Ravshan Ashurov

National University of Uzbekistan · Department of differential equations

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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),...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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.

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...

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...

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...

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...

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...

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...

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...

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 ?...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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.

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...

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...

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...

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...

We obtain sufficient conditions for the Riesz means of spectral expansions converge to the function to be expanded.

One of the effective methods to find explicit solutions of differential equations is the method based on the operator representation of solutions. The essence of this method is to construct a series, whose members are the relevant iteration operators acting to some classes of sufficiently smooth functions. This method is widely used in the works of...

In this article, the capability of discrete wavelet transform (DWT) to discriminate tree species with different ages using airborne hyperspectral remote sensing is investigated. The performance of DWT is compared against commonly used traditional methods, i.e. original reflectance and first and second derivatives. The hyperspectral data are obtaine...

In this article, the capability of discrete wavelet transform (DWT) to discriminate tree species with different ages using airborne hyperspectral remote sensing is investigated. The performance of DWT is compared against commonly used traditional methods, i.e. original reflectance and first and second derivatives. The hyperspectral data are obtaine...

Discriminating tropical rainforest tree species is still a challenging task due to a variety of species with high spectral similarity and due to very limited studies conducted in this area. We are investigating the effect of discrete wavelet transform (DWT) on enhancing discrimination of tropical rainforest tree species. For this purpose, airborne...

In this work we investigate the Neumann boundary value problem in the unit ball for a non-homogeneous biharmonic equation. It is well known, that even for the Poisson equation this problem does not have a solution for an arbitrary smooth right hand side and boundary functions; it follows from the Green formula, that these given functions should sat...

The processing of remotely sensed data includes compression, noise reduction, classification, feature extraction, change detection and any improvement associated with the problems at hand. In the literature, wavelet methods have been widely used for analysing remote sensing images and signals. The second-generation of wavelets, which is designed ba...

In this work the Neumann boundary value problem for a non-homogeneous polyharmonic equation is studied in a unit ball. Necessary and sufficient conditions for solvability of this problem are found. To do this we first reduce the Neumann problem to the Dirichlet problem for a different non-homogeneous polyharmonic equation and then use the Green fun...

Almost-everywhere convergence of wavelet transforms of Lp -functions under minimal conditions on wavelets was proved by Rao et al. in 1994. However, results on convergence almost everywhere do not provide any information regarding the exceptional set (of Lebesgue measure zero), where convergence does not hold. We prove that if a wavelet ψ satisfies...

We study the behavior of Fourier integrals summed by the symbols of elliptic operators and pointwise convergence of Fourier inversion. We consider generalized localization principle which in classical L p spaces was investigated by Sjölin (1983), Carbery and Soria (1988, 1997) and Alimov (1993). Proceeding these studies, in this paper, we establi...

Detection and mapping the impervious surface accurately is one of the important tasks in urban remote sensing. In this study, airborne hyperspectral data and Worldview-2 image were used to classify urban area .The main goal of this study are to compare the hyperspectral data and worldview 2 images and shows the potential of worldview 2 images for d...

Cities are centers of human activity and more than half of the world's population live in metropolitan areas. Urban areas are characterized by a large variety of artificial and natural surface materials influencing ecological, climatic and energetic conditions. With advent of new sensors in remote sensing fields that capture the data in high spatia...

The partial integrals of the N-fold Fourier integrals connected with elliptic polynomials (not necessarily homogeneous; principal part of which has a strictly convex level surface) are considered. It is proved that if a + s > (N - 1)/2 and ap = N then the Riesz means of the nonnegative order s of the N-fold Fourier integrals of continuous finite fu...

Wavelet analysis is a universal and promising tool with very rich mathematical content and great potential for applications in various scientific fields, in particular, in signal (image) processing and the theory of differential equations. On the other hand distributions are widely used in these fields. And to apply wavelet analysis in these areas...

The almost everywhere convergence of wavelets transforms of L p-functions under minimal conditions on wavelets is well known. But this result does not provide any information about the exceptional set (of Lebesgue measure zero), where convergence does not hold. In this paper, under slightly stronger conditions on wavelets, we prove convergence of w...

We consider spectral expansions of functions from Nikol’skii classes H p a (ℝ N ), related to selfadjoint extensions of elliptic differential operators A(D) of order m in ℝ N . We construct a continuous function from Nikol’skii class with pa<N, such that the Riesz means of spectral expansion of which diverge at the origin. This result demonstrates...

The partial integrals of the N-fold Fourier integrals connected with elliptic polynomials (with a strictly convex level surface) are considered. It is proved that if a + s > (N -1)/2 and ap = N, then the Riesz means of the nonnegative order s of the N-fold Fourier integrals of continuous finite functions from the Sobolev spaces Wpa (R ) converge un...

It is well-known that only a single condition (called the admissibility condition) is sufficient for L2-convergence of multiple continuous wavelet transforms (MCWT). However known results suggest that to guarantee the pointwise convergence of MCWT for Lp-functions wavelets should vanish quite rapidly at infinity. In this article, we consider the cl...

In this paper we consider an initial-value problem for diffusion equation in three dimensional Euclidean space. The initial value is a piecewise smooth function. To solve this problem we apply Fourier transform method and since Fourier integrals of a piecewise smooth function do not converge everywhere, we make use of Riesz summation method.

It is well known, that if N ≥ 3, then spherical partial sums of N-fold Fourier integrals (eigenfunction expansions of Laplace operator) of the characteristic function of the unit ball diverge at the origin. Note, here level surface of Laplace operator and the surface of discontinuity of the considered piecewise smooth function are both spheres. It...