271 research outputs found
Existence, uniqueness and numerical solution of a fractional PDE with integral conditions
This paper is devoted to the solution of one-dimensional Fractional Partial Differential Equation (FPDE) with nonlocal integral conditions. These FPDEs have been of considerable interest in the recent literature because fractional-order derivatives and integrals enable the description of the memory and hereditary properties of different substances. Existence and uniqueness of the solution of this FPDE are demonstrated. As for the numerical approach, a Galerkin method based on least squares is considered. The numerical examples illustrate the fast convergence of this technique and show the efficiency of the proposed method
An hp-version discontinuous Galerkin method for integro-differential equations of parabolic type
We study the numerical solution of a class of parabolic integro-differential equations with weakly singular kernels. We use an -version discontinuous Galerkin (DG) method for the discretization in time. We derive optimal -version error estimates and show that exponential rates of convergence can be achieved for solutions with singular (temporal) behavior near caused by the weakly singular kernel. Moreover, we prove that by using nonuniformly refined time steps, optimal algebraic convergence rates can be achieved for the -version DG method. We then combine the DG time-stepping method with a standard finite element discretization in space, and present an optimal error analysis of the resulting fully discrete scheme. Our theoretical results are numerically validated in a series of test problems
Numerical methods for hyperbolic and parabolic integro-differential equations
An analysis by energy methods is given for fully discrete numerical methods for time-dependent partial integro-differential equations. Stability and error estimates are derived in H1 and L2. The methods considered pay attention to the storage needs during time-stepping
The He\u27s Variational Iteration Method for Solving the Integro-differential Parabolic Problem with Integral Conditions
In this paper, the variational iteration method is applied for finding the solution of an Integro-differential parabolic problem with integral conditions. Convergence of the proposed method is also discussed. Finally, some numerical examples are given to show the effectiveness of the proposed method
Invariant Measures for Dissipative Dynamical Systems: Abstract Results and Applications
In this work we study certain invariant measures that can be associated to
the time averaged observation of a broad class of dissipative semigroups via
the notion of a generalized Banach limit. Consider an arbitrary complete
separable metric space which is acted on by any continuous semigroup
. Suppose that possesses a global
attractor . We show that, for any generalized Banach limit
and any distribution of initial
conditions , that there exists an invariant probability measure
, whose support is contained in , such that for all
observables living in a suitable function space of continuous mappings
on .
This work is based on a functional analytic framework simplifying and
generalizing previous works in this direction. In particular our results rely
on the novel use of a general but elementary topological observation, valid in
any metric space, which concerns the growth of continuous functions in the
neighborhood of compact sets. In the case when does not
possess a compact absorbing set, this lemma allows us to sidestep the use of
weak compactness arguments which require the imposition of cumbersome weak
continuity conditions and limits the phase space to the case of a reflexive
Banach space. Two examples of concrete dynamical systems where the semigroup is
known to be non-compact are examined in detail.Comment: To appear in Communications in Mathematical Physic
A thermodynamically consistent model of magneto-elastic materials under diffusion at large strains and its analysis
The theory of elastic magnets is formulated under possible diffusion and heat
flow governed by Fick's and Fourier's laws in the deformed (Eulerian)
configuration, respectively. The concepts of nonlocal nonsimple materials and
viscous Cahn-Hilliard equations are used. The formulation of the problem uses
Lagrangian (reference) configuration while the transport processes are pulled
back. Except the static problem, the demagnetizing energy is ignored and only
local non-selfpenetration is considered. The analysis as far as existence of
weak solutions of the (thermo)dynamical problem is performed by a careful
regularization and approximation by a Galerkin method, suggesting also a
numerical strategy. Either ignoring or combining particular aspects, the model
has numerous applications as ferro-to-paramagnetic transformation in elastic
ferromagnets, diffusion of solvents in polymers possibly accompanied by
magnetic effects (magnetic gels), or metal-hydride phase transformation in some
intermetalics under diffusion of hydrogen accompanied possibly by magnetic
effects (and in particular ferro-to-antiferromagnetic phase transformation),
all in the full thermodynamical context under large strains
Plasmonic nanoparticle monomers and dimers: From nano-antennas to chiral metamaterials
We review the basic physics behind light interaction with plasmonic
nanoparticles. The theoretical foundations of light scattering on one metallic
particle (a plasmonic monomer) and two interacting particles (a plasmonic
dimer) are systematically investigated. Expressions for effective particle
susceptibility (polarizability) are derived, and applications of these results
to plasmonic nanoantennas are outlined. In the long-wavelength limit, the
effective macroscopic parameters of an array of plasmonic dimers are
calculated. These parameters are attributable to an effective medium
corresponding to a dilute arrangement of nanoparticles, i.e., a metamaterial
where plasmonic monomers or dimers have the function of "meta-atoms". It is
shown that planar dimers consisting of rod-like particles generally possess
elliptical dichroism and function as atoms for planar chiral metamaterials. The
fabricational simplicity of the proposed rod-dimer geometry can be used in the
design of more cost-effective chiral metamaterials in the optical domain.Comment: submitted to Appl. Phys.
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