Integral Equations Of Second Kind Research Articles

The Regularization Technique in Modeling of the Plane E-Polarized EM Wave Scattering by Coplanar System of Electrically Conducting Flat Strips

The integral equations technique has been successfully modified for studying the plane E-polarized electromagnetic wave scattering by multilevel coplanar systems of zero-thickness impedance strips. Reformulation of the scattering problem in the form of the second-kind regular integral equations has been realized on the base of the Carleman regularization technique. Two novel and original classes of specific Cantor functions have been presented and analyzed. Using new Cantor functions, it is easy to create a lot of non-classical orderings for specific multilevel coplanar strips systems. That sort of system can be useful for modeling irregular natural processes and zones in the future. Considerable attention was focused on the plane E-polarized electromagnetic wave scattering by sparsely filled coplanar systems of electrically narrow impedance strips. An explicit solution of the scattering problem has been obtained for such case of strips system.

Definition of the solution of the homogeneous Volterra integral equation

In this paper, we study the solvability of a second-kind Volterra integral equation. By replacing the right-hand side and the unknown function, the integral equation is reduced to an integral equation, the kernel of which is not «compressiblek. Using the Laplace transform, the obtained equation is reduced to an ordinary first-order differential equation (linear). Its solution is found. The solution of the homogeneous integral equation corresponding to the original nonhomogeneous integral equation found in explicit form. Special cases of a homogeneous integral equation and its solutions are written for different values of the parameter k. Classes are indicated in which the integral equation has a solution. Singular integral equations were considered in works [1–3]. Their kernels were also «incompressiblek, but kernels had an another form. In this connection, the weight classes of the solution existence differ from the class of the solution existence for the equation considered in this work.

Piecewise Polynomial in solving Fredholm Integral Equation of Second Kind by using Successive over Relaxation method

Piecewise Polynomial in solving Fredholm Integral Equation of Second Kind by using Successive over Relaxation method

Solution of Linear Volterra Integral Equation of Second Kind via Rishi Transform

The solution of various problems of engineering and science can easily determined by representing these problems in integral equations. There are numerous analytical and numerical methods which can be used for solving different kinds of integral equations. In this paper, authors used recently developed integral transform “Rishi Transform” for obtaining the analytical solution of linear Volterra integral equation of second kind (LVIESK). For this, the kernel of LVIESK has assumed a convolution type kernel. Five numerical examples are considered for demonstrating the complete procedure of determining the solution. Results of these problems suggest that Rishi transform provides the exact analytical solution of LVIESK without doing complicated calculation work.

TRIPLE SERIES EQUATIONS INVOLVING GENERALIZED LAGUERRE POLYNOMIALS

In this paper, the solutions of two sets of triple series equations involving generalized Laguerre polynomials have been obtained by reducing them to a Fredholm integral equation of second kind. In each case, the problem is reduced to the solution of a Fredholm integral equation of the second kind. We consider certain triple series equations involving generalized Laguerre polynomials which are generalization of those considerd by Lawndes-Srivastava .Connected to this work solutions have been considered by Sneddon, Lowndes and Srivastava ,Dwivedi and Trivedi, Singh et al., Narain, Srivastava - Panda etc

Windowed Green Function MoM for Second-Kind Surface Integral Equation Formulations of Layered Media Electromagnetic Scattering Problems

This paper presents a second-kind surface integral equation method for the numerical solution of frequency-domain electromagnetic scattering problems by locally perturbed layered media in three spatial dimensions. Unlike standard approaches, the proposed methodology does not involve the use of layer Green functions. It instead leverages an indirect M\"uller formulation in terms of free-space Green functions that entails integration over the entire unbounded penetrable boundary. The integral equation domain is effectively reduced to a small-area surface by means of the windowed Green function method, which exhibits high-order convergence as the size of the truncated surface increases. The resulting (second-kind) windowed integral equation is then numerically solved by means of the standard Galerkin method of moments (MoM) using RWG basis functions. The methodology is validated by comparison with Mie-series and Sommerfeld-integral exact solutions as well as against a layer Green function-based MoM. Challenging examples including realistic structures relevant to the design of plasmonic solar cells and all-dielectric metasurfaces, demonstrate the applicability, efficiency, and accuracy of the proposed methodology.

