On Hankel Transformable Spaces and a Cauchy Problem

1985 ◽  
Vol 37 (1) ◽  
pp. 84-106 ◽  
Author(s):  
R. S. Pathak
Keyword(s):  
Cauchy Problem ◽  
Adjoint Method ◽  
Kernel Method ◽  
Direct Method ◽  
Integral Transforms ◽  
Hankel Transform ◽  
Generalized Functions ◽  
One Dimension ◽  
Special Cases ◽  
Image Position

The classical Hankel transform of a conventional function ϕ on (0, ∞) defined formally bywas extended by Zemanian [21-23] to certain generalized functions of one dimension. Koh [9, 10] extended the work of [21] to n-dimensions, and that of [22] to arbitrary real values of μ. Motivated from the work of Gelfand and Shilov [6], Lee [11] introduced spaces of type Hμ and studied their Hankel transforms. The results of Lee [11] and Zemanian [21] are special cases of recent results obtained by the author and Pandey [14]. The aforesaid extensions are accomplished by using the so-called adjoint method of extending integral transforms to generalized functions. Dube and Pandey [2], Pathak and Pandey [15, 16] applied a more direct method, the so-called kernel method, for extending the Hankel and other related transforms.

scholarly journals A Class of Integral Transforms

1964 ◽  
Vol 14 (1) ◽  
pp. 33-40 ◽  
Author(s):  
Jet Wimp
Keyword(s):  
Heat Conduction ◽  
Boundary Value Problems ◽  
Inversion Formula ◽  
Heat Conduction Equation ◽  
Boundary Value ◽  
Integral Transforms ◽  
Conduction Equation ◽  
New Class ◽  
Special Cases ◽  
Image Position

In this paper we discuss a new class of integral transforms and their inversion formula. The kernel in the transform is a G-function (for a treatment of this function, see ((1), 5.3) and integration is performed with respect to the argument of that function. In the inversion formula, the kernel is likewise a G-function, but there integration is performed with respect to a parameter. Known special cases of our results are the Kontorovitch-Lebedev transform pair ((2), v. 2; (3))and the generalised Mehler transform pair (7)These transforms are used in solving certain boundary value problems of the wave or heat conduction equation involving wedge or conically-shaped boundaries, and are extensively tabulated in (6).

Distributional Watson Transforms

1972 ◽  
Vol 24 (6) ◽  
pp. 1191-1197 ◽  
Author(s):  
Hsing-Yuan Hsu
Keyword(s):  
Hilbert Transform ◽  
Mellin Transform ◽  
Hankel Transform ◽  
Special Cases ◽  
Image Position ◽  
Watson Transform

All our notation is as denned in [2] with the restriction to n = 1. However, for our purposes, we introduce a sequence of norms byin It is not difficult to see that turns out to be a fundamental space.It is a well-known fact that the Watson transform and the Mellin transform are connected by the fact thatandif and only if K(s)K(l — s) = 1, where K(s) is the Mellin transform of k(x). Further, the Hankel transform and Hilbert transform can be considered as special cases of Watson transforms.

On Self-reciprocal functions for Fourier-Bessel integral transforms

1961 ◽  
Vol 57 (4) ◽  
pp. 778-781
Author(s):  
Afzal Ahmad ◽  
V. Lakshmikanth
Keyword(s):  
Bessel Function ◽  
Integral Transform ◽  
Integral Transforms ◽  
Special Cases ◽  
Image Position

Following Hardy and Titchmarsh(1) a function f(x) is said to be self-reciprocal if it satisfies the Fourier-Bessel integral transformwhere Jp(x) is a Bessel function of order P ≥ –½. This integral is denoted by Rp. The special cases P ½ and P ½, we denote by Rs and Rc, respectively.

scholarly journals The H-function transform

1970 ◽  
Vol 11 (2) ◽  
pp. 142-148 ◽  
Author(s):  
K. C. Gupta ◽  
P. K. Mittal
Keyword(s):  
Laplace Transform ◽  
Inversion Formula ◽  
Integral Transform ◽  
Applied Mathematics ◽  
Integral Transforms ◽  
Hankel Transform ◽  
Stieltjes Transform ◽  
Special Cases ◽  
Mathematics And Physics

Here we introduce a new integral transform whose kernel is the H-function. Since most of the important functions occurring in Applied Mathematics and Physics are special cases of the H-function, various integral transforms involving these functions as kernels follow as special cases of our transform. We mention some of them here and observe that a study of this transform gives general and useful results which serve as key formulae for several important integral transforms viz. Laplace transform, Hankel transform. Stieltjes transform and the various generalizations of these transforms. In the end we establish an inversion formula for the new transform and point out its special cases which are generalizations of results found recently.

Mkhitar Djrbashian and his contribution to Fractional Calculus

2020 ◽  
Vol 23 (6) ◽  
pp. 1797-1809
Author(s):  
Sergei Rogosin ◽  
Maryna Dubatovskaya
Keyword(s):  
Cauchy Problem ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Approximation Theory ◽  
Fractional Derivatives ◽  
Complex Analysis ◽  
Integral Transforms ◽  
Modern Theory ◽  
Survey Paper ◽  
The Cauchy Problem

Abstract This survey paper is devoted to the description of the results by M.M. Djrbashian related to the modern theory of Fractional Calculus. M.M. Djrbashian (1918-1994) is a well-known expert in complex analysis, harmonic analysis and approximation theory. Anyway, his contributions to fractional calculus, to boundary value problems for fractional order operators, to the investigation of properties of the Queen function of Fractional Calculus (the Mittag-Leffler function), to integral transforms’ theory has to be understood on a better level. Unfortunately, most of his works are not enough popular as in that time were published in Russian. The aim of this survey is to fill in the gap in the clear recognition of M.M. Djrbashian’s results in these areas. For same purpose, we decided also to translate in English one of his basic papers [21] of 1968 (joint with A.B. Nersesian, “Fractional derivatives and the Cauchy problem for differential equations of fractional order”), and were invited by the “FCAA” editors to publish its re-edited version in this same issue of the journal.

