599 research outputs found
Cloaking via mapping for the heat equation
This paper explores the concept of near-cloaking in the context of
time-dependent heat propagation. We show that after the lapse of a certain
threshold time instance, the boundary measurements for the homogeneous heat
equation are close to the cloaked heat problem in a certain Sobolev space norm
irrespective of the density-conductivity pair in the cloaked region. A
regularised transformation media theory is employed to arrive at our results.
Our proof relies on the study of the long time behaviour of solutions to the
parabolic problems with high contrast in density and conductivity coefficients.
It further relies on the study of boundary measurement estimates in the
presence of small defects in the context of steady conduction problem. We then
present some numerical examples to illustrate our theoretical results.Comment: 31 pages, 7 figures, 1 tabl
Large-distance and long-time asymptotic behavior of the reduced density matrix in the non-linear Schr\"{o}dinger model
Starting from the form factor expansion in finite volume, we derive the
multidimensional generalization of the so-called Natte series for the
zero-temperature, time and distance dependent reduced density matrix in the
non-linear Schr\"{o}dinger model. This representation allows one to
\textit{read-off} straightforwardly the long-time/large-distance asymptotic
behavior of this correlator. This method of analysis reduces the complexity of
the computation of the asymptotic behavior of correlation functions in the
so-called interacting integrable models, to the one appearing in free fermion
equivalent models. We compute explicitly the first few terms appearing in the
asymptotic expansion. Part of these terms stems from excitations lying away
from the Fermi boundary, and hence go beyond what can be obtained by using the
CFT/Luttinger liquid based predictions.Comment: 75 pages, 6 figures, V2, some comments adde
Fast numerical methods for waves in periodic media
Periodic media problems widely exist in many modern
application areas like
semiconductor nanostructures (e.g.\ quantum dots and nanocrystals),
semi-conductor superlattices,
photonic crystals (PC) structures,
meta materials or Bragg gratings of surface
plasmon polariton (SPP) waveguides, etc.
Often these application problems are modeled by partial differential
equations with periodic coefficients and/or periodic geometries.
In order to numerically solve these periodic structure problems efficiently one usually confines the spatial domain to a bounded computational domain
(i.e.\ in a neighborhood of the region of physical interest).
Hereby, the usual strategy is to introduce so-called
\emph{artificial boundaries} and impose suitable boundary conditions.
For wave-like equations, the ideal boundary conditions should not only lead to well-posed problems,
but also mimic the perfect absorption of waves traveling out of the computational domain
through the artificial boundaries.
In the first part of this chapter we present a novel analytical impedance expression
for general second order ODE problems with periodic coefficients.
This new expression for the kernel of the Dirichlet-to-Neumann mapping of the artificial boundary
conditions is then used for computing the bound states of the Schr\"odinger operator with
periodic potentials at infinity.
Other potential applications are associated with the exact artificial boundary conditions
for some time-dependent problems with periodic structures.
As an example, a two-dimensional hyperbolic equation modeling the TM polarization of
the electromagnetic field with a periodic dielectric permittivity is considered.
In the second part of this chapter we present a new numerical technique for solving periodic structure problems. This novel approach possesses several advantages.
First, it allows for a fast evaluation of the Sommerfeld-to-Sommerfeld operator for periodic
array problems. Secondly,
this computational method can also be used for bi-periodic structure problems with local defects.
In the sequel we consider several problems, such as the exterior elliptic problems with
strong coercivity, the time-dependent Schr\"odinger equation and the Helmholtz equation
with damping.
Finally, in the third part we consider
periodic arrays that are structures consisting of geometrically identical
subdomains, usually called periodic cells.
We use the Helmholtz equation as a model equation and consider
the definition and evaluation of the exact boundary mappings for general
semi-infinite arrays that are periodic in one direction for any real wavenumber.
The well-posedness of the Helmholtz equation is established via the
\emph{limiting absorption principle} (LABP).
An algorithm based on the doubling procedure of the second part of this chapter
and an extrapolation method is proposed to construct the
exact Sommerfeld-to-Sommerfeld boundary mapping.
