On the Zap Integral Operators over Fourier Transforms

2021 ◽  
Author(s):  
Juan Manuel Velazquez Arcos ◽  
Ricardo Teodoro Paez Hernandez ◽  
Alejandro Perez Ricardez ◽  
Jaime Granados Samaniego ◽  
Alicia Cid Reborido
Keyword(s):  
Integral Equation ◽  
Quantum Mechanics ◽  
Fourier Transforms ◽  
Integral Operators ◽  
Integral Transforms ◽  
Fourier Integral ◽  
Fredholm Equation ◽  
Projection Operators ◽  
Inhomogeneous Term ◽  
Generalized Projection Operators

We devote the current chapter to describe a class of integral operators with properties equivalent to a killer operator of the quantum mechanics theory acting over a determined state, literally killing the state but now operating over some kind of Fourier integral transforms that satisfies a certain Fredholm integral equation, we call this operators Zap Integral Operators (ZIO). The result of this action is to eliminate the inhomogeneous term and recover a homogeneous integral equation. We show that thanks to this class of operators we can explain the presence of two extremely different solutions of the same Generalized Inhomogeneous Fredholm equation. So we can regard the Generalized Inhomogeneous Fredholm Equation as a Super-Equation with two kinds of solutions, the resonant and the conventional but coexisting simultaneously. Also, we remember the generalized projection operators and we show they are the precursors of the ZIO. We present simultaneous academic examples for both kinds of solutions.

The mellin integral transform in fractional calculus

2013 ◽  
Vol 16 (2) ◽  
Author(s):  
Yuri Luchko ◽  
Virginia Kiryakova
Keyword(s):  
Fractional Calculus ◽  
Fourier Transforms ◽  
Integral Transform ◽  
Special Functions ◽  
Integral Transforms ◽  
Fourier Integral ◽  
Monotone Functions ◽  
Completely Monotone ◽  
Mellin Convolution ◽  
New Applications

AbstractIn Fractional Calculus (FC), the Laplace and the Fourier integral transforms are traditionally employed for solving different problems. In this paper, we demonstrate the role of the Mellin integral transform in FC. We note that the Laplace integral transform, the sin- and cos-Fourier transforms, and the FC operators can all be represented as Mellin convolution type integral transforms. Moreover, the special functions of FC are all particular cases of the Fox H-function that is defined as an inverse Mellin transform of a quotient of some products of the Gamma functions.In this paper, several known and some new applications of the Mellin integral transform to different problems in FC are exemplarily presented. The Mellin integral transform is employed to derive the inversion formulas for the FC operators and to evaluate some FC related integrals and in particular, the Laplace transforms and the sin- and cos-Fourier transforms of some special functions of FC. We show how to use the Mellin integral transform to prove the Post-Widder formula (and to obtain its new modi-fication), to derive some new Leibniz type rules for the FC operators, and to get new completely monotone functions from the known ones.

scholarly journals Stationary Dynamic Stress Solutions for a Rectangular Load Applied within a 3D Viscoelastic Isotropic Full-Space

2019 ◽  
Vol 2019 ◽  
pp. 1-12 ◽  
Author(s):  
E. Romanini ◽  
J. Labaki ◽  
E. Mesquita ◽  
R. C. Silva
Keyword(s):  
Fourier Transforms ◽  
Damping Ratio ◽  
Three Dimensional ◽  
Integral Transforms ◽  
Fourier Integral ◽  
Wave Number ◽  
Influence Functions ◽  
Time Harmonic ◽  
Final Stress ◽  
Stress Influence

This paper presents stress influence functions for uniformly distributed, time-harmonic rectangular loads within a three-dimensional, viscoelastic, isotropic full-space. The coupled differential equations relating displacements and stresses in the full-space are solved through double Fourier integral transforms in the wave number domain, in which they can be solved algebraically. The final stress fields are expressed in terms of double indefinite integrals arising from the Fourier transforms. The paper presents numerical schemes with which to integrate these functions accurately. The article presents numerical validation of the synthesized stress kernels and their behavior for high frequencies and large distances from the excitation source. The influence of damping ratio on the dynamic results is also investigated. This article is complementary to previous results of the authors in which the corresponding displacement solutions were derived. Stress influence functions, together with their displacement counterparts, are a fundamental part of many numerical methods of discretization such the boundary element method.

