151 research outputs found
Heat transfer in a complex medium
The heat equation is considered in the complex medium consisting of many
small bodies (particles) embedded in a given material. On the surfaces of the
small bodies an impedance boundary condition is imposed. An equation for the
limiting field is derived when the characteristic size of the small bodies
tends to zero, their total number tends to infinity at a
suitable rate, and the distance between neighboring small bodies
tends to zero: , . No periodicity is
assumed about the distribution of the small bodies. These results are basic for
a method of creating a medium in which heat signals are transmitted along a
given line. The technical part for this method is based on an inverse problem
of finding potential with prescribed eigenvalues.Comment: arXiv admin note: text overlap with arXiv:1207.056
Electromagnetic Wave Scattering by Small Impedance Particles of an Arbitrary Shape
Scattering of electromagnetic (EM) waves by one and many small ()
impedance particles of an arbitrary shape, embedded in a homogeneous
medium, is studied. Analytic formula for the field, scattered by one particle,
is derived. The scattered field is of the order , where
is a number. This field is much larger than in the
Rayleigh-type scattering. An equation is derived for the effective EM field
scattered by many small impedance particles distributed in a bounded domain.
Novel physical effects in this domain are described and discussed
Dynamical Systems Gradient method for solving nonlinear equations with monotone operators
A version of the Dynamical Systems Gradient Method for solving ill-posed
nonlinear monotone operator equations is studied in this paper. A discrepancy
principle is proposed and justified. A numerical experiment was carried out
with the new stopping rule. Numerical experiments show that the proposed
stopping rule is efficient. Equations with monotone operators are of interest
in many applications.Comment: 2 figure
Dynamical Systems Method for ill-posed equations with monotone operators
Consider an operator equation (*) in a real Hilbert space.
Let us call this equation ill-posed if the operator is not boundedly
invertible, and well-posed otherwise. The DSM (dynamical systems method) for
solving equation (*) consists of a construction of a Cauchy problem, which has
the following properties:
1) it has a global solution for an arbitrary initial data,
2) this solution tends to a limit as time tends to infinity,
3) the limit is the minimal-norm solution to the equation .
A global convergence theorem is proved for DSM for equation (*) with monotone
operators
A recipe for making materials with negative refraction in acoustics
A recipe is given for making materials with negative refraction in acoustics,
i.e., materials in which the group velocity is directed opposite to the phase
velocity.
The recipe consists of injecting many small particles into a bounded domain,
filled with a material whose refraction coefficient is known. The number of
small particles to be injected per unit volume around any point is
calculated as well as the boundary impedances of the embedded particles
A new discrepancy principle
The aim of this note is to prove a new discrepancy principle. The advantage
of the new discrepancy principle compared with the known one consists of
solving a minimization problem approximately, rather than exactly, and in the
proof of a stability result
Wave scattering by small impedance particles in a medium
The theory of acoustic wave scattering by many small bodies is developed for
bodies with impedance boundary condition. It is shown that if one embeds many
small particles in a bounded domain, filled with a known material, then one can
create a new material with the properties very different from the properties of
the original material. Moreover, these very different properties occur although
the total volume of the embedded small particles is negligible compared with
the volume of the original material
Equations for the self-consistent field in random medium
An integral-differential equation is derived for the self-consistent
(effective) field in the medium consisting of many small bodies randomly
distributed in some region. Acoustic and electromagnetic fields are considered
in such a medium. Each body has a characteristic dimension , where
is the wavelength in the free space.
The minimal distance between any of the two bodies satisfies the
condition , but it may also satisfy the condition . Using
Ramm's theory of wave scattering by small bodies of arbitrary shapes, the
author derives an integral-differential equation for the self-consistent
acoustic or electromagnetic fields in the above medium
On a new notion of the solution to an ill-posed problem
A new understanding of the notion of the stable solution to ill-posed
problems is proposed. The new notion is more realistic than the old one and
better fits the practical computational needs. A method for constructing stable
solutions in the new sense is proposed and justified. The basic point is: in
the traditional definition of the stable solution to an ill-posed problem
, where is a linear or nonlinear operator in a Hilbert space , it
is assumed that the noisy data are given,
, and a stable solution u_\d:=R_\d f_\d is defined
by the relation \lim_{\d\to 0}||R_\d f_\d-y||=0, where solves the
equation , i.e., . In this definition and are unknown. Any
f\in B(f_\d,\d) can be the exact data, where B(f_\d,\d):=\{f:
||f-f_\delta||\leq \delta\}.The new notion of the stable solution excludes the
unknown and from the definition of the solution
Electromagnetic wave scattering by many small particles
Scattering of electromagnetic waves by many small particles of arbitrary
shapes is reduced rigorously to solving linear algebraic system of equations
bypassing the usual usage of integral equations.
The matrix elements of this linear algebraic system have physical meaning.
They are expressed in terms of the electric and magnetic polarizability
tensors. Analytical formulas are given for calculation of these tensors with
any desired accuracy for homogeneous bodies of arbitrary shapes. An idea to
create a "smart" material by embedding many small particles in a given region
is formulated
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