Electrodynamic analysis of ring array of vertical asymmetric vibrators

The numerical solution of integral equations or a system of integral equations (for a system of conductors) consists in their discretization and reduction to a system of linear algebraic equations for the desired function. For discretization, it is possible to use projection methods, for example, the Galerkin method or the collocation method. A system of linear algebraic equations for the class of problems under consideration is characterized by a complete (filled) complex matrix. For conductor antenna structures that are quite arbitrary in geometry and size, systems of linear algebraic equations turn out to be of a high order, and special computational algorithms are required to solve them. The purpose of the work is to study adequate mathematical models for wire antennas with a subsequent description of uniform numerical algorithms based on methods of sampling and approximation of antenna current. With this method, the error in solving integral equations is determined by the error in calculating the elements of the matrix of a system of linear algebraic equations for a given piecewise polynomial approximation of the solution at step h. If the quadrature formula for numerical integration is chosen, then there are two ways to reduce the solution error. An increase in the number of collocation points leads to a rapid increase in the volume of calculations and, consequently, to a rapid increase in the amount of occupied computer memory. Therefore, the question arises of choosing the most profitable method of piecewise polynomial interpolation and discretization step h from the point of view of using computer resources, ensuring an acceptable number of solutions.

The aim of the present work is to consider a mixed boundary value problem for the biharmonic equation in a strip. The problem may be interpreted as a deflection surface of a strip plate with the edges y=0,y = h having clamped conditions on intervals |x|≥a and hinged support conditions for |x|<a. Using the Fourier transform, the problem is reduced to studying a system of dual integral equations on the edges of the strip. The uniqueness and existence theorems of solution of system of dual integral equations are established in appropriate Sobolev spaces. A method for reducing the dual integral equation to infinite system of linear algebraic equations is also proposed.

A scalar three-dimensional boundary value problem of wave diffraction for the Helmholtz equation with transmission conditions that assume the presence of an infinitely thin material at the media interface is considered. Uniqueness and existence theorems for solutions are proved. The original problem is reduced to a system of integral equations over the media interface. Calculation formulas for the system of linear algebraic equations obtained after applying the collocation method and numerical results for solving the problem when the domain is a ball with certain transmission conditions are given.

This article proposes a simple efficient method for solving a Volterra integral equations system of the first kind. By using block pulse functions and their operational matrix of integration, a first kind integral equations system can be reduced to a linear system of algebraic equations. The coefficient matrix of this system is a block matrix with lower triangular blocks. Numerical examples show that the approximate solutions have a good degree of accuracy.

In the present article a diffraction problem on an ideally conducting parallelepiped the upper face of which is coated with a ferrite layer is solved by means of the method of integral equations. A general approach to the compilation of a system of linear integral vector equations is formulated, and the process of reducing the latter system to a system of linear algebraic equations for a family of problems of the given type is presented.

This technical note presents an analytical method, accompanied by a numerical scheme, to evaluate the response of beams on nonuniform elastic foundations, namely, when the foundation modulus is k=k(x). The method employs a Green's function formulation, which results in a system of nonsingular integral equations for the distributed reaction q(x). These equations can be discretized in a straightforward manner to yield a system of linear algebraic equations that can be solved by elementary numerical techniques.

In this paper we will develop procedures for solving the system of non-linear Volterra integral equations, which describe the relations between the complex wave functions in the exit pupil and the Fourier transforms of two intensity distributions in the image plane, corresponding to two images of the same object taken at different values of the defocusing of the optical system. The first method reduces the Volterra integral equations to a finite system of linear algebraic equations which can be solved successively. The second algorithm is based on the Newton-Kantorovich approximation for functional equations, which gives a system of recurrence relations by means of which the Volterra equations can be solved.

Based on a nonclassical criterial approach associated with the mechanism for local loss of stability in the neighborhood of a defect, a method is proposed for solving problems related to the fracture of a compressible elastic cylinder with a cylindrical crack near its surface for large subcritical deformations. The theory of integral Fourier transforms and series expansions are used to reduce these problems to a system of paired integral equations and then to a uniform system of linear algebraic equations in the contraction parameter.

We consider a method for computing approximate solutions to systems of two-dimensional Volterra integral equations. The approximate solution is sought in the form of a linear combination of two-variable shifted Legendre functions. The operational matrices technique is used to reduce the problem to a system of linear algebraic equations. Some numerical tests have been carried out and the results show that this method has a good performance, even in the case when the system matrix is singular in the whole considered domain.

