Spectral Analysis to the Derivation of Green's Function of Non – Homogeneous Acoustic Helmholtz Integral Equation Via Dirac - Delta Function and Cauchy Residual Approach

Chapter 12 introduces Graphene, which is a two-dimensional “Dirac-like” material in the sense that its energy spectrum resembles that of a relativistic electron/positron (hole) described by the Dirac equation (having zero mass in this case). Its device-friendly properties of high electron mobility and excellent sensitivity as a sensor have attracted a huge world-wide research effort since its discovery about ten years ago. Here, the associated retarded Graphene Green’s function is treated and the dynamic, non-local dielectric function is discussed in the degenerate limit. The effects of a quantizing magnetic field on the Green’s function of a Graphene sheet and on its energy spectrum are derived in detail: Also the magnetic-field Green’s function and energy spectrum of a Graphene sheet with a quantum dot (modelled by a 2D Dirac delta-function potential) are thoroughly examined. Furthermore, Chapter 12 similarly addresses the problem of a Graphene anti-dot lattice in a magnetic field, discussing the Green’s function for propagation along the lattice axis, with a formulation of the associated eigen-energy dispersion relation. Finally, magnetic Landau quantization effects on the statistical thermodynamics of Graphene, including its Free Energy and magnetic moment, are also treated in Chapter 12 and are seen to exhibit magnetic oscillatory features.

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The propagator matrix related to the Kirchhoff‐Helmholtz integral in inverse wavefield extrapolation

Geophysics ◽

10.1190/1.1443577 ◽

1994 ◽

Vol 59 (12) ◽

pp. 1902-1910 ◽

Cited By ~ 7

Author(s):

Lasse Amundsen

Keyword(s):

Integral Equation ◽

Green’S Function ◽

Green's Function ◽

Matrix Method ◽

Pressure Field ◽

Normal Derivative ◽

Closed Surface ◽

Observation Point ◽

Propagator Matrix ◽

Helmholtz Integral

The Kirchhoff‐Helmholtz formula for the wavefield inside a closed surface surrounding a volume is most commonly given as a surface integral over the field and its normal derivative, given the Green’s function of the problem. In this case the source point of the Green’s function, or the observation point, is located inside the volume enclosed by the surface. However, when locating the observation point outside the closed surface, the Kirchhoff‐Helmholtz formula constitutes a functional relationship between the field and its normal derivative on the surface, and thereby defines an integral equation for the fields. By dividing the closed surface into two parts, one being identical to the (infinite) data measurement surface and the other identical to the (infinite) surface onto which we want to extrapolate the data, the solution of the Kirchhoff‐Helmholtz integral equation mathematically gives exact inverse extrapolation of the field when constructing a Green’s function that generates either a null pressure field or a null normal gradient of the pressure field on the latter surface. In the case when the surfaces are plane and horizontal and the medium parameters are constant between the surfaces, analysis in the wavenumber domain shows that the Kirchhoff‐Helmholtz integral equation is equivalent to the Thomson‐Haskell acoustic propagator matrix method. When the medium parameters have smooth vertical gradients, the Kirchhoff‐Helmholtz integral equation in the high‐frequency approximation is equivalent to the WKBJ propagator matrix method, which also is equivalent to the extrapolation method denoted by extrapolation by analytic continuation.

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Retarded Green’s Functions

10.1093/oso/9780198791942.003.0005 ◽

2018 ◽

Author(s):

Norman J. Morgenstern Horing

Keyword(s):

Magnetic Fields ◽

Green’S Function ◽

Green's Function ◽

Green's Functions ◽

Green’S Functions ◽

Impurity Scattering ◽

Delta Function ◽

Two Dimensions ◽

Three Dimensions ◽

Dirac Delta

Chapter 5 introduces single-particle retarded Green’s functions, which provide the probability amplitude that a particle created at (x, t) is later annihilated at (x′,t′). Partial Green’s functions, which represent the time development of one (or a few) state(s) that may be understood as localized but are in interaction with a continuum of states, are discussed and applied to chemisorption. Introductions are also made to the Dyson integral equation, T-matrix and the Dirac delta-function potential, with the latter applied to random impurity scattering. The retarded Green’s function in the presence of random impurity scattering is exhibited in the Born and self-consistent Born approximations, with application to Ando’s semi-elliptic density of states for the 2D Landau-quantized electron-impurity system. Important retarded Green’s functions and their methods of derivation are discussed. These include Green’s functions for electrons in magnetic fields in both three dimensions and two dimensions, also a Hamilton equation-of-motion method for the determination of Green’s functions with application to a 2D saddle potential in a time-dependent electric field. Moreover, separable Hamiltonians and their product Green’s functions are discussed with application to a one-dimensional superlattice in axial electric and magnetic fields. Green’s function matching/joining techniques are introduced and applied to spatially varying mass (heterostructures) and non-local electrostatics (surface plasmons).

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On the Fractional Derivative of Dirac Delta Function and Its Application

Advances in Mathematical Physics ◽

10.1155/2020/1842945 ◽

2020 ◽

Vol 2020 ◽

pp. 1-7

Author(s):

Zaiyong Feng ◽

Linghua Ye ◽

Yi Zhang

Keyword(s):

Integral Equation ◽

Laplace Transform ◽

Fractional Derivative ◽

Fractional Order ◽

Delta Function ◽

Dirac Delta Function ◽

Fractional Order System ◽

Order System ◽

Dirac Delta ◽

Integer Order

The Dirac delta function and its integer-order derivative are widely used to solve integer-order differential/integral equation and integer-order system in related fields. On the other hand, the fractional-order system gets more and more attention. This paper investigates the fractional derivative of the Dirac delta function and its Laplace transform to explore the solution for fractional-order system. The paper presents the Riemann-Liouville and the Caputo fractional derivative of the Dirac delta function, and their analytic expression. The Laplace transform of the fractional derivative of the Dirac delta function is given later. The proposed fractional derivative of the Dirac delta function and its Laplace transform are effectively used to solve fractional-order integral equation and fractional-order system, the correctness of each solution is also verified.

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Computing the Green's Function of the Initial Boundary Value Problem for the Wave Equation in a Radially Layered Cylinder

International Journal of Computational Methods ◽

10.1142/s0219876215500279 ◽

2015 ◽

Vol 12 (05) ◽

pp. 1550027

Author(s):

V. Yakhno ◽

D. Ozdek

Keyword(s):

Fourier Series ◽

Boundary Value Problem ◽

Finite Number ◽

Green’S Function ◽

Green's Function ◽

Initial Boundary ◽

Initial Boundary Value Problem ◽

Delta Function ◽

Dirac Delta ◽

Initial Boundary Value

In this paper, a method for construction of the time-dependent approximate Green's function for the initial boundary value problem in a radially multilayered cylinder is suggested. This method is based on determination of the eigenvalues and the orthogonal set of the eigenfunctions; regularization of the Dirac delta function in the form of the Fourier series with a finite number of terms; expansion of the unknown Green's function in the form of Fourier series with unknown coefficients and computation of a finite number of unknown Fourier coefficients. Computational experiment confirms the robustness of the method for the approximate computation of the Dirac delta function and Green's function.

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