Une nouvelle équation intégrale pour l'étude de la radiation scalaire dans une cavité

1978 ◽  
Vol 56 (3) ◽  
pp. 387-394 ◽  
Author(s):  
Byron T. Darling ◽  
Jacques A. Imbeau
Keyword(s):  
Integral Equation ◽  
Analytic Solution ◽  
Normal Derivative ◽  
Prolate Spheroidal ◽  
Equation Intégrale ◽  
Closed Cavity ◽  
Field Point ◽  
Special Cases ◽  
Limit Form ◽  
Dipole Sources

We derive an integral equation of the first kind connecting the surface values and the normal derivative for a regular solution inside a closed cavity of the Helmholtz equation. This integral equation has two advantages over the usual limit form of integral equations where the field point must lie on the boundary and the kernel is singular, namely, the field point may be anywhere inside or outside the cavity, and the kernel is regular. Analytic solution of our integral equation is obtained for the special cases of monopole and of dipole sources at the center of a sphere (Dirichlet's condition). The next paper will apply this integral equation to prolate spheroidal cavities.

Boundary Integral Equation Formulation in Generalized Linear Thermo-Viscoelasticity With Rheological Volume

2003 ◽  
Vol 70 (5) ◽  
pp. 661-667 ◽  
Author(s):  
A. S. El-Karamany
Keyword(s):  
Integral Equation ◽  
Rheological Properties ◽  
Boundary Integral Equation ◽  
Relaxation Times ◽  
Fundamental Solutions ◽  
Boundary Integral ◽  
Phase Lag ◽  
Integral Equation Formulation ◽  
Special Cases ◽  
The Given

A general model of generalized linear thermo-viscoelasticity for isotropic material is established taking into consideration the rheological properties of the volume. The given model is applicable to three generalized theories of thermoelasticity: the generalized theory with one (Lord-Shulman theory) or with two relaxation times (Green-Lindsay theory) and with dual phase-lag (Chandrasekharaiah-Tzou theory) as well as to the dynamic coupled theory. The cases of thermo-viscoelasticity of Kelvin-Voigt model or thermoviscoelasticity ignoring the rheological properties of the volume can be obtained from the given model. The equations of the corresponding thermoelasticity theories result from the given model as special cases. A formulation of the boundary integral equation (BIE) method, fundamental solutions of the corresponding differential equations are obtained and an example illustrating the BIE formulation is given.

scholarly journals Solving Mixed Volterra - Fredholm Integral Equation (MVFIE) by Designing Neural Network

2019 ◽  
Vol 16 (1) ◽  
pp. 0116
Author(s):  
Al-Saif Et al.
Keyword(s):  
Neural Network ◽  
Integral Equation ◽  
Analytic Solution ◽  
Fredholm Integral Equations ◽  
Training Algorithm ◽  
Output Unit ◽  
Feed Forward Neural Network ◽  
Levenberg Marquardt ◽  
Hidden Layer ◽  
And Training

       In this paper, we focus on designing feed forward neural network (FFNN) for solving Mixed Volterra – Fredholm Integral Equations (MVFIEs) of second kind in 2–dimensions. in our method, we present a multi – layers model consisting of a hidden layer which has five hidden units (neurons) and one linear output unit. Transfer function (Log – sigmoid) and training algorithm (Levenberg – Marquardt) are used as a sigmoid activation of each unit. A comparison between the results of numerical experiment and the analytic solution of some examples has been carried out in order to justify the efficiency and the accuracy of our method.                                  

