374 research outputs found
Inverse source problems for positive operators. I: Hypoelliptic diffusion and subdiffusion equations
A class of inverse problems for restoring the right-hand side of a parabolic
equation for a large class of positive operators with discrete spectrum is
considered. The results on existence and uniqueness of solutions of these
problems as well as on the fractional time diffusion (subdiffusion) equations
are presented. Consequently, the obtained results are applied for the similar
inverse problems for a large class of subelliptic diffusion and subdiffusion
equations (with continuous spectrum). Such problems are modelled by using
general homogeneous left-invariant hypoelliptic operators on general graded Lie
groups. A list of examples is discussed, including Sturm-Liouville problems,
differential models with involution, fractional Sturm-Liouville operators,
harmonic and anharmonic oscillators, Landau Hamiltonians, fractional
Laplacians, and harmonic and anharmonic operators on the Heisenberg group. The
rod cooling problem for the diffusion with involution is modelled numerically,
showing how to find a "cooling function", and how the involution normally slows
down the cooling speed of the rod.Comment: 26 pages, 7 figures. arXiv admin note: text overlap with
arXiv:1812.0133
Applied Mathematics and Fractional Calculus
In the last three decades, fractional calculus has broken into the field of mathematical analysis, both at the theoretical level and at the level of its applications. In essence, the fractional calculus theory is a mathematical analysis tool applied to the study of integrals and derivatives of arbitrary order, which unifies and generalizes the classical notions of differentiation and integration. These fractional and derivative integrals, which until not many years ago had been used in purely mathematical contexts, have been revealed as instruments with great potential to model problems in various scientific fields, such as: fluid mechanics, viscoelasticity, physics, biology, chemistry, dynamical systems, signal processing or entropy theory. Since the differential and integral operators of fractional order are nonlinear operators, fractional calculus theory provides a tool for modeling physical processes, which in many cases is more useful than classical formulations. This is why the application of fractional calculus theory has become a focus of international academic research. This Special Issue "Applied Mathematics and Fractional Calculus" has published excellent research studies in the field of applied mathematics and fractional calculus, authored by many well-known mathematicians and scientists from diverse countries worldwide such as China, USA, Canada, Germany, Mexico, Spain, Poland, Portugal, Iran, Tunisia, South Africa, Albania, Thailand, Iraq, Egypt, Italy, India, Russia, Pakistan, Taiwan, Korea, Turkey, and Saudi Arabia
A Discovery Tour in Random Riemannian Geometry
We study random perturbations of Riemannian manifolds
by means of so-called Fractional Gaussian Fields,
which are defined intrinsically by the given manifold. The fields will act on the manifolds via conformal transformation
. Our
focus will be on the regular case with Hurst parameter , the celebrated
Liouville geometry in two dimensions being borderline. We want to understand
how basic geometric and functional analytic quantities like diameter, volume,
heat kernel, Brownian motion, spectral bound, or spectral gap will change under
the influence of the noise. And if so, is it possible to quantify these
dependencies in terms of key parameters of the noise. Another goal is to define
and analyze in detail the Fractional Gaussian Fields on a general Riemannian
manifold, a fascinating object of independent interest.Comment: 38 pages, 9 figures. Version 2: new proof of Prop. 3.
Abstract book
Welcome at the International Conference on Differential and Difference Equations
& Applications 2015.
The main aim of this conference is to promote, encourage, cooperate, and bring
together researchers in the fields of differential and difference equations. All areas
of differential & difference equations will be represented with special emphasis on
applications. It will be mathematically enriching and socially exciting event.
List of registered participants consists of 169 persons from 45 countries.
The five-day scientific program runs from May 18 (Monday) till May 22, 2015
(Friday). It consists of invited lectures (plenary lectures and invited lectures in
sections) and contributed talks in the following areas:
Ordinary differential equations,
Partial differential equations,
Numerical methods and applications, other topics
A space-time pseudospectral discretization method for solving diffusion optimal control problems with two-sided fractional derivatives
We propose a direct numerical method for the solution of an optimal control
problem governed by a two-side space-fractional diffusion equation. The
presented method contains two main steps. In the first step, the space variable
is discretized by using the Jacobi-Gauss pseudospectral discretization and, in
this way, the original problem is transformed into a classical integer-order
optimal control problem. The main challenge, which we faced in this step, is to
derive the left and right fractional differentiation matrices. In this respect,
novel techniques for derivation of these matrices are presented. In the second
step, the Legendre-Gauss-Radau pseudospectral method is employed. With these
two steps, the original problem is converted into a convex quadratic
optimization problem, which can be solved efficiently by available methods. Our
approach can be easily implemented and extended to cover fractional optimal
control problems with state constraints. Five test examples are provided to
demonstrate the efficiency and validity of the presented method. The results
show that our method reaches the solutions with good accuracy and a low CPU
time.Comment: This is a preprint of a paper whose final and definite form is with
'Journal of Vibration and Control', available from
[http://journals.sagepub.com/home/jvc]. Submitted 02-June-2018; Revised
03-Sept-2018; Accepted 12-Oct-201
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