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.

Slender-body theory for the generation of micrometre-sized emulsions through tip streaming

2012 ◽  
Vol 698 ◽  
pp. 423-445 ◽  
Author(s):  
E. Castro-Hernández ◽  
F. Campo-Cortés ◽  
José Manuel Gordillo
Keyword(s):  
Differential Equation ◽  
Velocity Ratio ◽  
Immiscible Liquid ◽  
Boundary Integral ◽  
Slender Body ◽  
Physical Situation ◽  
Dimensionless Numbers ◽  
Third Order ◽  
Simple Method ◽  
Slender Body Theory

AbstractWe report experiments in which a flow rate ${Q}_{i} $ of a fluid with a viscosity ${\ensuremath{\mu} }_{i} $ discharges into an immiscible liquid of viscosity ${\ensuremath{\mu} }_{o} $ that flows in parallel with the axis of the injector. When the outer capillary number verifies the condition ${\mathit{Ca}}_{o} = {\ensuremath{\mu} }_{o} {U}_{o} / \sigma ~\geqslant ~5$, where ${U}_{o} $ and $\sigma $ indicate, respectively, the outer velocity and the interfacial tension coefficient, and if the inner-to-outer velocity ratio is such that ${U}_{i} / {U}_{o} = {Q}_{i} / (\lrm{\pi} {U}_{o} { R}_{i}^{2} )\ll 1$, with ${R}_{i} $ the inner radius of the injector, a jet is formed with the same type of cone–jet geometry as predicted by the numerical results of Suryo & Basaran (Phys. Fluids, vol. 18, 2006, p. 082102). For extremely low values of the velocity ratio ${U}_{i} / {U}_{o} $, we find that the diameter of the jet emanating from the tip of the cone is so small that drops with sizes below $1~\lrm{\ensuremath{\mu}} \mathrm{m} $ can be formed. We also show that, through this simple method, concentrated emulsions composed of micrometre-sized drops with a narrow size distribution can be generated. Moreover, thanks to the information extracted from numerical simulations of boundary-integral type and using the slender-body approximation due to Taylor (Proceedings of the 11th International Congress of Applied Mechanics, Munich, 1964, pp. 790–796), we deduce a third-order, ordinary differential equation that predicts, for arbitrary values of the three dimensionless numbers that control this physical situation, namely, ${\mathit{Ca}}_{o} $, ${\ensuremath{\mu} }_{i} / {\ensuremath{\mu} }_{o} $ and ${U}_{i} / {U}_{o} $, the shape of the jet and the sizes of the drops generated. Most interestingly, the influence of the geometry of the injector system on the jet shape and drop size enters explicitly into the third-order differential equation through two functions that can be easily calculated numerically. Therefore, our theory can be used as an efficient tool for the design of new emulsification devices.

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.

A New Method and Applications of the Boundary Value Problem of Differential Equation

2014 ◽  
Vol 937 ◽  
pp. 695-699
Author(s):  
Hong E Li ◽  
Xiao Xu Dong ◽  
Shun Chu Li ◽  
Dong Dong Gui ◽  
Cong Yin Fan
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Kernel Function ◽  
Boundary Value ◽  
Outer Boundary ◽  
Seepage Flow ◽  
Petroleum Engineering ◽  
Constructive Method ◽  
Linear Homogeneous Differential Equation ◽  
The Mathematical Model

The similar structure of solution for the boundary value problem of second order linear homogeneous differential equation has been studied. Based on the analysis of the relationship between similar structure of solution, its kernel function, the equation and boundary conditions, similar constructive method (shortened as SCM) of solution is obtained. According to the SCM, the similar structure of solution and its kernel function are constructed for the mathematical model of homogeneous reservoir which considers the influence of bottom-hole storage and skin effect under the infinite outer boundary condition. The SCM is a new and innovative way to solve boundary value problem of differential equation and seepage flow theory, which is especially used in Petroleum Engineering.

A Simple Method for Deriving the Formulas of Relativity

2012 ◽  
Vol 433-440 ◽  
pp. 3218-3222
Author(s):  
Ye Lin Xu
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Nuclear Physics ◽  
Mass Formula ◽  
Simple Method ◽  
Light Velocity ◽  
Particle Speed ◽  
Energy Formula ◽  
Mathematical Calculation ◽  
Expression Form

In this paper, a new and simple method for deriving the formulas of Relativity is introduced. Einstein derived the formulas of Relativity by applying two hypotheses about light velocity, but this process was rather complicated and hard to understand. Nuclear physics has been developing for nearly a hundred years and has accumulated plenty of experiment results. Now it is feasible to derive Einstein’s formulas by employing the experiment results of nuclear physics. According to the result that energy is an expression form of mass, E=m can be obtained. We differential the above formula, adopt several steps of simple mathematical calculation, a simple differential equation can be obtained. Take the result of “light velocity is the velocity limit of a moving object” as the boundary condition of the differential equation, we get energy formula and mass formula. Length formula and time formula can also be easily derived based on the experiment result of particle speed in a cyclotron. As all the formulas of Relativity in this paper were derived by applying true experimental results from nuclear physics rather than hypotheses, they are reliable and easy to understand.

