scholarly journals Scattering of electromagnetic waves by a discrete octahedron from resonant spheres

Radiotekhnika ◽  
2020 ◽  
pp. 181-185
Author(s):  
A.I. Kozar
Keyword(s):  
Electromagnetic Waves ◽  
Nonlocal Boundary ◽  
Scattered Wave ◽  
Nonlocal Boundary Conditions ◽  
Internal Field ◽  
Fredholm Integral Equations ◽  
X Ray ◽  
Equations Of Electrodynamics ◽  
Resonance Properties ◽  
Discrete Convex

A solution is given to the problem of scattering of electromagnetic waves by a discrete convex polyhedron – an octahedron of resonant magnetodielectric spheres based on a complex rhombic crystal lattice. Here we consider a case equivalent to the X-ray optics of crystals, when α / λ՛<<1 and can be α / λg ~ 1; d, h, l / λ՛ ~ 1, where α is the radius of the spheres; λ՛, λg are the lengths of the scattered wave outside and inside the spheres; d, h, l are constant lattices. The solution of the problem is obtained based on the Fredholm integral equations of electrodynamics of the second kind with nonlocal boundary conditions. The expressions found in this work for a metacrystal in the form of an octahedron can be used to study the fields scattered by the crystal in the Fresnel and Fraunhofer zones, as well as to study its internal field. The relations obtained in this work can find application in the study of the scattering of waves of various kinds by convex polyhedrons, the creation on their basis of new types of limited metacrystals, including nanocrystals with resonance properties, and in the study of their behavior in various external media. As well as in the development of methods for modeling electromagnetic phenomena that can occur in real crystals in resonance regions in the optical and X-ray wavelength ranges.

scholarly journals An inverse phaseless problem for electrodynamic equations in an anisotropic medium

2019 ◽  
Vol 488 (4) ◽  
pp. 367-371
Author(s):  
V. G. Romanov
Keyword(s):  
Inverse Problem ◽  
Bounded Domain ◽  
Incomplete Data ◽  
Electromagnetic Waves ◽  
Original Problem ◽  
Plane Waves ◽  
Inverse Kinematic ◽  
System Of Equations ◽  
X Ray ◽  
Equations Of Electrodynamics

For the system of equations of electrodynamics which has the anisotropy of the permittivity, an inverse problem of determining the permittivity is studied. It is supposed that the permittivity is characterized by the diagonal matrix = diag (1(x), 1(x), 2(x)) and 1 and 2 are positive constants anywhere outside of a bounded domain 0 3. Periodic in time solutions of the system of Maxwells equations related to two modes of plane waves falled down from infinity on the local non-homogeneity located in 0 is considered. For determining functions 1(x) and 2(x) some information on the module of the vector of the electric strength of two interfered waves is given. It is demonstrated that this information reduces the original problem to two inverse kinematic problems with incomplete data about travel times of the electromagnetic waves. An investigation of the linearized statement for these problems is given. It is shown that in the linear approximation the problem of the determining 1(x) and 2(x) is reduced to two X-ray tomography problems.

ON THE SOLVABILITY OF A MIXED PROBLEM WITH DEGREE LAPLACE OPERATORS WITH NONLOCAL BOUNDARY CONDITIONS

2020 ◽  
pp. 85-104
Author(s):  
Shakirbai G. Kasimov ◽  
◽  
Mahkambek M. Babaev ◽  
◽  
Keyword(s):  
Boundary Conditions ◽  
Spectral Problem ◽  
Nonlocal Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlocal Boundary Conditions ◽  
Laplace Operators ◽  
Initial Boundary Problem ◽  
Initial Boundary Value ◽  
Delayed Time

The paper studies a problem with initial functions and boundary conditions for partial differential partial equations of fractional order in partial derivatives with a delayed time argument, with degree Laplace operators with spatial variables and nonlocal boundary conditions in Sobolev classes. The solution of the initial boundary-value problem is constructed as the series’ sum in the eigenfunction system of the multidimensional spectral problem. The eigenvalues are found for the spectral problem and the corresponding system of eigenfunctions is constructed. It is shown that the system of eigenfunctions is complete and forms a Riesz basis in the Sobolev subspace. Based on the completeness of the eigenfunctions system the uniqueness theorem for solving the problem is proved. In the Sobolev subspaces the existence of a regular solution to the stated initial-boundary problem is proved.

scholarly journals Multi-term fractional differential equations with nonlocal boundary conditions

2018 ◽  
Vol 16 (1) ◽  
pp. 1519-1536
Author(s):  
Bashir Ahmad ◽  
Najla Alghamdi ◽  
Ahmed Alsaedi ◽  
Sotiris K. Ntouyas
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Fractional Differential ◽  
The Given

AbstractWe introduce and study a new kind of nonlocal boundary value problems of multi-term fractional differential equations. The existence and uniqueness results for the given problem are obtained by applying standard fixed point theorems. We also construct some examples for demonstrating the application of the main results.

Solving a linear fractional equation with nonlocal boundary conditions based on multiscale orthonormal bases method in the reproducing kernel space

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Wei Jiang ◽  
Zhong Chen ◽  
Ning Hu ◽  
Yali Chen
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Differential Equations ◽  
Reproducing Kernel ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Kernel Space ◽  
Reproducing Kernel Space ◽  
Orthonormal Bases ◽  
Fractional Differential

AbstractIn recent years, the study of fractional differential equations has become a hot spot. It is more difficult to solve fractional differential equations with nonlocal boundary conditions. In this article, we propose a multiscale orthonormal bases collocation method for linear fractional-order nonlocal boundary value problems. In algorithm construction, the solution is expanded by the multiscale orthonormal bases of a reproducing kernel space. The nonlocal boundary conditions are transformed into operator equations, which are involved in finding the collocation coefficients as constrain conditions. In theory, the convergent order and stability analysis of the proposed method are presented rigorously. Finally, numerical examples show the stability, accuracy and effectiveness of the method.

scholarly journals New general decay result for a system of two singular nonlocal viscoelastic equations with general source terms and a wide class of relaxation functions

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Mohammad M. Al-Gharabli ◽  
Adel M. Al-Mahdi ◽  
Salim A. Messaoudi
Keyword(s):  
Wide Class ◽  
Relaxation Function ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
General Assumption ◽  
General Decay ◽  
Source Terms ◽  
Relaxation Functions ◽  
Viscoelastic Equations ◽  
General Source

Abstract This work is concerned with a system of two singular viscoelastic equations with general source terms and nonlocal boundary conditions. We discuss the stabilization of this system under a very general assumption on the behavior of the relaxation function $k_{i}$ k i , namely, $$\begin{aligned} k_{i}^{\prime }(t)\le -\xi _{i}(t) \Psi _{i} \bigl(k_{i}(t)\bigr),\quad i=1,2. \end{aligned}$$ k i ′ ( t ) ≤ − ξ i ( t ) Ψ i ( k i ( t ) ) , i = 1 , 2 . We establish a new general decay result that improves most of the existing results in the literature related to this system. Our result allows for a wider class of relaxation functions, from which we can recover the exponential and polynomial rates when $k_{i}(s) = s^{p}$ k i ( s ) = s p and p covers the full admissible range $[1, 2)$ [ 1 , 2 ) .

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