268 research outputs found
Schrödinger operators in the twentieth century
This paper reviews the past fifty years of work on spectral theory and related issues in nonrelativistic quantum mechanics
Fractional Laguerre spectral methods and their applications to fractional differential equations on unbounded domain
In this article, we first introduce a singular fractional Sturm-Liouville problem (SFSLP) on unbounded domain. The associated fractional differential operator is both Weyl and Caputo type. The properties of spectral data for fractional operator on unbounded domain have been investigated. Moreover, it has been shown that the eigenvalues of the singular problem are real-valued and the corresponding eigenfunctions are orthogonal. The analytical eigensolutions of SFSLP are obtained and defined as generalized Laguerre fractional-polynomials. The optimal approximation of such generalized Laguerre fractional-polynomials in suitably weighted Sobolev spaces involving fractional derivatives has been derived. We construct an efficient generalized Laguerre fractional-polynomials-Petrov–Galerkin methods for a class of fractional initial value problems and fractional boundary value problems. As a numerical example, we examine space fractional advection–diffusion equation. Our theoretical results are confirmed by associated numerical results
A tutorial on inverse problems for anomalous diffusion processes
Over the last two decades, anomalous diffusion processes in which the mean squares variance grows slower or faster than that in a Gaussian process have found many applications. At a macroscopic level, these processes are adequately described by fractional differential equations, which involves fractional derivatives in time or/and space. The fractional derivatives describe either history mechanism or long range interactions of particle motions at a microscopic level. The new physics can change dramatically the behavior of the forward problems. For example, the solution operator of the time fractional diffusion diffusion equation has only limited smoothing property, whereas the solution for the space fractional diffusion equation may contain weak singularity. Naturally one expects that the new physics will impact related inverse problems in terms of uniqueness, stability, and degree of ill-posedness. The last aspect is especially important from a practical point of view, i.e., stably reconstructing the quantities of interest. In this paper, we employ a formal analytic and numerical way, especially the two-parameter Mittag-Leffler function and singular value decomposition, to examine the degree of ill-posedness of several 'classical' inverse problems for fractional differential equations involving a Djrbashian–Caputo fractional derivative in either time or space, which represent the fractional analogues of that for classical integral order differential equations. We discuss four inverse problems, i.e., backward fractional diffusion, sideways problem, inverse source problem and inverse potential problem for time fractional diffusion, and inverse Sturm–Liouville problem, Cauchy problem, backward fractional diffusion and sideways problem for space fractional diffusion. It is found that contrary to the wide belief, the influence of anomalous diffusion on the degree of ill-posedness is not definitive: it can either significantly improve or worsen the conditioning of related inverse problems, depending crucially on the specific type of given data and quantity of interest. Further, the study exhibits distinct new features of 'fractional' inverse problems, and a partial list of surprising observations is given below. (a) Classical backward diffusion is exponentially ill-posed, whereas time fractional backward diffusion is only mildly ill-posed in the sense of norms on the domain and range spaces. However, this does not imply that the latter always allows a more effective reconstruction. (b) Theoretically, the time fractional sideways problem is severely ill-posed like its classical counterpart, but numerically can be nearly well-posed. (c) The classical Sturm–Liouville problem requires two pieces of spectral data to uniquely determine a general potential, but in the fractional case, one single Dirichlet spectrum may suffice. (d) The space fractional sideways problem can be far more or far less ill-posed than the classical counterpart, depending on the location of the lateral Cauchy data. In many cases, the precise mechanism of these surprising observations is unclear, and awaits further analytical and numerical exploration, which requires new mathematical tools and ingenuities. Further, our findings indicate fractional diffusion inverse problems also provide an excellent case study in the differences between theoretical ill-conditioning involving domain and range norms and the numerical analysis of a finite-dimensional reconstruction procedure. Throughout we will also describe known analytical and numerical results in the literature
-small ball asymptotics for Gaussian random functions: a survey
This article is a survey of the results on asymptotic behavior of small ball
probabilities in -norm. Recent progress in this field is mainly based on
the methods of spectral theory of differential and integral operators.Comment: 65 pages, 2 figure
An inverse problem for a one-dimensional time-fractional diffusion problem
Over the last two decades, anomalous diusion processes in which the mean squares variance grows
slower or faster than that in a Gaussian process have found many applications. At a macroscopic level, these
processes are adequately described by fractional dierential equations, which involves fractional derivatives in
time or/and space. The fractional derivatives describe either history mechanism or long range interactions
of particle motions at a microscopic level. The new physics can change dramatically the behavior of the
forward problems. For example, the solution operator of the time fractional diusion diusion equation has
only limited smoothing property, whereas the solution for the space fractional diusion equation may contain
weakly singularity. Naturally one expects that the new physics will impact related inverse problems in terms
of uniqueness, stability, and degree of ill-posedness. The last aspect is especially important from a practical
point of view, i.e., stably reconstructing the quantities of interest.
In this paper, we employ a formal analytic and numerical way, especially the two-parameter Mittag-Leer
function and singular value decomposition, to examine the degree of ill-posedness of several \classical" inverse
problems for fractional dierential equations involving a Djrbashian-Caputo fractional derivative in either time
or space, which represent the fractional analogues of that for classical integral order dierential equations. We
discuss four inverse problems, i.e., backward fractional diusion, sideways problem, inverse source problem and
inverse potential problem for time fractional diusion, and inverse Sturm-Liouville problem, Cauchy problem,
backward fractional diusion and sideways problem for space fractional diusion. It is found that contrary
to the wide belief, the in
uence of anomalous diusion on the degree of ill-posedness is not denitive: it can
either signicantly improve or worsen the conditioning of related inverse problems, depending crucially on
the specic type of given data and quantity of interest. Further, the study exhibits distinct new features of
\fractional" inverse problems, and a partial list of surprising observations is given below.
(a) Classical backward diusion is exponentially ill-posed, whereas time fractional backward diusion is only
mildly ill-posed in the sense of norms on the domain and range spaces. However, this does not imply
that the latter always allows a more eective reconstruction.
(b) Theoretically, the time fractional sideways problem is severely ill-posed like its classical counterpart, but
numerically can be nearly well-posed.
(c) The classical Sturm-Liouville problem requires two pieces of spectral data to uniquely determine a general
potential, but in the fractional case, one single Dirichlet spectrum may suce.
(d) The space fractional sideways problem can be far more or far less ill-posed than the classical counterpart,
depending on the location of the lateral Cauchy data.
In many cases, the precise mechanism of these surprising observations is unclear, and awaits further analytical
and numerical exploration, which requires new mathematical tools and ingenuities. Further, our ndings
indicate fractional diusion inverse problems also provide an excellent case study in the dierences between
theoretical ill-conditioning involving domain and range norms and the numerical analysis of a nite-dimensional
reconstruction procedure. Throughout we will also describe known analytical and numerical results in the literature
Spectral geometry of partial differential operators
Access; Differential; Durvudkhan; Geometry; Makhmud; Michael; OA; Open; Operators; Partial; Ruzhansky; Sadybekov; Spectral; Suraga
- …