Integral Equation Analysis Research Articles

Efficient Integral Equation Analysis of 3-D Rectangular Waveguide Microwave Circuits by Using Green’s Functions Accelerated With the Ewald Method

Efficient Integral Equation Analysis of 3-D Rectangular Waveguide Microwave Circuits by Using Green’s Functions Accelerated With the Ewald Method

Uniqueness, regularity, continuity, and stability of solutions of integral equations in irregular domains

This study explores the uniqueness, regularity, continuity, and stability of solutions to integral equations related to elliptical boundary value problems in irregular domains. Traditional methods usually consider soft boundaries, but this work extends these results to include domains with rough boundaries. By employing Sobolev spaces, especially fractional Sobolev spaces, we develop a comprehensive theoretical framework. Under appropriate conditions, the studied integral equation has a unique solution in fractional Sobolev spaces, maintaining the regularity properties of the input function. Two new theorems are introduced to strengthen the findings. The first theorem establishes that solutions are continuous with respect to perturbations in the input data. This means small changes in the input do not cause significant deviations in the solution, ensuring robustness in practical applications. The second theorem demonstrates the stability of the solutions when the domain geometry undergoes small changes. This stability is crucial for real-world scenarios where the exact shapes of domains may not be perfectly known or may vary slightly. These results significantly improve the mathematical understanding of boundary value problems in non-smooth domains. The proposed extended framework allows for more generalized applications, addressing challenges in various fields of physics and engineering. The ability to handle irregular domains with rough boundaries opens up new possibilities for modeling and solving complex problems where traditional soft boundary assumptions are not valid. This research provides a robust basis for future studies and applications, highlighting the importance of considering fractional Sobolev spaces in the analysis of integral equations.

Isogeometric Analysis of Electric Field Integral Equation on Multipatch NURBS Surfaces With Discontinuous Galerkin

Isogeometric Analysis of Electric Field Integral Equation on Multipatch NURBS Surfaces With Discontinuous Galerkin

Theoretical and numerical analysis of a first-kind linear Volterra functional integral equation with weakly singular kernel and vanishing delay

Theoretical and numerical analysis of a first-kind linear Volterra functional integral equation with weakly singular kernel and vanishing delay

Analysis of two-operator boundary-domain integral equations for variable-coefficient Dirichlet and Neumann problems in 2D with general right-hand side

Analysis of two-operator boundary-domain integral equations for variable-coefficient Dirichlet and Neumann problems in 2D with general right-hand side

Convergence analysis for the fractional decomposition method applied to class of nonlinear fractional Fredholm integro-differential equation

For scientists conducting research, fractional integral differential equation analysis is crucial. Therefore, in this study, we investigate analysis utilizing a novel method called the fractional decomposition method, which is applicable to fractional nonlinear fractional Fredholm integro-differential equations. Then, we apply the approach to five test problems for a general fractional derivative [Formula: see text] involving fractional Fredholm integro-differential equations. To the best of our knowledge, we are the first to ever do so because of the very complicated calculations involved when dealing with the general case [Formula: see text]. For fractional Fredholm integral-differential equations, we provide both exact and approximate solutions. Throughout this work, the fractional Caputo derivative is discussed. This technique leads us to say that the method is precise, accurate, and efficient, according to the theoretical analysis.

Integral equation analysis of multiport H‐plane microwave circuits by using 2D rectangular cavity Green's functions accelerated by the Ewald Method

An integral equation technique for the analysis of multiport H-plane microwave circuits composed of an unlimited number of arbitrarily shaped metallic and/or dielectric elements is proposed. The structure under study is divided into different regions after the application of the surface equivalence principle. Following this approach, the access waveguide ports are modelled by means of parallel plate Green's functions, whereas the central region is characterised by 2D rectangular cavity Green's functions. The use of this kind of Green's functions represents a novelty for solving H-plane integral equation problems. For the first time, the Ewald method has been employed in order to accelerate the computation of the 2D rectangular cavity Green's functions and their derivatives, needed in the integral equation. In addition, convergence studies show the numerical convenience of using the Ewald method to achieve a fast and accurate evaluation of these Green's functions. Results show that the analysis can be very efficient if the structure is segmented in a proper way based on fitting the cavity resonator within the geometry of the structure under analysis. In fact, all the walls of the structure that are coincident with the auxiliary resonator need not to be discretised during the numerical solution. Finally, several simulation examples of practical inductive microwave circuits are presented and discussed, showing a good agreement and higher computational efficiency when compared to results provided by commercial full-wave software tools and measurements.

