operator A is considered. This problem is ill-posed in the sense of Hadamard due to the lack of stability of the solution. Nevertheless, we will show that if we consider sufficiently smooth...
Article
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We deal with boundary value problems for equations with the ∂y 2 operator ∂ 2 − A (x, D), where A (x, D) is a nonnegative elliptic differential operator and with boundary operators depending on a positive real parameter ρ. In particular, boundary conditions can be given through the one-sided Marchaud, Gru¨nwald-Letnikov or Liouville-Weyl fractional...
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The paper investigates the inverse problem of determining the order of the fractional derivative in the sense of Gerasimov-Caputo in the wave equation with an arbitrary positive self-adjoint operator A with a discrete spectrum. It is proved by the classical Fourier method that the value of the projection of the solution on some eigenfunction at a fi...
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In this work, an analogue of the Tricomi problem for equations of mixed type with a fractional derivative is investigated. In one part of the domain, the considered equation is a subdiffusion equation with a fractional derivative of order ? 2 (0; 1) in the sense of Riemann-Liouville, and in the other it is a wave equation. Assuming the parameter ?...
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In this paper the inverse problem of determining the fractional orders in mixed-type equations is considered. In one part of the domain the considered equation is the subdiffusion equation with a fractional derivative in the sense of Gerasimov-Caputo of the order 0<a<1 , and in the other part - a wave equation with a fractional derivative of the or...
Article
An initial-boundary value problem for a time-fractional subdiffusion equation with an arbitrary order elliptic differential operator is considered. Uniqueness and existence of the classical solution of the posed problem are proved by the classical Fourier method. Sufficient conditions for the initial function and for the right-hand side of the equa...
Article
In this paper we consider of a multidimensional loaded mixed type equation of the second order with some seminonlocal conditions on the coefficients. The existence and uniqueness of solution of the seminonlocal boundary value problem of second kind is proved in the Sobolev space (Formula presented.). In proof of the theorems are used the methods of...
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This paper is devoted to the general theory of linear systems of fractional order pseudo-differential equations. Single fractional order differential and pseudo-differential equations are studied by many authors and several monographs and handbooks have been published devoted to its theory and applications. However, the state of systems of fraction...
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В работе изучается сходимость почти всюду сферических частичных сумм кратных рядов Фурье функций из классов Соболева. Доказано, что сходимость почти всюду будет иметь место при тех же условиях на порядок гладкости разлагаемой функции, что и при кратных интегралов Фурье, что установлено в известной работе Карбери и Сория (1988). Наши рассуждения во...
Article
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The identification of the right order of the equation in applied fractional modeling plays an important role. In this paper we consider an inverse problem for determining the order of time fractional derivative in a subdiffusion equation with an arbitrary second order elliptic differential operator. We prove that the additional information about th...
Article
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An inverse problem for determining the order of the Caputo time-fractional derivative in a subdiffusion equation with an arbitrary positive self-adjoint operator A with discrete spectrum is considered. By the Fourier method it is proved that the value of {\|Au(t)\|} , where {u(t)} is the solution of the forward problem, at a fixed time instance rec...
Preprint
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An initial-boundary value problem for a subdiffusion equation with an elliptic operator A(D) in R^n is considered. Uniqueness and existence theorems for a solution of this problem are proved by the Fourier method. Considering the order of the Caputo time-fractional derivative as an unknown parameter, the corresponding inverse problem of determining...
Preprint
An initial-boundary value problem for a time-fractional subdiffusion equation with an arbitrary order elliptic differential operator is considered. Uniqueness and existence of the classical solution of the posed problem are proved by the classical Fourier method. Sufficient conditions for the initial function and for the right-hand side of the equa...
Preprint
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The identification of the right order of the equation in applied fractional modeling plays an important role. In this paper we consider an inverse problem for determining the order of time fractional derivative in a subdiffusion equation with an arbitrary second order elliptic differential operator. We prove that the additional information about th...
Article
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В работе рассматриваются сферически симметричные непрерывные всплеск-разложения и вводится для них понятие средних Рисса. В классах $L_p$ без каких-либо ограничений на всплески, доказана справедливость обобщенной локализации для рассматриваемых разложений. Далее обобщенная локализация изучена для средних Рисса всплеск-разложений распределений из со...
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Let $S_\lambda F(x)$ be the spherical partial sums of the multiple Fourier series of function $F\in L_2(\mathbb{T}^N)$. We prove almost-everywhere convergence $S_\lambda F(x)\rightarrow F(x)$ for functions in Sobolev spaces $H_p^a(\mathbb{T}^N)$ provided $1< p \leq 2$ and $a> (N-1)(\frac{1}{p}-\frac{1}{2})$. For multiple Fourier integrals this is w...
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It is well known, that Luzin's conjecture has a positive solution for one dimensional trigonometric Fourier series and it is still open for the spherical partial sums $S_\lambda f(x)$, $f\in L_2(\mathbb{T}^N)$, of multiple Fourier series, while it has the solution for square and rectangular partial sums. Historically progress with solving Luzin's c...
Article
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In this paper the generalized localization principle for the spherical partial sums of the multiple Fourier series in the \(L_2\)-class is proved, that is, if \(f\in L_2({\mathbb {T}}^N)\) and \(f=0\) on an open set \(\Omega \subset {\mathbb {T}}^N\), then it is shown that the spherical partial sums of this function converge to zero almost-everywhe...
Article
In this paper the generalized localization principle for the spherical partial sums of the multiple Fourier series in the L2-class is proved, that is, if f L2 (ТN) and f = 0 on an open set ТN then it is shown that the spherical partial sums of this function converge to zero almost - everywhere on . It has been previously known that the generalized...
Article
Spherically symmetric continuous wavelet decompositions are considered, and the notion of Riesz means is introduced for them. Generalized localization is proved for the decompositions under study in Lp classes without any restrictions on the wavelets. Further, generalized localization is studied for the Riesz means of wavelet decompositions of dist...
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In this paper, we study the well-posedness of a linear inverse problem for a mixed-type second-order multidimensional equation of the first kind. We prove the unique solvability of this problem in a certain function class with the help of “ε-regularization”, a priori estimates, and successive approximation methods.
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In this paper it is proposed a very simple method for estimating the maximal operator in $L_1$. Using this method one can considerably improve the existing theorems on convergence almost-everywhere of eigenfunction expansions of an arbitrary elliptic differential operators with a point spectrum. In particular, it is obtained a new result on converg...
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In this paper the generalized localization principle for the spherical partial sums of the multiple Fourier series in the $L_2$ - class is proved, that is, if $f\in L_2(T^N)$ and $f=0$ on an open set $\Omega \subset T^N$, then it is shown that the spherical partial sums of this function converge to zero almost - everywhere on $\Omega$. It has been...
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We study the well-posedness of a linear inverse problem for a multidimensional mixed-type equation including the classical equations of elliptic, hyperbolic, and parabolic types as special cases. For this problem, using the “ε-regularization,” a priori estimate, and successive approximationmethods, we prove the existence and uniqueness theorems for...
Article
We obtain sufficient conditions for the Riesz means of spectral expansions converge to the function to be expanded.

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