10,915 research outputs found
Numerical minimization of dirichlet laplacian eigenvalues of four-dimensional geometries
We develop the first numerical study in four dimensions of optimal eigenmodes associated with the Dirichlet Laplacian. We describe an extension of the method of fundamental solutions adapted to the four-dimensional context. Based on our numerical simulation and a postprocessing adapted to the identification of relevant symmetries, we provide and discuss the numerical description of the eighth first optimal domains.The work of the first author was partially supported by FCT, Portugal, through the program “Investigador FCT” with reference IF/00177/2013 and the scientific project PTDC/MATCAL/4334/2014.
The work of the second author was supported by the ANR, through the projects COMEDIC, PGMO, and OPTIFORMinfo:eu-repo/semantics/publishedVersio
Analysis of a class of boundary value problems depending on left and right Caputo fractional derivatives
In this work we study boundary value problems associated to a nonlinear fractional ordinary differential equation involving left and right Caputo derivatives. We discuss the regularity of the solutions of such problems and, in particular, give precise necessary conditions so that the solutions are C1([0, 1]). Taking into account our analytical results, we address the numerical solution of those problems by the augmented-RBF method. Several examples illustrate the good performance of the numerical method.P.A. is partially supported by FCT, Portugal, through the program “Investigador FCT” with reference IF/00177/2013 and the scientific projects PEstOE/MAT/UI0208/2013 and PTDC/MAT-CAL/4334/2014. R.F. was supported by the “Fundação para a Ciência e a Tecnologia (FCT)” through the program “Investigador FCT” with reference IF/01345/2014.info:eu-repo/semantics/publishedVersio
Bounds and extremal domains for Robin eigenvalues with negative boundary parameter
We present some new bounds for the first Robin eigenvalue with a negative
boundary parameter. These include the constant volume problem, where the bounds
are based on the shrinking coordinate method, and a proof that in the fixed
perimeter case the disk maximises the first eigenvalue for all values of the
parameter. This is in contrast with what happens in the constant area problem,
where the disk is the maximiser only for small values of the boundary
parameter. We also present sharp upper and lower bounds for the first
eigenvalue of the ball and spherical shells.
These results are complemented by the numerical optimisation of the first
four and two eigenvalues in 2 and 3 dimensions, respectively, and an evaluation
of the quality of the upper bounds obtained. We also study the bifurcations
from the ball as the boundary parameter becomes large (negative).Comment: 26 pages, 20 figure
On the behavior of clamped plates under large compression
We determine the asymptotic behavior of eigenvalues of clamped plates under large compression by relating this problem to eigenvalues of the Laplacian with Robin boundary conditions. Using the method of fundamental solutions, we then carry out a numerical study of the extremal domains for the first eigenvalue, from which we see that these depend on the value of the compression, and start developing a boundary structure as this parameter is increased. The corresponding number of nodal domains of the first eigenfunction of the extremal domain also increases with the compression.This work was partially supported by the Funda ̧c ̃ao para a Ciˆencia e a Tecnologia(Portugal) through the program “Investigador FCT” with reference IF/00177/2013 and the projectExtremal spectral quantities and related problems(PTDC/MAT-CAL/4334/2014).info:eu-repo/semantics/publishedVersio
Absence of Gluonic Components in Axial and Tensor Mesons
A quarkonium-gluonium mixing scheme previously developed to describe the
characteristic of the pseudoscalar mesons is applied to axial and tensor
mesons. The parameters of the model are determined by fitting the eigenvalues
of a mass matrix. The corresponding eigenvectors give the proportion of light
quarks, strange quarks and glueball in each meson. However the predictions of
the model for branching ratios and electromagnetic decays are incompatible with
the experimental results. These results suggest the absence of gluonic
components in the states of axial and tensor isosinglet mesons analyzed here.Comment: 12 page
Detection of holes in an elastic body based on eigenvalues and traces of eigenmodes
We consider the numerical solution of an inverse problem of finding the shape and location of holes in an elastic body. The problem is solved by minimizing a functional depending on the eigenvalues and traces of corresponding eigenmodes. We use the adjoint method to calculate the shape derivative of this functional. The optimization is performed by BFGS, using a genetic algorithm as a preprocessor and the Method of Fundamental Solutions as a solver for the direct problem. We address several numerical simulations that illustrate the good performance of the method.info:eu-repo/semantics/publishedVersio
Asymptotic behaviour and numerical approximation of optimal eigenvalues of the Robin Laplacian
We consider the problem of minimising the eigenvalue of the Robin
Laplacian in . Although for and a positive boundary
parameter it is known that the minimisers do not depend on ,
we demonstrate numerically that this will not always be the case and illustrate
how the optimiser will depend on . We derive a Wolf-Keller type result
for this problem and show that optimal eigenvalues grow at most with ,
which is in sharp contrast with the Weyl asymptotics for a fixed domain. We
further show that the gap between consecutive eigenvalues does go to zero as
goes to infinity. Numerical results then support the conjecture that for
each there exists a positive value of such that the eigenvalue is minimised by disks for all and,
combined with analytic estimates, that this value is expected to grow with
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