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A. G. Ramm

Publications and source records attributed to A. G. Ramm.

At least 55 records · Page 3Linked to original sources
arXiv2012

Numerical solution of many-body wave scattering problem for small particles

A numerical approach to the problem of wave scattering by many small particles is developed under the assumptions k<<1, d>>a, where a is the size of the particles and d is the distance between the neighboring particles. On the wavelength one may have many small particles. An impedance boundary conditions are assumed on the boundaries of small particles. The results of numerical simulation show good agreement with the theory. They open a way to numerical simulation of the method for creating materials with a desired refraction coefficient.

physics.comp-ph↗
arXiv2012

Numerical modeling in wave scattering problem for small particles

A numerical solution to the problem of wave scattering by many small particles is studied under the assumption k<<1, d>>a, where a is the size of the particles and d is the distance between the neighboring particles. Impedance boundary conditions are assumed on the boundaries of small particles. The results of numerical simulation show good agreement with the theory. They open a way to numerical simulation of the method for creating materials with a desired refraction coefficient.

physics.comp-ph↗
arXiv2012

Scattering of electromagnetic waves by many thin cylinders: theory and computational modeling

Electromagnetic (EM) wave scattering by many parallel infinite cylinders is studied asymptotically as a tends to 0, where a is the radius of the cylinders. It is assumed that the centres of the cylinders are distributed so that their numbers is determined by some positive function N(x). The function N(x) >= 0 is a given continuous function. An equation for the self-consistent (limiting) field is derived as a tends to 0. The cylinders are assumed perfectly conducting. Formula for the effective refraction coefficient of the new medium, obtained by embedding many thin cylinders into a given region, is derived. The numerical results presented demonstrate the validity of the proposed approach and its efficiency for solving the many-body scattering problems, as well as the possibility to create media with negative refraction coefficients.

physics.comp-ph↗
arXiv2011

Permeabilities of metamaterials

Scattering of electromagnetic (EM) waves by many small particles, embedded in a given medium, is studied. Physical properties of the particles are described by their boundary impedances. The limiting equation is obtained for the effective EM field in the limiting medium, in the limit $a\to 0$, where $a$ is the characteristic size of a particle and the number $M(a)$ of the particles tends to infinity at a suitable rate. An analytical formula for the permeability $μ(x)$ of the limiting medium is given. Analysis of this formula allows one to find out the range of the values of the permeability in the material, obtained by embedding many small particles.

physics.optics↗
arXiv2011

Wave scattering by small bodies and creating materials with a desired refraction coefficient

Asymptotic solution to many-body wave scattering problem is given in the case of many small scatterers. The small scatterers can be particles whose physical properties are described by the boundary impedances, or they can be small inhomogeneities, whose physical properties are described by their refraction coefficients. Equations for the effective field in the limiting medium are derived. The limit is considered as the size $a$ of the particles or inhomogeneities tends to zero while their number $M(a)$ tends to infinity. These results are applied to the problem of creating materials with a desired refraction coefficient. For example, the refraction coefficient may have wave-focusing property, or it may have negative refraction, i.e., the group velocity may be directed opposite to the phase velocity. This paper is a review of the author's results presented in MR2442305 (2009g:78016), MR2354140 (2008g:82123), MR2317263 (2008a:35040), MR2362884 (2008j:78010), and contains new results.

math-ph↗
arXiv2011

Electromagnetic wave scattering by a thin layer in which many small particles are embedded

Scattering of electromagnetic (EM) waves by many small particles (bodies), embedded in a thin layer, is studied. Physical properties of the particles are described by their boundary impedances. The thin layer of depth of the order $O(a)$ with many embedded in it small particles of characteristic size $a$, is described by a boundary condition on the surface of the layer. The limiting interface boundary condition is obtained for the effective EM field in the limiting medium, in the limit $a\to 0$, where the number $M(a)$ of the particles tends to infinity at a suitable rate.

math-ph↗
arXiv2011

Electromagnetic wave scattering by many small particles and creating materials with a desired permeability