Theory and Computations for the System of Integral Equations via the use of Optimal Auxiliary Function Method

This paper extends the application of Optimal Auxiliary Function Method (OAFM) to the system of integral equations. The system of Volterra integral equations of second kind are taken as test examples. The results obtained by proposed method are compared with different methods i.e., Biorthogonal systems in a Banach space Fixed point, the implicit Trapezoidal rule in conjunction with Newton's method, and Relaxed Monte Carlo method (RMCM). The results revealed that OAFM is more efficient, simple to apply, and fast convergent. The auxiliary functions used in the method control its convergence. The values of arbitrary constants involved in the auxiliary functions are calculated optimally using the method of least square

Spectrally-Accurate Analysis of EM Scattering by Material and Metamaterial, Isotropic or Anisotropic, Radially Inhomogeneous Spherical Objects

We study electromagnetic (EM) plane wave scattering by material and metamaterial radially inhomogeneous spherical objects that are either isotropic or uniaxially anisotropic with radial anisotropy. Using field representations in terms of scalar Hertz potentials, we reformulate Maxwell’s equations into two systems of first-order ordinary differential equations, which we then convert into equivalent second-kind linear Volterra matrix integral equations. The discretization scheme uses a high-order, entire domain, Nystrom method that employs the Gauss–Legendre–Radau quadrature rule. The proposed algorithms have spectral convergence and apply to either single-layer or multilayer structures with continuous or sectionally continuous profiles.

Numerical algorithms for water waves with background flow over obstacles and topography

We present two accurate and efficient algorithms for solving the incompressible, irrotational Euler equations with a free surface in two dimensions with background flow over a periodic, multiply connected fluid domain that includes stationary obstacles and variable bottom topography. One approach is formulated in terms of the surface velocity potential while the other evolves the vortex sheet strength. Both methods employ layer potentials in the form of periodized Cauchy integrals to compute the normal velocity of the free surface, are compatible with arbitrary parameterizations of the free surface and boundaries, and allow for circulation around each obstacle, which leads to multiple-valued velocity potentials but single-valued stream functions. We prove that the resulting second-kind Fredholm integral equations are invertible, possibly after a physically motivated finite-rank correction. In an angle-arclength setting, we show how to avoid curve reconstruction errors that are incompatible with spatial periodicity. We use the proposed methods to study gravity-capillary waves generated by flow around several elliptical obstacles above a flat or variable bottom boundary. In each case, the free surface eventually self-intersects in a splash singularity or collides with a boundary. We also show how to evaluate the velocity and pressure with spectral accuracy throughout the fluid, including near the free surface and solid boundaries. To assess the accuracy of the time evolution, we monitor energy conservation and the decay of Fourier modes and compare the numerical results of the two methods to each other. We implement several solvers for the discretized linear systems and compare their performance. The fastest approach employs a graphics processing unit (GPU) to construct the matrices and carry out iterations of the generalized minimal residual method (GMRES).

Numerical Investigation of Volterra Integral Equations of Second Kind using Optimal Homotopy Asymptotic Method

This investigation is concerned with the solutions of Volterra integral equations of second kind that have been determined by employing Optimal Homotopy Asymptotic method (OHAM). The existence and uniqueness of solutions are proved in this work. The obtained solutions are novel, and previous literature lacks such derivations. The convergence of the approximate solutions using the proposed method is investigated. Error’s estimation to the corresponding numerical scheme is also carried out. The reliability and accuracy of OHAM have been shown by comparison of our derived solutions with solutions obtained by other existing methods. The efficiency of the proposed numerical technique is exhibited through graphical illustrations, and results are drafted in tabular form for specific values of parameter to validate the numerical investigation.

The Numerical Solution of Volterra Integral Equation of Second Kind Using Quartic Non-Polynomial Spline Function

In this study, the non-polynomial spline function of fourth order is given to solve the second kind of the nonlinear Volterra integral equation. Numerical examples are show to clarify the applications of the proposed method and to compare the calculated results with other methods. In addition, there is a comparison of the absolute value of the error that estimated by of the suggested method is compared by those that estimated through other methods and the exact solution. The obtained results are very encouraging compared by other numerical methods.

Ten non-polynomial cubic splines for some classes of Fredholm integral equations

In this paper, a new general form for the non-polynomial cubic spline function is introduced. This form enables us to generate about ten different formulas of the proposed spline function. Although some of these functions have been used before to solve some types of differential and integral equations, the others are considered new. These ten functions are used to approximate the solution of linear, nonlinear, and fuzzy Fredholm integral equations of second kind. Moreover, the convergence analysis of the proposed method is established which indicates that its order of convergence is four. Also, some examples are solved and compared with the previous methods which showed that the present method is more accurate.

Discontinuous Galerkin approximations to second-kind Volterra integral equations with weakly singular kernel

The discontinuous Galerkin (DG) method is employed to solve second-kind Volterra integral equations (VIEs) with weakly singular kernels. It is proved that the quadrature DG (QDG) method obtained from the DG method by approximating the inner products by suitable numerical quadrature formulas, is equivalent to the piecewise discontinuous polynomial collocation method. The convergence theory is established for VIEs for both uniform and graded meshes. Some numerical experiments are given to illustrate the obtained theoretical results.