Analysis of the boundary integral equation for the growth of a parabolic/paraboloidal dendrite with convection

The growth of a parabolic/paraboloidal dendrite streamlined by viscous and potential flows in an undercooled one-component melt is analyzed using the boundary integral equation. The total melt undercooling is found as a function of the Péclet, Reynolds, and Prandtl numbers in two- and three-dimensional cases. The solution obtained coincides with the modified Ivantsov solution known from previous theories of crystal growth. Varying Péclet and Reynolds numbers we show that the melt undercooling practically coincides in cases of viscous and potential flows for a small Prandtl number, which is typical for metals. In cases of water solutions and non-metallic alloys, the Prandtl number is not small enough and the melt undercooling is substantially different for viscous and potential flows. In other words, a simpler potential flow hydrodynamic model can be used instead of a more complicated viscous flow model when studying the solidification of undercooled metals with convection.

Strong convergence of the Euler-Maruyama method for the generalized stochastic Volterra integral equations driven by Lévy noise

In this paper, the theoretical and numerical analysis of the stochastic Volterra integral equations (SVIEs) driven by L?vy noise are considered. We investigate the existence, uniqueness, boundedness and H?lder continuity of the analytic solutions for SVIEs driven by L?vy noise. The Euler-Maruyama method for SVIEs driven by L?vy noise is proposed. The boundedness of the numerical solution is proved, and the strong convergence order is obtained. Some numerical examples are given to support the theoretical results.

Analysis of two-operator boundary-domain integral equations for variable-coefficient mixed BVP in 2D with general right-hand side

The mixed (Dirichlet–Neumann) boundary value problem (BVP) for the linear second-order scalar elliptic differential equation with variable coefficients in a bounded two-dimensional domain is considered. The PDE on the right-hand side belongs to H−1(Ω) or H˜−1(Ω), when neither classical nor canonical conormal derivatives of solutions are well defined. The two-operator approach and appropriate parametrix (Levi function) are used to reduce this BVP to four systems of boundary-domain integral equations (BDIEs). Although the theory of BDIEs in 3D is well developed, the BDIEs in 2D need a special consideration due to their different equivalence properties. As a result, we need to set conditions on the domain or on the associated Sobolev spaces to ensure the invertibility of corresponding parametrix-based integral layer potentials, and hence the unique solvability of BDIEs. The equivalence of the BDIE systems to the original BVP is shown. The invertibility of the associated operators is proved in the corresponding Sobolev spaces.

Application of Sobolev-Volterra projection and finite element numerical analysis of integral differential equations in modern art design

Abstract The article uses the Spss statistical analysis software to establish a multiple linear regression model of short-term stock price changes in domestic agricultural listed companies. It uses a stable time series based on the ARMA model for stable agricultural value-added, fiscal expenditure and market interest rates. The regression method is used to study its impact on the stock price index. Compared with the existing stock forecasting methods, this method has simple data collection and no specific requirements for data selection, and the prediction results have a high degree of fit. Therefore, this method is suitable for most stocks.

Stability and convergence analysis of singular integral equations for unequal arms branch crack problems in plane elasticity

In this paper, an unequal arms branch crack problem in a plane elasticity is treated. Using distribution dislocation function and complex variable potential method, the problem is formulated into a singular integral equation. The appropriate integration scheme, in which a point dislocation is set at the origin and the distribution dislocation, is applied through all arms of the branch crack to solve the obtained singular integral equations numerically. Stability, convergence, the order of convergence, and the error term of the solution are analyzed. Some numerical examples are examined to describe the behavior of stress intensity factors at the endpoints of each branch crack.

Homotopy scattering series for seismic forward modelling with variable density and velocity

ABSTRACTWe have derived a convergent scattering series solution for the frequency‐domain wave equation in acoustic media with variable density and velocity. The convergent scattering series solution is based on the homotopy analysis of a vectorial integral equation of the Lippmann–Schwinger type. By using the Green's function and partial integration, we have derived the vectorial integral equation of the Lippmann–Schwinger type that involves the pressure gradient field as well as the pressure field from the wave equation. The vectorial Lippmann–Schwinger equation can in principle be solved via matrix inversion, but the computational cost of matrix inversion scales like , where is the number of grid blocks. The computational cost can be significantly reduced if one solves the vectorial Lippmann–Schwinger equation iteratively. A simple iterative solution is the Born series, but it is only convergent when the scattering potential is sufficiently small. In this study, we have used the so‐called homotopy analysis method to derive an iterative solution for the vectorial Lippmann–Schwinger equation which can be made convergent even in strongly scattering media. The computational cost of our convergent scattering series scales as . Our algorithm, which is based on the homotopy analysis method, involves a convergence control operator that we select using hierarchical matrices. We use a three‐layer model and a resampled version of the SEG/EAGE salt model to show the performance of the developed convergent scattering series.