Scattering of electromagnetic (EM) waves by many small particles (bodies) embedded in a homogeneous medium is studied. Physical properties of the particles are described by their boundary impedances. The limiting equation is obtained for the effective EM field in the limiting medium, in the limit $a\to 0$, where $a$ is the characteristic size of a particle and the number $M(a)$ of the particles tends to infinity at a suitable rate. The proposed theory allows one to create a medium with an inhomogeneous permeability. The main new physical result is the explicit analytical formula for the permeability $μ(x)$ of the limiting medium. While the initial medium has a constant permeability $μ_0$, the limiting medium, obtained as a result of embedding many small particles with prescribed boundary impedances, has a non-homogeneous permeability which is expressed analytically in terms of the density of the distribution of the small particles and their boundary impedances.

math-ph↗
arXiv2010

Justification of the Dynamical Systems Method (DSM) for global homeomorphisms

The Dynamical Systems Method (DSM) is justified for solving operator equations $F(u)=f$, where $F$ is a nonlinear operator in a Hilbert space $H$. It is assumed that $F$ is a global homeomorphism of $H$ onto $H$, that $F\in C^1_{loc}$, that is, it has a continuous with respect to $u$ Fréchet derivative $F'(u)$, that the operator $[F'(u)]^{-1}$ exists for all $u\in H$ and is bounded, $||[F'(u)]^{-1}||\leq m(u)$, where $m(u)>0$ is a constant, depending on $u$, and not necessarily uniformly bounded with respect to $u$. It is proved under these assumptions that the continuous analog of the Newton's method $\dot{u}=-[F'(u)]^{-1}(F(u)-f), \quad u(0)=u_0, \quad (*)$ converges strongly to the solution of the equation $F(u)=f$ for any $f\in H$ and any $u_0\in H$. The global (and even local) existence of the solution to the Cauchy problem (*) was not established earlier without assuming that $F'(u)$ is Lipschitz-continuous. The case when $F$ is not a global homeomorphism but a monotone operator in $H$ is also considered.

math.NA↗
arXiv2010

Stability of solutions to some evolution problem

Large time behavior of solutions to abstract differential equations is studied. The corresponding evolution problem is: $$\dot{u}=A(t)u+F(t,u)+b(t), \quad t\ge 0; \quad u(0)=u_0. \qquad (*)$$ Here $\dot{u}:=\frac {du}{dt}$, $u=u(t)\in H$, $t\in \R_+:=[0,\infty)$, $A(t)$ is a linear dissipative operator: Re$(A(t)u,u)\le -γ(t)(u,u)$, $γ(t)\ge 0$, $F(t,u)$ is a nonlinear operator, $\|F(t,u)\|\le c_0\|u\|^p$, $p>1$, $c_0,p$ are constants, $\|b(t)\|\le β(t),$ $β(t)\ge 0$ is a continuous function. Sufficient conditions are given for the solution $u(t)$ to problem (*) to exist for all $t\ge0$, to be bounded uniformly on $\R_+$, and a bound on $\|u(t)\|$ is given. This bound implies the relation $\lim_{t\to \infty}\|u(t)\|=0$ under suitable conditions on $γ(t)$ and $β(t)$. The basic technical tool in this work is the following nonlinear inequality: $$ \dot{g}(t)\leq -γ(t)g(t)+α(t,g(t))+β(t),\ t\geq 0;\quad g(0)=g_0. $$

math.DS↗
arXiv2010

Scattering by many small inhomogeneities and applications

Many-body quantum-mechanical scattering problem is solved asymptotically when the size of the scatterers (inhomogeneities) tends to zero and their number tends to infinity. A method is given for calculation of the number of small inhomogeneities per unit volume and their intensities such that embedding of these inhomogeneities in a bounded region results in creating a new system, described by a desired potential. The governing equation for this system is a non-relativistic Schrödinger equation described by a desired potential. Similar ideas were developed by the author for acoustic and electromagnetic (EM) wave scattering problems.

math-ph↗
arXiv2010

Asymptotic stability of solutions to abstract differential equations

An evolution problem for abstract differential equations is studied. The typical problem is: $$\dot{u}=A(t)u+F(t,u), \quad t\geq 0; \,\, u(0)=u_0;\quad \dot{u}=\frac {du}{dt}\qquad (*)$$ Here $A(t)$ is a linear bounded operator in a Hilbert space $H$, and $F$ is a nonlinear operator, $\|F(t,u)\|\leq c_0\|u\|^p,\,\,p>1$, $c_0, p=const>0$. It is assumed that Re$(A(t)u,u)\leq -γ(t)\|u\|^2$ $\forall u\in H$, where $γ(t)>0$, and the case when $\lim_{t\to \infty}γ(t)=0$ is also considered. An estimate of the rate of decay of solutions to problem (*) is given. The derivation of this estimate uses a nonlinear differential inequality.

