arXiv subjects
Publications and source records attributed to Ravshan Ashurov.
The Cauchy problem for the telegraph equation $(D_{t}^{ρ})^{2}u(t)+2αD_{t}^{ρ}u(t)+Au(t)=f(t)$ ($0<t\leq T, \, 0<ρ<1$), 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 to the problem under consideration is proved. Inequalities of stability are obtained.
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 are proved.
A nonlocal boundary value problem for the fractional version of the well known in fluid dynamics Rayleigh-Stokes equation is studied. Namely, the condition $u(x,T)=βu(x,0)+φ(x)$, where $β$ is an arbitrary real number, is proposed instead of the initial condition. If $β=0$, then we get the inverse problem in time, called the backward problem. It is well known that the backward problem is ill-posed in the sense of Hadamard. If $β=1$, then the corresponding non-local problem becomes well-posed in the sense of Hadamard, and moreover, in this case a coercive estimate for the solution can be established. The aim of this work is to find values of the parameter $β$, which separates two types of behavior of the semi-backward problem under consideration. We prove the following statements: if $β\ge 1,$ or $β<0$, then the problem is well-posed; if $β\in (0,1)$, then depending on the eigenvalues of the elliptic part of the equation, for the existence of a solution an additional condition on orthogonality of the right-hand side of the equation and the boundary function to some eigenfunctions of the corresponding elliptic operator may emerge.
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 derivative of order $α$, which is used to describe the viscoelastic behavior of the flow. In this paper, we study the behavior of the solution of such equations depending on the parameter $α$. In particular, it is proved that for sufficiently large $t$ the norm $||u(x,t)||_{L_2(Ω)}$ of the solution decreases with respect to $α$. Moreover the inverse problem of determining the order of the derivative $α$ is solved uniquely.
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)= ψ(x) $ is taken as the over-determination condition, where $t_0$ is some interior point of the considering domain and $ψ(x) $ is a given function. It is proved by the Fourier method that under certain conditions on the functions $g(t)$ and $ψ(x) $ the solution of the inverse problem exists and is unique. An example is given showing the violation of the uniqueness of the solution of the inverse problem for some sign-changing functions $g(t)$. It is shown that for the existence of a solution to the inverse problem for such functions $g(t)$, certain orthogonality conditions for the given functions and some eigenfunctions of the elliptic part of the equation must be satisfied.
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: $u(x,T)=βu(x,0)+φ(x)$, where $β$ is either zero or one. It is well known that if $β=0$, then the corresponding problem, called the backward problem, is ill-posed in the sense of Hadamard, i.e. a small change in $u(x,T)$ leads to large changes in the initial data. Nevertheless, we will show that if we consider sufficiently smooth current information, then the solution exists and it is unique and stable. It will also be shown that if $β=1$, then the corresponding non-local problem is well-posed and coercive type inequalities are valid.
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 $ϕ$ in the non-local boundary value problem $D_t^ρ u(t) + Au(t) = f(t)$ ($0 < ρ< 1, 0 < t \leq T$), $u(ξ) = αu(0) + ϕ$, ($α$ is a constant and $0 < ξ\leq T)$, is considered. Operator $D_t$ on the left-hand side of the equation expresses the Caputo derivative. For both inverse problems $u(ξ_1) = V$ is taken as the over-determination condition. Existence and uniqueness theorems for solutions of the problems under consideration are proved. The influence of the constant $α$ on the existence and uniqueness of a solution to problems is investigated. An interesting effect was discovered: when solving the forward problem, the uniqueness of the solution $u(t)$ was violated, while when solving the inverse problem for the same values of $α$, the solution $u(t)$ became unique.
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 instance recovers uniquely the order of derivative. A list of examples is discussed, including a linear system of fractional di_erential equations, di_erential models with involution, fractional Sturm-Liouville operators, and many others.
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 (vector) order can be found utilizing the available data. In this paper we consider this inverse problem for linear systems of fractional order pseudo-differential equations. We prove that the Fourier transform of the vector-solution $\widehat{U}(t, ξ)$ evaluated at a fixed time instance, which becomes possible due to the available data, recovers uniquely the unknown vector-order of the system of governing pseudo-differential equations.
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 of the equation are indicated, under which the corresponding Fourier series converges absolutely and uniformly. It should be noted, that the condition on the initial function found in this paper is less restrictive than the analogous condition in the case of an equation with derivatives in the sense of Caputo
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 current information, then the solution exists and it is unique. Using this result, we study the inverse problem of initial value identification for subdiffusion equation. The results obtained differ significantly from the corresponding results for the classical diffusion equation (i.e. ? = 1) and even for the subdiffusion equation with the Caputo derivative. A list of examples of operator A is discussed, including linear systems of fractional differential equations, differential models with involution, fractional Sturm-Liouville operators, and many others.
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 equation are indicated, under which the corresponding Fourier series converge absolutely and uniformly. In the case of an initial-boundary value problem on N-dimensional torus, one can easily see that these conditions are not only sufficient, but also necessary.
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 the solution at a fixed time instant at a monitoring location, as "the observation data", identifies uniquely the order of the fractional derivative.
Let $S_λF(x)$ be the spherical partial sums of the multiple Fourier series of function $F\in L_2(\mathbb{T}^N)$. We prove almost-everywhere convergence $S_λF(x)\rightarrow F(x)$ 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 well known result of Carbery and Soria (1988). To prove our result, we first extend the transplantation technic of Kenig and Tomas (1980) from $L_p$ spaces to $H_p^a$ spaces, then apply it to the Carbery and Soria result.
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 $S_λf(x)$, $f\in L_2(\mathbb{T}^N)$, of multiple Fourier series, while it has the solution for square and rectangular partial sums. Historically progress with solving Luzin's conjecture has been made by considering easier problems. One of such easier problems for $S_λf(x)$ was suggested by V. A. Il'in in 1968 and this problem is called the generalized localization principle. In this paper we first give a short survey on convergence almost-everywhere of Fourier series and on generalized localization of Fourier integrals, then present a positive solution for the generalized localization problem for $S_λf(x)$.
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 convergence almost-everywhere of spherical partial sums of the multiple Fourier series of smooth functions.
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 $Ω\subset T^N$, then it is shown that the spherical partial sums of this function converge to zero almost - everywhere on $Ω$. It has been previously known that the generalized localization is not valid in $L_p(T^N)$ when $1\leq p<2$. Thus the problem of generalized localization for the spherical partial sums is completely solved in $L_p(T^N)$, $p\geq 1$: if $p\geq2$ then we have the generalized localization and if $p<2$, then the generalized localization fails.