This new algorithm benefits from its robustness and the
simplicity of implementation.
But it also suffers from the high computational cost and the
resonance wave numbers.
To overcome these shortcomings, we propose another algorithm based
on a conjecture about the asymptotic behaviour of
limiting absorption principle solutions.
The price we have to pay is the resolution of some generalized eigenvalue problem,
but still the overall computational cost is significantly reduced.
Numerical evidences show that this algorithm presents theoretically
the same results as the first algorithm.
Moreover, some quantitative comparisons between these two algorithms are given
Numerical computing approach for solving Hunter-Saxton equation arising in liquid crystal model through sinc collocation method
In this study, numerical treatment of liquid crystal model described through Hunter-Saxton equation (HSE) has been presented by sinc collocation technique through theta weighted scheme due to its enormous applications including, defects, phase diagrams, self-assembly, rheology, phase transitions, interfaces, and integrated biological applications in mesophase materials and processes. Sinc functions provide the procedure for function approximation over all types of domains containing singularities, semi-infinite or infinite domains. Sinc functions have been used to reduce HSE into an algebraic system of equations that makes the solution quite superficial. These algebraic equations have been interpreted as matrices. This projected that sinc collocation technique is considerably efficacious on computational ground for higher accuracy and convergence of numerical solutions. Stability analysis of the proposed technique has ensured the accuracy and reliability of the method, moreover, as the stability parameter satisfied the condition the proposed solution of the problem converges. The solution of the HSE is presented through graphical figures and tables for different cases that are constructed on various values of θ and collocation points. The accuracy and efficiency of the proposed technique is analyzed on the basis of absolute errors.This research has been partially supported by Ministerio de Ciencia, Innovación y Universidades grant number PGC2018-0971-B-100 and Fundación Séneca -Agencia de Ciencia y Tecnología de la Región de Murcia grant number 20783/PI/18. Also, It has been supported by the National Research Program for Universities (NRPU), Higher Education Commission, Pakistan, No. 8103/Punjab/NRPU/R and D/HEC/2017
Fast numerical methods for waves in periodic media
Periodic media problems widely exist in many modern application areas
like semiconductor nanostructures (e.g. quantum dots and nanocrystals),
semi-conductor superlattices, photonic crystals (PC) structures, meta
materials or Bragg gratings of surface plasmon polariton (SPP) waveguides,
etc. Often these application problems are modeled by partial differential
equations with periodic coefficients and/or periodic geometries. In order to
numerically solve these periodic structure problems efficiently one usually
confines the spatial domain to a bounded computational domain (i.e. in a
neighborhood of the region of physical interest). Hereby, the usual strategy
is to introduce so-called artificial boundaries and impose suitable boundary
conditions. For wave-like equations, the ideal boundary conditions should not
only lead to w ell-posed problems, but also mimic the perfect absorption of
waves traveling out of the computational domain through the artificial
boundaries ..
A Brief Review on Mathematical Tools Applicable to Quantum Computing for Modelling and Optimization Problems in Engineering
Since its emergence, quantum computing has enabled a wide spectrum of new possibilities and advantages, including its efficiency in accelerating computational processes exponentially. This has directed much research towards completely novel ways of solving a wide variety of engineering problems, especially through describing quantum versions of many mathematical tools such as Fourier and Laplace transforms, differential equations, systems of linear equations, and optimization techniques, among others. Exploration and development in this direction will revolutionize the world of engineering. In this manuscript, we review the state of the art of these emerging techniques from the perspective of quantum computer development and performance optimization, with a focus on the most common mathematical tools that support engineering applications. This review focuses on the application of these mathematical tools to quantum computer development and performance improvement/optimization. It also identifies the challenges and limitations related to the exploitation of quantum computing and outlines the main opportunities for future contributions. This review aims at offering a valuable reference for researchers in fields of engineering that are likely to turn to quantum computing for solutions. Doi: 10.28991/ESJ-2023-07-01-020 Full Text: PD
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