A Short Method to Compute Nusselt Numbers in Rectangular and Annular Channels With any Ratio of Constant Heat Rate

2010 ◽  
Author(s):  
Alexandre Malon ◽  
Thierry Muller
Keyword(s):  
Differential Equation ◽  
Nusselt Number ◽  
Heat Rate ◽  
Equation System ◽  
Radius Ratio ◽  
Differential Equation System ◽  
Simple Method ◽  
Nusselt Numbers ◽  
Annular Channels ◽  
Wide Range

An analytic investigation of the thermal exchanges in channels is conducted with the prospect of building a simple method to determine the Nusselt number in steady, laminar or turbulent and monodimensional flow through rectangular and annular spaces with any ratio of constant and uniform heat rate. The study of the laminar case leads to explicit laws for the Nusselt number while the turbulent case is solved using a Reichardt turbulent viscosity model resulting in easy to solve one-dimensional ordinary differential equation system. This differential equation system is solved using a Matlab based boundary value problems solver (bvp4c). A wide range of Reynolds, Prandtl and radius ratio is explored with the prospect of building correlation laws allowing the computing of Nusselt numbers for any radius ratio. Those correlations are in good agreement with the results obtained by W.M. Kays and E.Y. Leung in 1963 [1]. They conduced a similar analysis but with an experimental basis, they explored a greater range of Prandtl but only a few discreet radius ratio. The correlations are also compared with a CFD analysis made on a case extracted from the Re´acteur Jules Horowitz.

Controllability results of fractional integro-differential equation with non-instantaneous impulses on time scales

2020 ◽  
Author(s):  
Vipin Kumar ◽  
Muslim Malik
Keyword(s):  
Differential Equation ◽  
Functional Analysis ◽  
Time Scales ◽  
Linear Functional ◽  
Integro Differential Equation ◽  
Numerical Example ◽  
Contraction Theorem ◽  
Banach Contraction ◽  
Banach Contraction Theorem ◽  
Different Time Scales

Abstract In this work, we investigate the controllability results of a fractional integro-differential equation with non-instantaneous impulses on time scales. Banach contraction theorem and the non-linear functional analysis have been used to establish these results. In support, a numerical example with simulation for different time scales is given to validate the obtained analytical outcomes.

III. On the differential equations of dynamics. A sequel to a paper on simultaneous differential equations

1863 ◽  
Vol 12 ◽  
pp. 420-424
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Dynamical Problem ◽  
General Character ◽  
Partial Differential ◽  
Central Function ◽  
First Order ◽  
Linear Partial Differential Equations

Jacobi in a posthumous memoir, which has only this year appeared, has developed two remarkable methods (agreeing in their general character, but differing in details) of solving non-linear partial differential equations of the first order, and has applied them in connexion with that theory of the differential equations of dynamics which was established by Sir W. R. Hamilton in the 'Philosophical Transactions’ for 1834-35. The knowledge, indeed, that the solution of the equation of a dynamical problem is involved in the discovery of a single central function, defined by a single partial differential equation of the first order, does not appear to have been hitherto (perhaps it will never be) very fruitful in practical results.

The final size of a nearly critical epidemic, and the first passage time of a Wiener process to a parabolic barrier

1998 ◽  
Vol 35 (3) ◽  
pp. 671-682 ◽  
Author(s):  
Anders Martin-Löf
Keyword(s):  
Differential Equation ◽  
Wiener Process ◽  
Critical Parameter ◽  
Spectral Representation ◽  
First Passage Time ◽  
Passage Time ◽  
Initial Number ◽  
Airy Functions ◽  
Final Size ◽  
First Passage

The distribution of the final size, K, in a general SIR epidemic model is considered in a situation when the critical parameter λ is close to 1. It is shown that with a ‘critical scaling’ λ ≈ 1 + a / n1/3, m ≈ bn1/3, where n is the initial number of susceptibles and m is the initial number of infected, then K / n2/3 has a limit distribution when n → ∞. It can be described as that of T, the first passage time of a Wiener process to a parabolic barrier b + at − t2/2. The proof is based on a diffusion approximation. Moreover, it is shown that the distribution of T can be expressed analytically in terms of Airy functions using the spectral representation connected with Airy's differential equation.

scholarly journals A Note on Solitary Wave Solutions of the Nonlinear Generalized Camassa-Holm Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Lei Zhang ◽  
Xing Tao Wang
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Solitary Wave ◽  
Numerical Simulations ◽  
Solitary Wave Solutions ◽  
Simple Method ◽  
Wave Solutions ◽  
Holm Equation

We give a simple method for applying ordinary differential equation to solve the nonlinear generalized Camassa-Holm equation ut+2kux−uxxt+aumux−2uxuxx+uuxxx=0. Furthermore we give a new ansätz. In the cases where m=1,2,3, the numerical simulations demonstrate the results.

scholarly journals Solvability of nonlinear fractional integro-differential equation with nonlocal condition

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Sakhri Aicha ◽  
Ahcene Merad
Keyword(s):  
Differential Equation ◽  
Functional Analysis ◽  
Fractional Order ◽  
Existence And Uniqueness ◽  
A Priori ◽  
Nonlocal Conditions ◽  
Analysis Method ◽  
Content Type ◽  
Fractional Order Differential Equations ◽  
Estimate Method

PurposeThis study describes the applicability of the a priori estimate method on a nonlocal nonlinear fractional differential equation for which the weak solution's existence and uniqueness are proved. The authors divide the proof into two sections for the linear associated problem; the authors derive the a priori bound and demonstrate the operator range density that is generated. The authors solve the nonlinear problem by introducing an iterative process depending on the preceding results.Design/methodology/approachThe functional analysis method is the a priori estimate method or energy inequality method.FindingsThe results show the efficiency of a priori estimate method in the case of time-fractional order differential equations with nonlocal conditions. Our results also illustrate the existence and uniqueness of the continuous dependence of solutions on fractional order differential equations with nonlocal conditions.Research limitations/implicationsThe authors’ work can be considered a contribution to the development of the functional analysis method that is used to prove well-positioned problems with fractional order.Originality/valueThe authors confirm that this work is original and has not been published elsewhere, nor is it currently under consideration for publication elsewhere.

Sign in / Sign up

Export Citation Format

Share Document