The theory of weak functions. I

1963 ◽  
Vol 276 (1365) ◽  
pp. 149-167 ◽  
Keyword(s):  
Continuous Function ◽  
Weak Convergence ◽  
Fourier Transforms ◽  
Generalized Functions ◽  
One Dimension ◽  
Test Functions ◽  
Weak Derivative

This paper develops the theory of distributions or generalized functions without any reference to test functions and with no appeal to topology, apart from the concept of weak convergence. In the calculus of weak functions, which is so obtained, a weak function is always a weak derivative of a numerical continuous function, and the fundamental techniques of multiplication, division and passage to a limit are considerably simplified. The theory is illustrated by application to Fourier transforms. The present paper is restricted to weak functions in one dimension. The extension to several dimensions will be published later.

Determination of Depth of Surface Cracks in Asphalt Pavements

2003 ◽  
Vol 1853 (1) ◽  
pp. 143-149 ◽  
Author(s):  
Shane Underwood ◽  
Y. Richard Kim
Keyword(s):  
Surface Wave ◽  
Wave Velocity ◽  
Ultrasonic Testing ◽  
Kernel Method ◽  
Direct Method ◽  
Velocity Ratio ◽  
Crack Depth ◽  
Asphalt Pavements ◽  
Quantitative Prediction ◽  
Processing Techniques

Nondestructive measurement of crack depths of asphalt pavements in situ could be a valuable tool for engineers in rehabilitation planning. Such measurements currently must be made by first coring or trenching a pavement and then measuring the crack by hand. Two methods for performing this task nondestructively are presented. The two methods, surface wave and ultrasonic, use the slowing effect that a crack has on a wave. Two signal-processing techniques were used to analyze the surface wave method—the fast Fourier transform (FFT) and the short kernel method (SKM). The FFT method provided a frequency spectrum that was used to find the energy carried by specific frequencies. The percent energy reduction (PER) was computed and plotted at each crack depth; this plot revealed that PER values increase as crack depth increases. The SKM method showed the wave velocity to decrease as the crack depth in creased. By comparing the wave velocity of the cracked pavement with that of the undamaged pavement, a phase velocity ratio plot was developed and was shown to be adequate for predicting crack depth. Ultrasonic testing proved to be a simpler and more direct method than surface wave testing. It was not necessary to know the wave properties of an undamaged pavement with this method, and a quantitative prediction of crack depth was obtained. While encouraging results were observed with both methods, ultrasonic testing showed the most promise for application because of the commercial availability of ultrasonic meters and the direct prediction of crack depth.

scholarly journals ON STOCHASTIC GENERALIZED FUNCTIONS

2011 ◽  
Vol 14 (02) ◽  
pp. 237-260 ◽  
Author(s):  
PEDRO CATUOGNO ◽  
CHRISTIAN OLIVERA
Keyword(s):  
Cauchy Problem ◽  
Existence And Uniqueness ◽  
Generalized Functions ◽  
Uniqueness Of The Solution ◽  
Stochastic Cauchy Problem

In this work we introduce a new algebra of stochastic generalized functions. The regular Hida distributions in [Formula: see text] are embedded in this algebra via their chaos expansions. As an application, we prove the existence and uniqueness of the solution of a stochastic Cauchy problem involving singularities.

scholarly journals A Relation between Laplace and Hankel Transforms

1962 ◽  
Vol 5 (3) ◽  
pp. 114-115 ◽  
Author(s):  
B. R. Bhonsle
Keyword(s):  
Laplace Transform ◽  
Hankel Transform ◽  
Laplace And Hankel Transforms ◽  
Hankel Transforms ◽  
The Laplace Transform ◽  
Image Position

The Laplace transform of a function f(t) ∈ L(0, ∞) is defined by the equationand its Hankel transform of order v is defined by the equationThe object of this note is to obtain a relation between the Laplace transform of tμf(t) and the Hankel transform of f(t), when ℛ(μ) > − 1. The result is stated in the form of a theorem which is then illustrated by an example.

On the Cauchy problem associated with the Brinkman flow in

2021 ◽  
pp. 1-20
Author(s):  
Michel Molina Del Sol ◽  
Eduardo Arbieto Alarcon ◽  
Rafael José Iorio
Keyword(s):  
Cauchy Problem ◽  
Porous Media ◽  
Fluid Flow ◽  
Comparison Principle ◽  
Well Posedness ◽  
Quasilinear Equations ◽  
Brinkman Equations ◽  
The Cauchy Problem ◽  
Model Fluid ◽  
Image Position

In this study, we continue our study of the Cauchy problem associated with the Brinkman equations [see (1.1) and (1.2) below] which model fluid flow in certain types of porous media. Here, we will consider the flow in the upper half-space \[ \mathbb{R}_{+}^{3}=\left\{\left(x,y,z\right) \in\mathbb{R}^{3}\left\vert z\geqslant 0\right.\right\}, \] under the assumption that the plane $z=0$ is impenetrable to the fluid. This means that we will have to introduce boundary conditions that must be attached to the Brinkman equations. We study local and global well-posedness in appropriate Sobolev spaces introduced below, using Kato's theory for quasilinear equations, parabolic regularization and a comparison principle for the solutions of the problem.

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