296 research outputs found
Thin waveguides with Robin boundary conditions
We consider the Laplace operator in a thin three dimensional tube with a
Robin type condition on its boundary and study, asymptotically, the spectrum of
such operator as the diameter of the tube's cross section becomes
infinitesimal. In contrast with the Dirichlet condition case, we evidence
different behaviors depending on a symmetry criterium for the fundamental mode
in the cross section. If that symmetry condition fails, then we prove the
localization of lower energy levels in the vicinity of the minimum point of a
suitable function on the tube's axis depending on the curvature and the
rotation angle. In the symmetric case, the behavior of lower energy modes is
shown to be ruled by a one dimensional Sturm-Liouville problem involving an
effective potential given in explicit form
Numerical scheme based on the spectral method for calculating nonlinear hyperbolic evolution equations
High-precision numerical scheme for nonlinear hyperbolic evolution equations
is proposed based on the spectral method. The detail discretization processes
are discussed in case of one-dimensional Klein-Gordon equations. In conclusion,
a numerical scheme with the order of total calculation cost is
proposed. As benchmark results, the relation between the numerical precision
and the discretization unit size are demonstrated.Comment: To appear in the proceedings of ICCM2020. Figure is modified from the
original versio
Generalized solutions and distributional shadows for Dirac equations
We discuss the application of recent results on generalized solutions to the
Cauchy problem for hyperbolic systems to Dirac equations with external fields.
In further analysis we focus on the question of existence of associated
distributional limits and derive their explicit form in case of free Dirac
fields with regularizations of initial values corresponding to point-like
probability densities
The enclosure method for the heat equation
This paper shows how the enclosure method which was originally introduced for
elliptic equations can be applied to inverse initial boundary value problems
for parabolic equations. For the purpose a prototype of inverse initial
boundary value problems whose governing equation is the heat equation is
considered. An explicit method to extract an approximation of the value of the
support function at a given direction of unknown discontinuity embedded in a
heat conductive body from the temperature for a suitable heat flux on the
lateral boundary for a fixed observation time is given.Comment: 12pages. This is the final versio
Generalized Qualification and Qualification Levels for Spectral Regularization Methods
The concept of qualification for spectral regularization methods for inverse
ill-posed problems is strongly associated to the optimal order of convergence
of the regularization error. In this article, the definition of qualification
is extended and three different levels are introduced: weak, strong and
optimal. It is shown that the weak qualification extends the definition
introduced by Mathe and Pereverzev in 2003, mainly in the sense that the
functions associated to orders of convergence and source sets need not be the
same. It is shown that certain methods possessing infinite classical
qualification, e.g. truncated singular value decomposition (TSVD), Landweber's
method and Showalter's method, also have generalized qualification leading to
an optimal order of convergence of the regularization error. Sufficient
conditions for a SRM to have weak qualification are provided and necessary and
sufficient conditions for a given order of convergence to be strong or optimal
qualification are found. Examples of all three qualification levels are
provided and the relationships between them as well as with the classical
concept of qualification and the qualification introduced by Mathe and
Perevezev are shown. In particular, spectral regularization methods having
extended qualification in each one of the three levels and having zero or
infinite classical qualification are presented. Finally several implications of
this theory in the context of orders of convergence, converse results and
maximal source sets for inverse ill-posed problems, are shown.Comment: 20 pages, 1 figur
Inverse Transport Theory of Photoacoustics
We consider the reconstruction of optical parameters in a domain of interest
from photoacoustic data. Photoacoustic tomography (PAT) radiates high frequency
electromagnetic waves into the domain and measures acoustic signals emitted by
the resulting thermal expansion. Acoustic signals are then used to construct
the deposited thermal energy map. The latter depends on the constitutive
optical parameters in a nontrivial manner. In this paper, we develop and use an
inverse transport theory with internal measurements to extract information on
the optical coefficients from knowledge of the deposited thermal energy map. We
consider the multi-measurement setting in which many electromagnetic radiation
patterns are used to probe the domain of interest. By developing an expansion
of the measurement operator into singular components, we show that the spatial
variations of the intrinsic attenuation and the scattering coefficients may be
reconstructed. We also reconstruct coefficients describing anisotropic
scattering of photons, such as the anisotropy coefficient in a
Henyey-Greenstein phase function model. Finally, we derive stability estimates
for the reconstructions
The weakly coupled fractional one-dimensional Schr\"{o}dinger operator with index
We study fundamental properties of the fractional, one-dimensional Weyl
operator densely defined on the Hilbert space
and determine the asymptotic behaviour of
both the free Green's function and its variation with respect to energy for
bound states. In the sequel we specify the Birman-Schwinger representation for
the Schr\"{o}dinger operator
and extract the finite-rank portion which is essential for the asymptotic
expansion of the ground state. Finally, we determine necessary and sufficient
conditions for there to be a bound state for small coupling constant .Comment: 16 pages, 1 figur
Analysis of some localized boundary-domain integral equations for transmission problems with variable coefficients
This is the post-print version of the Article. The official published version can be found at the links below - Copyright @ 2011 Birkhäuser Boston.Some segregated systems of direct localized boundary-domain integral equations (LBDIEs) associated with several transmission problems for scalar PDEs with variable coefficients are formulated and analyzed for a bounded domain composed of two subdomains with a coefficient jump over the interface. The main results established in the paper are the LBDIE equivalence to the original transmission problems and the invertibility of the corresponding localized boundary-domain integral operators in corresponding Sobolev spaces function spaces.This research was supported by the EPSRC grant EP/H020497/1: ”Mathematical analysis of Localized Boundary-Domain Integral
Equations for Variable-Coefficient Boundary Value Problems” and partly by the Georgian Technical University grant in the case of the third author
Analysis of segregated boundary-domain integral equations for mixed variable-coefficient BVPs in exterior domains
This is the post-print version of the Article. The official published version can be accessed from the link below - Copyright @ 2011 Birkhäuser Boston.Some direct segregated systems of boundary–domain integral equations (LBDIEs) associated with the mixed boundary value problems for scalar PDEs with variable coefficients in exterior domains are formulated and analyzed in the paper. The LBDIE equivalence to the original boundary value problems and the invertibility of the corresponding boundary–domain integral operators are proved in weighted Sobolev spaces suitable for exterior domains. This extends the results obtained by the authors for interior domains in non-weighted Sobolev spaces.The work was supported by the grant EP/H020497/1 ”Mathematical analysis of localised boundary-domain integral equations for BVPs with variable coefficients” of the EPSRC, UK
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