We use finite-difference modeling to illustrate direct-P and direct-SV radiation produced by land-based, vertical-displacement sources such as vertical vibrators and vertical impacts, sources that reflection seismologists have used for decades to image subsurface targets. We are particularly interested in understanding the nature of SV radiation that these sources create. Such insight will be important for those in the seismic interpretation community who are interested in S-wave reflection seismology. One distinction between our work and that of earlier investigators is that we go beyond traditional assumptions that a propagation medium has a flat surface and is isotropic and homogeneous throughout the model space. Instead, we use models composed of small finite-difference cells that allow earth anomalies with dimensions of only a few feet to be near a source station. These small irregularities are expressed as variations in propagation velocities in a small population of model cells, or as slight changes in surface elevation. These small, realistic, near-source irregularities cause robust direct-SV illumination to travel in near-vertical takeoff angles from a surface-based source station and to often travel in true-vertical takeoff angles. We also depart from the conventional practice of calculating direct-P and direct-SV radiation produced by only a single vertical vibrator. Instead, we model direct-P and direct-SV illumination produced by arrays of 2, 3, and 4 inline vertical vibrators. We find that vibrator arrays create SV-mode beam forming that causes a rich amount of direct-SV radiation to occur inside the critical takeoff-angle cone below a source station that extends from [Formula: see text] to [Formula: see text] from vertical. We consider our source-radiation models to be more realistic representations of conditions encountered in actual seismic field practice than previous models that use oversimplified earth conditions. Our purpose is to provide interpreters information that will let them decide if common, surface-based, land sources such as vertical vibrators can be used to illuminate geologic targets with direct-P and direct-SV illuminating wavefields rather than limiting the use of these sources to P-wave imaging only, as has been standard practice for several decades. Our results indicate that vertical vibrators and vertical impacts produce illuminating wavefields that are appropriate for imaging deep geologic targets. Our models apply to the total family of surface-based, vertical-displacement sources, which includes vertical vibrators and accelerated weight drops.

In this paper, the existence and uniqueness of solution of the Fredholm‐Volterra integral equation (F‐VIE), with a generalized singular kernel, are discussed and proved in the spaceL2(Ω)×C(0,T). The Fredholm integral term (FIT) is considered in position while the Volterra integral term (VIT) is considered in time. Using a numerical technique we have a system of Fredholm integral equations (SFIEs). This system of integral equations can be reduced to a linear algebraic system (LAS) of equations by using two different methods. These methods are: Toeplitz matrix method and Product Nyström method. A numerical examples are considered when the generalized kernel takes the following forms: Carleman function, logarithmic form, Cauchy kernel, and Hilbert kernel.

In this paper, the existence and uniqueness of solution of the twodimensional Fredholm integral equation with time, with singular kernel, arediscussed and proved. The Fredholm integral term (FIT) is considered inposition while the Volterra integral term (VIT) is considered in time. Usinga numerical technique we have a system of Fredholm integral equations(SFIEs) in two dimensions . This system of integral equations can be reducedto a linear algebraic system (LAS) of equations by using two differentmethods. These methods are: Toeplitz matrix method and Product Nystrommethod. A numerical examples are considered when the kernel takes thefollowing forms: Carleman function, logarithmic form, Cauchy kernel.

The Permian Basin of west Texas spans two major subbasins — the Midland Basin and the Delaware Basin. Both basins contain Wolfcampian- to Leonardian-age turbidites that form thick sections of prolific unconventional reservoirs. For the past several years, the most active drilling targets in the United States have been the Wolfberry turbidite interval of the Midland Basin and the Wolfbone turbidite interval in the Delaware Basin. We have used two new technologies to examine the internal architecture and fabric of a thick interval of these unconventional drilling targets with seismic reflection seismology. First, we used seismic interpretation software that uses unsupervised machine learning (ML), so that a higher level of detail could be extracted from seismic images. Second, we complemented our P-P imaging of Wolfberry turbidites with a new seismic imaging option, that being SV-P (or converted-P) imaging. Because vertical vibrators, particularly arrays of vertical vibrators, produce downgoing P and downgoing SV illuminating wavefields, SV-P reflections can usually be extracted from the same vertical-geophone responses as are P-P reflections. The combination of these two images essentially doubles the amount of information that can be extracted from data generated by P sources and recorded with vertical geophones. SV-P imaging with P sources has been ignored by reflection seismologists for decades, so we felt an obligation to illustrate the value of this ignored seismic mode. These two new tools — SV-P imaging and interpreting P-P and SV-P images via unsupervised ML — expanded our insights into the internal architecture and fabric of Wolfberry turbidites. Our work provides interpreters a much-needed example of applying unsupervised ML technology in a joint interpretation of P- and S-wave data.

In this paper, we deal with a system of linear and nonlinear integral algebraic equations (IAEs) of Hessenberg type. Convergence analysis of the discontinuous collocation methods is investigated for the large class of IAEs based on the new definitions. Finally, some numerical experiments are provided to support the theoretical results.

The mechanical, electric, and piezoelectric characteristics of fibrous piezocomposites with doubly periodic system of fibers are averaged within the framework of a model of regular structures under the conditions of plane deformation. It is shown that, for the general case of stacking of fibers with arbitrary cross section, the problem is reduced to the determination of certain functionals specified by a system of linear algebraic and singular integral equations of the corresponding boundary-value problem of electroelasticity for the analyzed structure.