Evaluation of static stress on a fault plane from a Green's function

1974 ◽  
Vol 64 (6) ◽  
pp. 1629-1633
Author(s):  
D. J. Andrews
Keyword(s):  
Shear Strain ◽  
Green's Function ◽  
Plane Stress ◽  
Representation Theorem ◽  
Stress Drop ◽  
Numerical Evaluation ◽  
Analytic Solution ◽  
Fault Plane ◽  
Static Stress ◽  
Field Point

abstract Direct numerical evaluation of shear strain on a fault plane using the representation theorem is not possible because source points near the field point give large and canceling contributions to the integral. The representation theorem for strain can be integrated by parts to obtain an expression valid everywhere and suitable for numerical evaluation on the fault plane. Stress-drop evaluated by this method for the circular dislocation of Keylis-Borok agrees well with the analytic solution.

scholarly journals The Burton and Miller Method: Unlocking Another Mystery of Its Coupling Parameter

2016 ◽  
Vol 24 (01) ◽  
pp. 1550016 ◽  
Author(s):  
Steffen Marburg
Keyword(s):  
Integral Equation ◽  
Literature Review ◽  
Coupling Parameter ◽  
Solution Process ◽  
Normal Derivative ◽  
The Neumann Problem ◽  
Spurious Modes ◽  
Specific Formulation ◽  
Wrong Sign ◽  
Helmholtz Integral

The phenomenon of irregular frequencies or spurious modes when solving the Kirchhoff–Helmholtz integral equation has been extensively studied over the last six or seven decades. A class of common methods to overcome this phenomenon uses the linear combination of the Kirchhoff–Helmholtz integral equation and its normal derivative. When solving the Neumann problem, this method is usually referred to as the Burton and Miller method. This method uses a coupling parameter which, theoretically, should be complex with nonvanishing imaginary part. In practice, it is usually chosen proportional or even equal to [Formula: see text]. A literature review of papers about the Burton and Miller method and its implementations revealed that, in some cases, it is better to use [Formula: see text] as coupling parameter. The better choice depends on the specific formulation, in particular, on the harmonic time dependence and on the fundamental solution or Green’s function, respectively. Surprisingly, an unexpectedly large number of studies is based on the wrong choice of the sign in the coupling parameter. Herein, it is described which sign of the coupling parameter should be used for different configurations. Furthermore, it will be shown that the wrong sign does not just make the solution process inefficient but can lead to completely wrong results in some cases.

Mean dyadic Green's function of a randomly rough surface

1994 ◽  
Vol 72 (1-2) ◽  
pp. 20-29 ◽  
Author(s):  
Saba Mudaliar
Keyword(s):  
Integral Equation ◽  
Green’S Function ◽  
Multiple Scattering ◽  
Rough Surface ◽  
Green's Function ◽  
Diagram Method ◽  
Dyadic Green’S Function ◽  
Dyadic Green's Function ◽  
Special Cases ◽  
Randomly Rough Surface

The problem of wave propagation and scattering over a randomly rough surface is considered from a multiple-scattering point of view. Assuming that the magnitude of irregularities are small an approximate boundary condition is specified. This enables us to derive an integral equation for the dyadic Green's function (DGF) in terms of the known unperturbed DGF. Successive iteration of this integral equation yields the Neumann series. On averaging this and using a diagram method the Dyson equation is derived. Bilocal approximation to the mass operator leads to an integral equation whose kernel is of the convolution type. This is then readily solved and the results are presented in a simplified and useful form. It is observed that the coherent reflection coefficients involve infinite series of multiple scattering. Two special cases are considered the results of which are in agreement with our expectations.

A NOTE ON DIFFRACTION BY AN INFINITE SLIT

1960 ◽  
Vol 38 (1) ◽  
pp. 38-47 ◽  
Author(s):  
R. F. Millar
Keyword(s):  
Integral Equation ◽  
Normal Derivative ◽  
Cylindrical Wave ◽  
Narrow Slit ◽  
Angle Of Incidence ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Boundary Values ◽  
Transmission Coefficients ◽  
Differential Integral Equation

The two-dimensional problem of diffraction of a plane wave by a narrow slit is considered. The assumed boundary values on the screen are the vanishing of either the total wave function or its normal derivative. In the former case, a differential–integral equation is obtained for the unknown function in the slit; in the latter, a pure integral equation is found. Solutions to these equations are given in the form of series in powers of ε (where ε/π is the ratio of slit width to wavelength), the coefficients of which depend on log ε. Expressions are found for the transmission coefficients as functions of ε and the angle of incidence; these are compared with previous determinations of other authors.A brief outline is given for the treatment of diffraction of a cylindrical wave by the slit.