Extending the unified transform: curvilinear polygons and variable coefficient PDEs

2018 ◽  
Vol 40 (2) ◽  
pp. 976-1004 ◽  
Author(s):  
Matthew J Colbrook
Keyword(s):  
Fourier Transforms ◽  
Elliptic Boundary ◽  
Neumann Boundary ◽  
Integral Transforms ◽  
Variable Coefficients ◽  
Central Component ◽  
Unknown Boundary ◽  
Boundary Values ◽  
Global Relation ◽  
Chebyshev Interpolation

Abstract We provide the first significant extension of the unified transform (also known as the Fokas method) applied to elliptic boundary value problems, namely, we extend the method to curvilinear polygons and partial differential equations (PDEs) with variable coefficients. This is used to solve the generalized Dirichlet-to-Neumann map. The central component of the unified transform is the coupling of certain integral transforms of the given boundary data and of the unknown boundary values. This has become known as the global relation and, in the case of constant coefficient PDEs, simply links the Fourier transforms of the Dirichlet and Neumann boundary values. We extend the global relation to PDEs with variable coefficients and to domains with curved boundaries. Furthermore, we provide a natural choice of global relations for separable PDEs. These generalizations are numerically implemented using a method based on Chebyshev interpolation for efficient and accurate computation of the integral transforms that appear in the global relation. Extensive numerical examples are provided, demonstrating that the method presented in this paper is both accurate and fast, yielding exponential convergence for sufficiently smooth solutions. Furthermore, the method is straightforward to use, involving just the construction of a simple linear system from the integral transforms, and does not require knowledge of Green’s functions of the PDE. Details on the implementation are discussed at length.

scholarly journals Asymptotic Behavior of Singular and Entropy Numbers for Some Riemann–Liouville Type Operators

2001 ◽  
Vol 8 (2) ◽  
pp. 323-332
Author(s):  
A. Meskhi
Keyword(s):  
Asymptotic Behavior ◽  
Integral Operators ◽  
Integral Transforms ◽  
Singular Value ◽  
Entropy Numbers ◽  
Liouville Type ◽  
Singular Value Decompositions

Abstract The asymptotic behavior of the singular and entropy numbers is established for the Erdelyi–Köber and Hadamard integral operators (see, e.g., [Samko, Kilbas and Marichev, Integrals and derivatives. Theoryand Applications, Gordon and Breach Science Publishers, 1993]) acting in weighted L 2 spaces. In some cases singular value decompositions are obtained as well for these integral transforms.

Wavelet packets associated with linear canonical transform on spectrum

2021 ◽  
pp. 2150030
Author(s):  
M. Younus Bhat ◽  
Aamir H. Dar
Keyword(s):  
Fourier Transform ◽  
Multiresolution Analysis ◽  
Fourier Transforms ◽  
Fractional Fourier Transform ◽  
Integral Transforms ◽  
Wavelet Packets ◽  
Linear Canonical Transform ◽  
Fresnel Transform ◽  
The Fourier Transform ◽  
Vector Valued

The linear canonical transform (LCT) provides a unified treatment of the generalized Fourier transforms in the sense that it is an embodiment of several well-known integral transforms including the Fourier transform, fractional Fourier transform, Fresnel transform. Using this fascinating property of LCT, we, in this paper, constructed associated wavelet packets. First, we construct wavelet packets corresponding to nonuniform Multiresolution analysis (MRA) associated with LCT and then those corresponding to vector-valued nonuniform MRA associated with LCT. We investigate their various properties by means of LCT.

A lithospheric velocity anomaly beneath the Shagan river Test Site. Part 2. Imaging and inversion with amplitude transmission tomography

1992 ◽  
Vol 82 (2) ◽  
pp. 999-1017
Author(s):  
K. L. McLaughlin ◽  
J. R. Murphy ◽  
B. W. Barker
Keyword(s):  
Fourier Transforms ◽  
Test Site ◽  
Small Scale ◽  
Projection Operators ◽  
Order Partial Differential Equation ◽  
Linear Inversion ◽  
Discrete Fourier Transforms ◽  
Velocity Anomaly ◽  
Inversion Procedure ◽  
River Test

Abstract A linear inversion procedure is introduced that images weak velocity anomalies using amplitudes of transmitted seismic waves. Using projection operators from geometrical ray theory, an image of an anomaly is constructed from amplitudes recorded at arrays of receivers using arrays of sources. The image is related to the velocity anomaly by a second-order partial-differential equation that is inverted using 2-D discrete Fourier transforms. As an example of the inversion procedure, magnitude residuals for European stations recording Shagan River explosions are used to image the deep lithospheric anomaly beneath the Shagan River test site described in Part 1. This formal inversion analysis confirms the existence of a small-scale lateral heterogeneity located 50 km west-northwest of the test site at a probable depth between 80 and 100 km and indicates that it is consistent with a deterministic 1.5% peak-to-peak (or 0.5% rms) velocity anomaly with a scale length of about 3 km. 3-D dynamic raytracing is then used to verify that the inferred laterally varying structure produces amplitude fluctuations consistent with observations.

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