Theory of vector equations of electromagnetic wave diffraction on a rectilinear tubular cylinder

The vector equation for the diffraction of electromagnetic waves on the surface of a rectilinear circular cylinder without ends with respect to surface currents is considered. As a result of transformations from the original equation, one-dimensional systems of integral equations are obtained. For all four integral operators describing the systems, the main parts are highlighted. Using the remarkable properties of one-dimensional diffraction operators, the Fredholm equation of the second kind in Sobolev spaces is obtained.

Measures of noncompactness and infinite systems of integral equations of Urysohn type in $L^\infty(\mathfrak{G})$

"In this article, applying the concept of measure of noncompactness, some fixed point theorems in the Fréchet space $L^\infty(\mathfrak{G})$ (where $\mathfrak{G}\subseteq \mathbb{R}^{\omega}$) have been proved. We handle our obtained consequences to inquiry the existence of solutions for infinite systems of Urysohn type integral equations. Our results extend some famous related results in the literature. Finally, to indicate the effectiveness of our results we present a genuine example."

An open-source framework for the implementation of large-scale integral operators with flexible, modern HPC solutions - Enabling 3D Marchenko imaging by least squares inversion

Numerical integral operators of convolution type form the basis of most wave-equation-based methods for processing and imaging of seismic data. As several of these methods require the solution of an inverse problem, multiple forward and adjoint passes of the modeling operator are generally required to converge to a satisfactory solution. This work highlights the memory requirements and computational challenges that arise when implementing such operators on 3D seismic datasets and their usage for solving large systems of integral equations. A Python framework is presented that leverages libraries for distributed storage and computing, and provides a high-level symbolic representation of linear operators. A driving goal for our work is not only to offer a widely deployable, ready-to-use high-performance computing (HPC) framework, but to demonstrate that it enables addressing research questions that are otherwise difficult to tackle. To this end, the first example of 3D full-wavefield target-oriented imaging, which comprises of two subsequent steps of seismic redatuming, is presented. The redatumed fields are estimated by means of gradient-based inversion using the full dataset as well as spatially decimated versions of the dataset as a way to investi-gate the robustness of both inverse problems to spatial aliasing in the input dataset. Our numerical example shows that when one spatial direction is finely sampled, satisfactory redatuming and imaging can be accomplished also when the sampling in other direction is coarser than a quarter of the dominant wavelength. While aliasing introduces noise in the redatumed fields, they are less sensitive to well-known spurious artefacts compared to cheaper, adjoint-based redatuming techniques. These observations are shown to hold for a relatively simple geologic structure, and while further testing is needed for more complex scenarios, we expect them to be generally valid while possibly breaking down for extreme cases

Resolution of system of Volterra integral equations of the first kind by derivation technique and modified decomposition methods

A solution method for various systems of integral equations of the first kind is presented in this paper. The method starts off by transforming the systems via the application of the Leibnitz’s derivation technique and then employs three different decomposition methods based on the Standard Adomian decomposition method (SADM) for solutions. To demonstrate the efficiency of the proposed method, some illustrative examples are considered and the obtained results indicate that the approach is indeed of practical interest.

Existence results for nonlinear coupled system of integral equations of Urysohn Volterra-Chandrasekhar mixed type

The combined systems of integral equations have become of great importance in various fields of sciences such as electromagnetic and nuclear physics. New classes of the merged type of Urysohn Volterra-Chandrasekhar quadratic integral equations are proposed in this paper. This proposed system involves fractional Urysohn Volterra kernels and also Chandrasekhar kernels. The solvability of a coupled system of integral equations of Urysohn Volterra-Chandrasekhar mixed type is studied. To realize the existence of a solution of those mixed systems, we use the Perov’s fixed point combined with the Leray-Schauder fixed point approach in generalized Banach algebra spaces.

Existence of a common solution to systems of integral equations via fixed point results

The goal of this article is to prove some coupled common fixed point results by using weakly increasing mappings with two variables. Several examples indicating the usability are provided. Also, we use the results obtained to demonstrate the existence of a common solution to a system of integral equations.

Применение метода продолженных граничных условий к решению задачи дифракции волн на рассеивателях сложной геометрии, расположенных в однородной и неоднородной средах

Based on the method of continued boundary conditions, a technique is proposed that allows modeling the scattering characteristics for bodies of arbitrary geometry. The two-dimensional problem of the diffraction of a plane wave by dielectric bodies with complex section geometry, in particular, by fractal-like bodies, is considered. Comparison of numerical algorithms for solving the diffraction problem based on systems of integral equations of the 1st and 2nd kind is carried out. The method is generalized to the problem of diffraction by a cylindrical body located in a homogeneous magnetodielectric half-space. The correctness of the method is confirmed by checking the fulfillment of the optical theorem for various bodies and by comparing it with the results of calculations obtained by the modified method of discrete sources.