General formulation of the electric stratified problem with a boundary integral equation

Geophysics ◽  
1995 ◽  
Vol 60 (6) ◽  
pp. 1656-1670 ◽  
Author(s):  
André Straub
Keyword(s):  
Differential Equation ◽  
Kernel Function ◽  
Reference Model ◽  
Boundary Integral Equations ◽  
Hankel Transform ◽  
Cylindrical Symmetry ◽  
Boundary Integral ◽  
Point Of View ◽  
Computational Point ◽  
Linear System Of Equations

The electric potential created by a point source in a stratified model is usually written, in a spectral representation, in terms of a Hankel transform because of the cylindrical symmetry of the model. The solution in the radial wavenumber domain is called the kernel function. This kernel function, as a function of the depth coordinate, is the solution of a 1-D differential equation. The conventional procedure for the calculation of the kernel function consists in applying a recursive scheme. This procedure is effective from a computational point of view but becomes cumbersome from an analytical point of view, especially in the case of an arbitrary number of layers for arbitrary positions of the source and measurement points. I reformulate the problem of the kernel function by establishing the equivalence between the 1-D differential equation and a set of two boundary integral equations. This equivalence lowers the dimension of the problem by one unit so that the integration is performed over a space of dimension zero. The equations thus obtained are called jump summation equations. They are derived from a weighted product of two distinct models. The explicit form of these equations with the use of Green’s kernel (i.e., the kernel function for a homogeneous reference model) leads to the introduction of two basic representations, monopolar and dipolar. Each representation is related to a specific integral operator, but the basic representations are equivalent. The kernel function is computed by solving a linear system of equations. Our formulation is also well adapted to the inverse problem. The relationship between a perturbation of the model and the resulting perturbation of the kernel function is expressed by a Fréchet derivative. This sensitivity quantity is obtained by means of the jump summation equation, and its computation appears straightforward with the basic representations. An application to a novel evaluation of the depth of investigation for usual arrays is given.

scholarly journals Solution of the Falkner–Skan Equation Using the Chebyshev Series in Matrix Form

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Abdelrady Okasha Elnady ◽  
M. Fayek Abd Rabbo ◽  
Hani M. Negm
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Numerical Method ◽  
Matrix Representation ◽  
Point Of View ◽  
Finite Domain ◽  
Algebraic Equations ◽  
Chebyshev Series ◽  
Infinite Domain ◽  
Computational Point

A numerical method for the solution of the Falkner–Skan equation, which is a nonlinear differential equation, is presented. The method has been derived by truncating the semi-infinite domain of the problem to a finite domain and then expanding the required approximate solution as the elements of the Chebyshev series. Using matrix representation of a function and their derivatives, the problem is reduced to a system of algebraic equations in a simple way. From the computational point of view, the results are in excellent agreement with those presented in published works.

Properties of the kernel function in electric stratified problems

Geophysics ◽  
1995 ◽  
Vol 60 (6) ◽  
pp. 1671-1681 ◽  
Author(s):  
André Straub
Keyword(s):  
Differential Equation ◽  
Functional Analysis ◽  
Kernel Function ◽  
Spectral Representation ◽  
Boundary Integral ◽  
General Character ◽  
Kernel Space ◽  
Electrical Tomography ◽  
Simple Method ◽  
Stratified Model

The kernel function plays an important role in the 1-D problem because of the spectral representation of the electric potential for a stratified model with a point source. Functional analysis establishes the equivalence between the differential equation (which governs the kernel function) and a boundary integral equation, called a jump summation equation for the 1-D case. In this equation, the jumps of the weighted Wronskian of two distinct models are summed over all the singular points. Numerous applications of this general equation demonstrate its flexibility. An appropriate choice of models and of the weight function leads to two splitting theorems and two imbedding theorems. The basic idea is to split the stratification into two models for the splitting theorems and into three models for the imbedding theorems. An application of these theorems concerns the handling of underground and underwater sounding measurements. Three possible configurations are examined and their performances are compared. With these examples, a simple method of layer stripping is introduced in the kernel space. These theorems are also used to establish the shift properties for a given set of layers surrounded by two homogeneous half‐spaces. The consequences of these shift properties especially concern electrical tomography, where a case of equivalence is shown. The general character of these theorems may generate other applications.