NONLINEAR PROPAGATION OF LEAKY TE-POLARIZED ELECTROMAGNETIC WAVES IN A METAMATERIAL GOUBAU LINE

Propagation of leaky TE-polarized electromagnetic waves in the Goubau line (a perfectly conducting cylinder covered by a concentric dielectric layer) filled with nonlinear metamaterial medium is studied. The problem is reduced to the analysis of a nonlinear integral equation with a kernel in the form of the Green function of an auxiliary boundary value problem on an interval. The existence of propagating nonlinear leaky TE waves for the chosen nonlinearity (Kerr law) is proved using the method of contraction. For the numerical solution, a method based on solving an auxiliary Cauchy problem (a version of the shooting method) is proposed. New propagation regimes are discovered.

Excitation of guided waves of grounded dielectric slab by a THz plane wave scattered from finite number of embedded graphene strips: Singular integral equation analysis

The scattering and absorption of the H-polarised wave by a finite number of graphene strips embedded into a grounded dielectric slab is considered in the terahertz frequency range. The rigorous and efficient method of singular integral equations is used here. The discretisation is based on the convergent meshless Nystrom-type algorithm. Graphene is modelled as a zero thickness layer with a complex conductivity obtained from the Kubo formalism. Dependences of the total scattering cross section, including the part that is associated with dielectric slab guided waves, and absorption cross section as well as near field patterns are studied. The behaviour of the scattering and absorption characteristics is explained in terms of resonances on the natural modes supported by the studied structure. Special attention is paid to the excitation of the dielectric waveguide eigenwaves near the grating-mode resonances.

On the numerical treatment and analysis of Hammerstein integral equation

In this paper, we study the quadratic rules for the numerical solution of Hammerstein integral equation based on spline quasi-interpolant. Also the convergence analysis of the methods are given. The theoretical behavior is tested on examples and it is shown that the numerical results confirm theoretical part.

Method of Singular Integral Equations for Analysis of Strip Structures and Experimental Confirmation

This paper presents a rigorous solution of the Helmholtz equation for regular waveguide structures with the finite sizes of all cross-section elements that may have an arbitrary shape. The solution is based on the theory of Singular Integral Equations (SIE). The SIE method proposed here is used to find a solution to differential equations with a point source. This fundamental solution of the equations is then applied in an integral representation of the general solution for our boundary problem. The integral representation always satisfies the differential equations derived from the Maxwell’s ones and has unknown functions μe and μh that are determined by the implementation of appropriate boundary conditions. The waveguide structures under consideration may contain homogeneous isotropic materials such as dielectrics, semiconductors, metals, and so forth. The proposed algorithm based on the SIE method also allows us to compute waveguide structures containing materials with high losses. The proposed solution allows us to satisfy all boundary conditions on the contour separating materials with different constitutive parameters and the condition at infinity for open structures as well as the wave equation. In our solution, the longitudinal components of the electric and magnetic fields are expressed in the integral form with the kernel consisting of an unknown function μe or μh and the Hankel function of the second kind. It is important to note that the above-mentioned integral representation is transformed into the Cauchy type integrals with the density function μe or μh at certain singular points of the contour of integration. The properties and values of these integrals are known under certain conditions. Contours that limit different materials of waveguide elements are divided into small segments. The number of segments can determine the accuracy of the solution of a problem. We assume for simplicity that the unknown functions μe and μh, which we are looking for, are located in the middle of each segment. After writing down the boundary conditions for the central point of every segment of all contours, we receive a well-conditioned algebraic system of linear equations, by solving which we will define functions μe and μh that correspond to these central points. Knowing the densities μe, μh, it is easy to calculate the dispersion characteristics of the structure as well as the electromagnetic (EM) field distributions inside and outside the structure. The comparison of our calculations by the SIE method with experimental data is also presented in this paper.

A modification of the method of magnetic field integral equations for analysis of wave scattering by PEC cylinder of arbitrary cross-section

A modification of the method of magnetic field integral equations for analysis of wave scattering by PEC cylinder of arbitrary cross-section

Reducing Volume Integrals to Line Integrals for Some Functions Associated With Schaubert–Wilton–Glisson Basis Function

An accurate method is proposed to calculate volume integrals arising from integral equation (IE) analysis using the Schaubert-Wilton-Glisson (SWG) basis function (BF). The volume integral is transformed into a summation of some of line integrals. The dimension reduction is achieved using the divergence theorem twice. Finally, each line integral is evaluated using an accuracy-controllable quadrature rule. We compare our results with those generated by Mathematica. Good agreement is found, which verifies the effectiveness and correctness of our method.

Integral Equation Analysis of Terahertz Backscattering From Circular Dielectric Rod With Partial Graphene Cover

We study how the backward scattering cross-section (BSCS) of a microsize dielectric rod illuminated by an H-polarized plane wave can be efficiently manipulated using a partial graphene cover and associated plasmon resonances. Our treatment is based on the hyper-singular boundary integral equation, discretized with a Nystrom-type algorithm, which has a mathematically grounded convergence. We compute the BSCS and the extinction cross-section (ECS) in the sub-THz and THz ranges with controlled accuracy and demonstrate that BSCS can be both enhanced and reduced due to the graphene strip.