A New Approach for the Solution of the Generalized Abel Integral Equation

2020 ◽  
pp. 145-151
Author(s):  
Tahir Cosgun ◽  
Murat Sari ◽  
Hande Uslu
Keyword(s):  
Integral Equation ◽  
New Approach ◽  
Abel Integral Equation

scholarly journals On the Modular Operator of Mutli-component Regions in Chiral CFT

2021 ◽  
Author(s):  
Stefan Hollands
Keyword(s):  
Field Theory ◽  
Integral Equation ◽  
Hilbert Problem ◽  
Matrix Elements ◽  
New Approach ◽  
Conformal Field ◽  
The Matrix ◽  
Kms Condition ◽  
Modular Operator ◽  
Riemann Hilbert Problem

AbstractWe introduce a new approach to find the Tomita–Takesaki modular flow for multi-component regions in general chiral conformal field theory. Our method is based on locality and analyticity of primary fields as well as the so-called Kubo–Martin–Schwinger (KMS) condition. These features can be used to transform the problem to a Riemann–Hilbert problem on a covering of the complex plane cut along the regions, which is equivalent to an integral equation for the matrix elements of the modular Hamiltonian. Examples are considered.

An efficient approach to the seismogram synthesis for a basin structure using propagation invariants

1996 ◽  
Vol 86 (2) ◽  
pp. 379-388 ◽  
Author(s):  
H. Takenaka ◽  
M. Ohori ◽  
K. Koketsu ◽  
B. L. N. Kennett
Keyword(s):  
Integral Equation ◽  
Inverse Fourier Transform ◽  
Linear Equations ◽  
Computation Time ◽  
Reduction Technique ◽  
Final Solution ◽  
Space Domain ◽  
New Approach ◽  
Simultaneous Linear Equations ◽  
Single Integral

Abstract The Aki-Larner method is one of the cheapest methods for synthetic seismograms in irregularly layered media. In this article, we propose a new approach for a two-dimensional SH problem, solved originally by Aki and Larner (1970). This new approach is not only based on the Rayleigh ansatz used in the original Aki-Larner method but also uses further information on wave fields, i.e., the propagation invariants. We reduce two coupled integral equations formulated in the original Aki-Larner method to a single integral equation. Applying the trapezoidal rule for numerical integration and collocation matching, this integral equation is discretized to yield a set of simultaneous linear equations. Throughout the derivation of these linear equations, we do not assume the periodicity of the interface, unlike the original Aki-Larner method. But the final solution in the space domain implicitly includes it due to use of the same discretization of the horizontal wavenumber as the discrete wavenumber technique for the inverse Fourier transform from the wavenumber domain to the space domain. The scheme presented in this article is more efficient than the original Aki-Larner method. The computation time and memory required for our scheme are nearly half and one-fourth of those for the original Aki-Larner method. We demonstrate that the band-reduction technique, approximation by considering only coupling between nearby wavenumbers, can accelerate the efficiency of our scheme, although it may degrade the accuracy.

scholarly journals On the numerical solution of an Abel integral equation

1979 ◽  
Vol 16 (3) ◽  
pp. 497-503
Author(s):  
R. Smarzewski ◽  
H. Malinowski
Keyword(s):  
Integral Equation ◽  
Numerical Solution ◽  
Abel Integral Equation

Numerical Method of Estimating Distance Between Wells

2021 ◽  
Author(s):  
Ayobami Ezekiel ◽  
Prince Oduh ◽  
Emmanuel Okoh ◽  
Collins Onah ◽  
Michael Ojah ◽  
...  
Keyword(s):  
Integral Equation ◽  
Pressure Drop ◽  
Polynomial Equation ◽  
Important Variable ◽  
Well Testing ◽  
Model Calculations ◽  
New Approach ◽  
Pressure Drops ◽  
Quantum Leap ◽  
Interference Test

Abstract In this study, a simpler numerical model for calculating inter-well distance was developed. This model was developed as an alternative to the Ei-function used for computing pressure drops. The mainobjective of developing this model is tomake resolution of pilfering issues easyto resolve. With the developed model, calculations relating to pressure drops and more specifically, inter-well distance, can be done with greater ease and accuracy. In developing this model, the integral equation of the Eifunction in the pressure drop equation was solved numerically. The numerical solution reduced thepressure drop equation to a polynomial equation which is much easier to solve. The developed model was used to solve real problems. Results generated from it were compared with those obtained using previous approaches. Important informationsuch as well configuration, region of the reservoir, and transient history wherethe work is valid are stated. The development of the correlations and tables forthe range of validity and values of the Ei-function is a major quantum leap in well testing and analysis. It will be quite cumbersome to resolve integrals with unknowns, hence, methods of trials and errors have been resorted to over the years. However, this new approach resolved the pressure drop equation into a systemof polynomials which is much easier to solve. Consequently, the distance betweenpossibly interfering wells (which is an important variable during interference test) can now be gotten with ease. The developed model is valid within the range of validity of the Ei-function. Without doubt, this work will help redefine the pressure drop equation into a polynomial equation which can easily be resolved using any of the known approaches to solving problems involving polynomials. More so, getting the correct distance betweenthe two wells in question is pivotal to the test. With the model developed in this work, getting inter-well distance is now easier and more accurate.

scholarly journals Newtonsche Aufheizung, Abelsche Integralgleichungen zweiter Art und Mittag-Leffler-Funktionen

1987 ◽  
Vol 42 (10) ◽  
pp. 1141-1146 ◽  
Author(s):  
Rudolf Gorenflo
Keyword(s):  
Integral Equation ◽  
Explicit Solution ◽  
Temporal Evolution ◽  
Plane Boundary ◽  
Qualitative Properties ◽  
Boundary Temperature ◽  
Abel Integral Equation ◽  
Homogeneous Half Space ◽  
Interior Boundary ◽  
The Difference

The problem of heating a homogeneous half-space by radiation from outside across the plane boundary is considered. Newtonian heating means that the heat flux across the boundary is proportional to the difference of outside temperature and interior boundary temperature. The outside temperature is assumed to be constant and positive, the initial inside temperature is zero everywhere. The problem is onedim ensional in space. The temporal evolution of the inward boundary temperature obeys an Abel integral equation of second kind for whose explicit solution three methods are described (one by Laplace transform , the other two by infinite series defining the Mittag-Leffler function of index 1/2). The explicit solution facilitates discussion of its qualitative properties. Finally, the general Abel integral equation of second kind is treated by Mittag-Leffler functions.

scholarly journals Harmonic Analysis via an Integral Equation: An Application to Dengue Transmission

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
R. G. U. I. Meththananda ◽  
N. C. Ganegoda ◽  
S. S. N. Perera
Keyword(s):  
Integral Equation ◽  
Parameter Estimation ◽  
Harmonic Analysis ◽  
Basis Functions ◽  
Periodic Waves ◽  
Data Set ◽  
New Approach ◽  
Dengue Transmission ◽  
Different Types ◽  
The Laplace Transform

A collection of oscillatory basis functions generated via an integral equation is investigated here. This is a new approach in the harmonic analysis as we are able to interpret phenomena with damping and amplifying oscillations other than classical Fourier-like periodic waves. The proposed technique is tested with a data set of dengue incidence, where different types of influences prevail. An intermediate transform supported by the Laplace transform is available. It facilitates parameter estimation and strengthens the extraction of hidden influencing accumulations. This mechanistic work can be extended as a tool in signal processing that encounters oscillatory and accumulated effects.

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