math.DS↗
arXiv2010

A nonlinear inequality and evolution problems

Assume that $g(t)\geq 0$, and $$\dot{g}(t)\leq -γ(t)g(t)+α(t,g(t))+β(t),\ t\geq 0;\quad g(0)=g_0;\quad \dot{g}:=\frac{dg}{dt}, $$ on any interval $[0,T)$ on which $g$ exists and has bounded derivative from the right, $\dot{g}(t):=\lim_{s\to +0}\frac{g(t+s)-g(t)}{s}$. It is assumed that $γ(t)$, and $β(t)$ are nonnegative continuous functions of $t$ defined on $\R_+:=[0,\infty)$, the function $α(t,g)$ is defined for all $t\in \R_+$, locally Lipschitz with respect to $g$ uniformly with respect to $t$ on any compact subsets$[0,T]$, $T<\infty$, and non-decreasing with respect to $g$, $α(t,g_1)\geq α(t,g_2)$ if $g_1\ge g_2$. If there exists a function $μ(t)>0$, $μ(t)\in C^1(\R_+)$, such that $$α\left(t,\frac{1}{μ(t)}\right)+β(t)\leq \frac{1}{μ(t)}\left(γ(t)-\frac{\dotμ(t)}{μ(t)}\right),\quad \forall t\ge 0;\quad μ(0)g(0)\leq 1,$$ then $g(t)$ exists on all of $\R_+$, that is $T=\infty$, and the following estimate holds: $$0\leq g(t)\le \frac 1{μ(t)},\quad \forall t\geq 0. $$ If $μ(0)g(0)< 1$, then $0\leq g(t)< \frac 1{μ(t)},\quad \forall t\geq 0. $ A discrete version of this result is obtained. The nonlinear inequality, obtained in this paper, is used in a study of the Lyapunov stability and asymptotic stability of solutions to differential equations in finite and infinite-dimensional spaces.

math.CA↗
arXiv2010

Stability of solutions to abstract differential equations

A sufficient condition for asymptotic stability of the zero solution to an abstract nonlinear evolution problem is given. The governing equation is $\dot{u}=A(t)u+F(t,u),$ where $A(t)$ is a bounded linear operator in Hilbert space $H$ and $F(t,u)$ is a nonlinear operator, $\|F(t,u)\|\leq c_0\|u\|^{1+p}$, $p=const >0$, $c_0=const>0$. It is not assumed that the spectrum $σ:=σ(A(t))$ of $A(t)$ lies in the fixed halfplane Re$z\leq -κ$, where $κ>0$ does not depend on $t$. As $t\to \infty$ the spectrum of $A(t)$ is allowed to tend to the imaginary axis.

math.CA↗
arXiv2010

Some nonlinear inequalities and applications

Sufficient conditions are given for the relation $\lim_{t\to\infty}y(t) = 0$ to hold, where $y(t)$ is a continuous nonnegative function on $[0,1)$ satisfying some nonlinear inequalities. The results are used for a study of large time behavior of the solutions to nonlinear evolution equations. Example of application is given for a solution to some evolution equation with a nonlinear partial differential operator.

math.CA↗
arXiv2010

Nonlinear differential inequality

A nonlinear inequality is formulated in the paper. An estimate of the rate of growth/decay of solutions to this inequality is obtained. This inequality is of interest in a study of dynamical systems and nonlinear evolution equations. It can be applied to a study of global existence of solutions to nonlinear PDE.

math.CA↗
arXiv2010

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 $Au=f$, where $A$ is a linear or nonlinear operator in a Hilbert space $H$, it is assumed that the noisy data $\{f_δ, δ\}$ are given, $||f-f_δ||\leq δ$, 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 $y$ solves the equation $Au=f$, i.e., $Ay=f$. In this definition $y$ and $f$ are unknown. Any $f\in B(f_\d,\d)$ can be the exact data, where $B(f_\d,\d):=\{f: ||f-f_δ||\leq δ\}$.The new notion of the stable solution excludes the unknown $y$ and $f$ from the definition of the solution.

math.NA↗