Reduction of dust extraction from an aspiration shelter due to mechanical shielding

2020 ◽  
pp. 65-71
Author(s):  
O. A. Averkova ◽  
A. B. Goltsov ◽  
K. I. Logachev ◽  
A. V. Minko
Keyword(s):  
Differential Equation ◽  
Velocity Field ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Air Flow ◽  
Boundary Integral ◽  
Dust Particles ◽  
Maximum Diameter ◽  
Dust Extraction ◽  
Dust Dynamics

The dust dynamics in an aspiration shelter equipped with various thin mechanical screens is considered. To calculate the trajectories of dust particles, the differential equation of their motion was used, the velocity field of the air flow was calculated by the method of boundary integral equations. The influence of different mechanical screens on the maximum diameter of dust particles is determined. It is shown that when using curved mechanical screens it is possible to significantly reduce dust extraction in the suction network.

COMBINATORIAL POLYNOMIALLY COMPUTABLE CHARACTERISTICS OF SUBSTITUTIONS AND THEIR PROPERTIES

2020 ◽  
Vol 7 (2) ◽  
pp. 34-41
Author(s):  
VLADIMIR NIKONOV ◽  
◽  
ANTON ZOBOV ◽  
Keyword(s):  
Mathematical Expectation ◽  
Point Of View ◽  
Small Range ◽  
Computational Point ◽  
Wide Range ◽  
Bijective Function ◽  
The Difference ◽  
Element Base ◽  
Selection Of

The construction and selection of a suitable bijective function, that is, substitution, is now becoming an important applied task, particularly for building block encryption systems. Many articles have suggested using different approaches to determining the quality of substitution, but most of them are highly computationally complex. The solution of this problem will significantly expand the range of methods for constructing and analyzing scheme in information protection systems. The purpose of research is to find easily measurable characteristics of substitutions, allowing to evaluate their quality, and also measures of the proximity of a particular substitutions to a random one, or its distance from it. For this purpose, several characteristics were proposed in this work: difference and polynomial, and their mathematical expectation was found, as well as variance for the difference characteristic. This allows us to make a conclusion about its quality by comparing the result of calculating the characteristic for a particular substitution with the calculated mathematical expectation. From a computational point of view, the thesises of the article are of exceptional interest due to the simplicity of the algorithm for quantifying the quality of bijective function substitutions. By its nature, the operation of calculating the difference characteristic carries out a simple summation of integer terms in a fixed and small range. Such an operation, both in the modern and in the prospective element base, is embedded in the logic of a wide range of functional elements, especially when implementing computational actions in the optical range, or on other carriers related to the field of nanotechnology.

Integral Representations for the Solution of Dynamic Bending of a Plate with Displacement-Traction Boundary Data

2003 ◽  
Vol 10 (3) ◽  
pp. 467-480
Author(s):  
Igor Chudinovich ◽  
Christian Constanda
Keyword(s):  
Existence And Uniqueness ◽  
Boundary Data ◽  
Boundary Integral Equations ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Boundary Integral ◽  
Integral Representations ◽  
Transverse Shear Deformation ◽  
Uniqueness Of Solutions ◽  
Nonstationary Boundary

Abstract The existence of distributional solutions is investigated for the time-dependent bending of a plate with transverse shear deformation under mixed boundary conditions. The problem is then reduced to nonstationary boundary integral equations and the existence and uniqueness of solutions to the latter are studied in appropriate Sobolev spaces.

scholarly journals Hyperbolastic Models from a Stochastic Differential Equation Point of View

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1835
Author(s):  
Antonio Barrera ◽  
Patricia Román-Román ◽  
Francisco Torres-Ruiz
Keyword(s):  
Differential Equation ◽  
Simulated Data ◽  
Real Data ◽  
Point Of View ◽  
Likelihood Method ◽  
Numerical Resolution ◽  
The Family ◽  
The Common ◽  
Linear Differential ◽  
Mean Function

A joint and unified vision of stochastic diffusion models associated with the family of hyperbolastic curves is presented. The motivation behind this approach stems from the fact that all hyperbolastic curves verify a linear differential equation of the Malthusian type. By virtue of this, and by adding a multiplicative noise to said ordinary differential equation, a diffusion process may be associated with each curve whose mean function is said curve. The inference in the resulting processes is presented jointly, as well as the strategies developed to obtain the initial solutions necessary for the numerical resolution of the system of equations resulting from the application of the maximum likelihood method. The common perspective presented is especially useful for the implementation of the necessary procedures for fitting the models to real data. Some examples based on simulated data support the suitability of the development described in the present paper.

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