An Integral Equation Method for the First-Passage Problem in Random Vibration

1984 ◽  
Vol 51 (3) ◽  
pp. 674-679 ◽  
Author(s):  
P. H. Madsen ◽  
S. Krenk
Keyword(s):  
Integral Equation ◽  
Random Vibration ◽  
Integral Equation Method ◽  
Nonstationary Process ◽  
Alternative Methods ◽  
Approximation Formula ◽  
First Passage ◽  
Special Cases ◽  
Nonstationary Stochastic Process ◽  
First Passage Problem

The first-passage problem for a nonstationary stochastic process is formulated as an integral identity, which produces known bounds and series expansions as special cases, while approximation of the kernel leads to an integral equation for the first-passage probability density function. An accurate, explicit approximation formula for the kernel is derived, and the influence of uni or multi modal frequency content of the process is investigated. Numerical results provide comparisons with simulation results and alternative methods for narrow band processes, and also the case of a multimodal, nonstationary process is dealt with.

Reduced surface integral equations for Laplacian fields in the presence of layered bodies

2006 ◽  
Vol 84 (12) ◽  
pp. 1049-1061 ◽  
Author(s):  
I R Ciric
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Unknown Function ◽  
Source Region ◽  
Normal Derivative ◽  
Surface Integral ◽  
Field Problem ◽  
Layered Bodies ◽  
High Computational Efficiency ◽  
Reduced Surface

Laplacian potential fields in stratified media are usually analyzed using an integral equation for an unknown function over the union of all the interfaces between regions with different homogeneous materials. In this paper, the field problem is solved using a reduced integral equation involving a single unknown function over only the boundary of the source region. The new integral equation is derived by introducing surface operators to express the potential and its normal derivative on each interface in terms of a single unknown function over the same interface. These operators and the corresponding single functions are obtained recursively, from one interface to the next. Thus, a substantial decrease in the amount of necessary numerical computation and computer memory is achieved especially for systems containing identical layered bodies where the reduction operators are only constructed for one of the bodies. The purpose of this paper is to derive reduced integral equations by directly applying the interface conditions and to show their high computational efficiency for systems of layered bodies.PACS Nos.: 02.30.Rz, 02.70.Pt, 41.20.Cv

scholarly journals Modeling and Simulation of stochastically deformed Waveguides using Schelkunoff's Method

2018 ◽  
Vol 16 ◽  
pp. 35-41
Author(s):  
Hoang Duc Pham ◽  
Soeren Ploennigs ◽  
Wolfgang Mathis
Keyword(s):  
Cross Sections ◽  
Electromagnetic Waves ◽  
Analytic Solution ◽  
Propagation Constant ◽  
Circular Waveguide ◽  
Transverse Field ◽  
Field Pattern ◽  
Coupled Mode ◽  
Special Cases ◽  
Cylindrical Waveguides

Abstract. This paper deals with the propagation of electromagnetic waves in cylindrical waveguides with irregularly deformed cross-sections. The general theory of electromagnetic waves is of high interest because of its practical use as a transmission medium. But only in a few special cases, an analytic solution of Maxwell's equations and the appropriate boundary conditions can be found (Spencer, 1951). The coupled-mode theory, also known as Schelkunoff's method, is a semi-numerical method for computing electromagnetic waves in hollow and cylindrical waveguides bounded by perfect electric walls (Saad, 1985). It allows to calculate the transverse field pattern and the propagation constant. The aim of this paper is to derive the so-called generalized telegraphist's equations for irregular deformed waveguides. Subsequently, the method's application will be used on a circular waveguide as an illustrating example.

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