The Structure of Fractional Spaces Generated by a Two-dimensional Difference Operator in a Half Plane

2018 ◽  
Vol 70 (8) ◽  
pp. 1176-1191
Author(s):  
A. Ashyralyev ◽  
S. Akturk
Keyword(s):  
Half Plane ◽  
Difference Operator ◽  
Two Dimensional ◽  
Fractional Spaces

A note on positivity of two-dimensional differential operators

Filomat ◽  
2017 ◽  
Vol 31 (14) ◽  
pp. 4651-4663 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Sema Akturk
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Half Plane ◽  
Differential Operators ◽  
Elliptic Problems ◽  
Well Posedness ◽  
Two Dimensional ◽  
Ellipticity Condition ◽  
Fractional Spaces ◽  
Uniform Ellipticity

We consider the two-dimensional differential operator A(t,x)u(t,x) = -a11 (t, x) utt -a22(t,x)uxx +?u defined on functions on the half-plane R+ x R with the boundary condition u(0,x) = 0, x ? R where aii(t,x), i = 1,2 are continuously differentiable and satisfy the uniform ellipticity condition a2 11(t,x) + a222(t,x)? ? > 0, ? > 0. The structure of fractional spaces E?,1 (L1 (R+ x R), A(t,x)) generated by the operator A(t,x) is investigated. The positivity of A(t,x) in L1 (W2?1(R+ x R)) spaces is established. In applications, theorems on well-posedness in L1 (W2?1 (R+ x R)) spaces of elliptic problems are obtained.

Diffraction of a pulse by a resistive half-plane. I. Normal incidence

1960 ◽  
Vol 255 (1283) ◽  
pp. 538-549 ◽  
Keyword(s):  
Time Dependence ◽  
Half Plane ◽  
Wave Diffraction ◽  
Diffraction Problem ◽  
Normal Incidence ◽  
Step Function ◽  
Two Dimensional ◽  
Dynamic Similarity ◽  
Function Time ◽  
Dimensional Wave

The two-dimensional wave diffraction problem, acoustic or electromagnetic, in which a pulse of step-function time dependence is diffracted by a resistive half-plane is solved by assuming dynamic similarity in the solution.

Two-Dimensional Contact Analysis Using Trigonometric Polynomials: Some Early Verification Problems

2018 ◽  
Author(s):  
Gaurav Chauda ◽  
Daniel J. Segalman
Keyword(s):  
Half Plane ◽  
Frictional Contact ◽  
Contact Analysis ◽  
Trigonometric Polynomials ◽  
Elastic Contact ◽  
Two Dimensional ◽  
Contact Algorithm ◽  
Numerical Predictions ◽  
Interface Mechanics ◽  
Friction Models

A discretization strategy for elastic contact on a half plane has been devised to explore the significance of different friction models on joint-like interface mechanics. It is necessary to verify that discretization and accompanying contact algorithm on known solutions. An extensive comparison of numerical predictions of this model with corresponding 2-D elastic, frictional contact solutions from the literature is presented.

Diffraction by a half plane perpendicular to the distinguished axis of a gyrotropic medium (oblique incidence)

1981 ◽  
Vol 59 (3) ◽  
pp. 403-424 ◽  
Author(s):  
S. Przeździecki ◽  
R. A. Hurd
Keyword(s):  
Half Plane ◽  
Fourier Transforms ◽  
Boundary Value ◽  
Diffraction Problem ◽  
Closed Form Solution ◽  
Second Order ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Dimensional Solution ◽  
Electromagnetic Plane Wave

An exact, closed form solution is found for the following half plane diffraction problem. (I) The medium surrounding the half plane is gyrotropic. (II) The scattering half plane is perfectly conducting and oriented perpendicular to the distinguished axis of the medium. (III) The direction of propagation of the incident electromagnetic plane wave is arbitrary (skew) with respect to the edge of the half plane. The result presented is a generalization of a solution for the same problem with incidence normal to the edge of the half plane (two-dimensional case).The fundamental, distinctive feature of the problem is that it constitutes a boundary value problem for a system of two coupled second order partial differential equations. All previously solved electromagnetic diffraction problems reduced to boundary value problems for either one or two uncoupled second order equations. (Exception: the two-dimensional case of the present problem.) The problem is formulated in terms of the (generalized) scalar Hertz potentials and leads to a set of two coupled Wiener–Hopf equations. This set, previously thought insoluble by quadratures, yields to the Wiener–Hopf–Hilbert method.The three-dimensional solution is synthesized from appropriate solutions to two-dimensional problems. Peculiar waves of ghost potentials, which correspond to zero electromagnetic fields play an essential role in this synthesis. The problem is two-moded: that is, superpositions of both ordinary and extraordinary waves are necessary for the spectral representation of the solution. An important simplifying feature of the problem is that the coupling of the modes is purely due to edge diffraction, there being no reflection coupling. The solution is simple in that the Fourier transforms of the potentials are just algebraic functions. Basic properties of the solution are briefly discussed.

scholarly journals The Diffraction of Transient Electro-Magnetic Waves by a Wedge

1958 ◽  
Vol 11 (2) ◽  
pp. 95-103 ◽  
Author(s):  
A. C. Butcher ◽  
J. S. Lowndes
Keyword(s):  
Wave Equation ◽  
Time Dependence ◽  
Half Plane ◽  
Line Source ◽  
Time Dependent ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Electro Magnetic ◽  
Infinite Wedge ◽  
Magnetic Waves

Much of the work on the theory of diffraction by an infinite wedge has been for cases of harmonic time-dependence. Oberhettinger (1) obtained an expression for the Green's function of the wave equation in the two dimensional case of a line source of oscillating current parallel to the edge of a wedge with perfectly conducting walls. Solutions of the time-dependent wave equation have been obtained by Keller and Blank (2), Kay (3) and more recently by Turner (4) who considered the diffraction of a cylindrical pulse by a half plane.

The half-plane diffraction problem for harmonic time dependence

1959 ◽  
Vol 252 (1270) ◽  
pp. 411-417 ◽  
Keyword(s):  
Electromagnetic Field ◽  
Time Dependence ◽  
Boundary Value Problems ◽  
Half Plane ◽  
Mixed Type ◽  
Boundary Value ◽  
Diffraction Problem ◽  
Two Dimensional ◽  
Conducting Screen ◽  
Plane Diffraction

Green’s functions are obtained for the boundary-value problems of mixed type describing the general two-dimensional diffraction problems at a screen in the form of a half-plane (Sommerfeld’s problem), applicable to acoustically rigid or soft screens, and to the full electromagnetic field at a perfectly conducting screen.

ON EUCLIDEAN G-MANIFOLDS WHICH HAVE TWO DIMENSIONAL ORBIT SPACES

2011 ◽  
Vol 22 (03) ◽  
pp. 399-406
Author(s):  
R. MIRZAIE
Keyword(s):  
Euclidean Space ◽  
Half Plane ◽  
Lie Group ◽  
Orbit Space ◽  
Two Dimensional ◽  
Orbit Spaces ◽  
Group Of Isometries ◽  
Connected Lie Group

We show that the orbit space of Euclidean space, under the action of a closed and connected Lie group of isometries is homeomorphic to a plane or closed half-plane, if the action is of cohomogeneity two.

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