Radiation of water waves by a heaving submerged disc in a three-layer fluid

A thin rigid disc is submerged in the infinitely deep lower layer of a three-layer fluid. Using linear water wave theory, the three-dimensional problem of wave radiation due to heaving motion of the disc is considered. The problem is first reduced to solving a hypersingular boundary integral equation. This integral equation is then reduced to a system of one-dimensional Fredholm integral equations of second kind. The solution of each integral equation is associated with a radial component of the difference of potential function across the plate. The solution of these integral equations has been used to compute added mass and damping coefficient numerically. Numerical results for a heaving disc submerged in a single layer, a two-layer, and a three-layer fluid have been depicted. Also, the effects of different parameters like depth of upper layer, middle layer, distance of the disc from the interface, and the density ratios of the fluid layers on these physical quantities have been demonstrated.

Collocation boundary value methods for auto-convolution Volterra integral equations

In this paper, we analyze the collocation boundary value solutions of the second-kind auto-convolution Volterra integral equations. In contrast to classical collocation methods, the full discretization of auto-convolution Volterra integral equations via the boundary value technique leads to a nonlinear system. Thus we reconsider the solvability of the proposed scheme and the boundedness of the numerical solution. Moreover, with the help of the Peano kernel theorem, the convergence order is developed, which is demonstrated by several numerical examples.

An efficient method for solving neutrosophic Fredholm integral equations of second kind

An efficient method for solving neutrosophic Fredholm integral equations of second kind

Asymptotic Behavior of Solutions of Integral Equations with Homogeneous Kernels

The multidimensional integral equation of second kind with a homogeneous of degree (−n) kernel is considered. The special class of continuous functions with a given asymptotic behavior in the neighborhood of zero is defined. It is proved that, if the free term of the integral equation belongs to this class and the equation itself is solvable, then its solution also belongs to this class. To solve this problem, a special research technique is used. The above-mentioned technique is based on the decomposition of both the solution and the free term in spherical harmonics.

Some fixed point results for <i>ξ</i>-chainable neutrosophic and generalized neutrosophic cone metric spaces with application

<abstract> <p>This manuscript is concerned for introducing novel concepts of <italic>ξ</italic>-chainable neutrosophic metric space and generalized neutrosophic cone metric spaces. We use four self-mappings to establish common fixed point theorem in the sense of <italic>ξ</italic>-chainable neutrosophic metric space and three self-mappings to establish common fixed point results in the sense of generalized neutrosophic metric spaces. Certain properties of <italic>ξ</italic>-chainable neutrosophic metric space and generalized neutrosophic metric spaces are defined and their examples are presented. An application to fuzzy Fredholm integral equation of second kind is developed to verify the validity of proposed results. These results boost the approaches of existing literature of fuzzy metric spaces and fuzzy fixed theory.</p> </abstract>

Coercivity, essential norms, and the Galerkin method for second-kind integral equations on polyhedral and Lipschitz domains

It is well known that, with a particular choice of norm, the classical double-layer potential operator D has essential norm <1/2 as an operator on the natural trace space H^{1/2}(Gamma ) whenever Gamma is the boundary of a bounded Lipschitz domain. This implies, for the standard second-kind boundary integral equations for the interior and exterior Dirichlet and Neumann problems in potential theory, convergence of the Galerkin method in H^{1/2}(Gamma ) for any sequence of finite-dimensional subspaces ({{mathcal {H}}}_N)_{N=1}^infty that is asymptotically dense in H^{1/2}(Gamma ). Long-standing open questions are whether the essential norm is also <1/2 for D as an operator on L^2(Gamma ) for all Lipschitz Gamma in 2-d; or whether, for all Lipschitz Gamma in 2-d and 3-d, or at least for the smaller class of Lipschitz polyhedra in 3-d, the weaker condition holds that the operators pm frac{1}{2}I+D are compact perturbations of coercive operators—this a necessary and sufficient condition for the convergence of the Galerkin method for every sequence of subspaces ({{mathcal {H}}}_N)_{N=1}^infty that is asymptotically dense in L^2(Gamma ). We settle these open questions negatively. We give examples of 2-d and 3-d Lipschitz domains with Lipschitz constant equal to one for which the essential norm of D is ge 1/2, and examples with Lipschitz constant two for which the operators pm frac{1}{2}I +D are not coercive plus compact. We also give, for every C>0, examples of Lipschitz polyhedra for which the essential norm is ge C and for which lambda I+D is not a compact perturbation of a coercive operator for any real or complex lambda with |lambda |le C. We then, via a new result on the Galerkin method in Hilbert spaces, explore the implications of these results for the convergence of Galerkin boundary element methods in the L^2(Gamma ) setting. Finally, we resolve negatively a related open question in the convergence theory for collocation methods, showing that, for our polyhedral examples, there is no weighted norm on C(Gamma ), equivalent to the standard supremum norm, for which the essential norm of D on C(Gamma ) is <1/2.

Approximation of functions by Bernoulli wavelet and its applications in solution of Volterra integral equation of second kind

In this paper, two new estimators E_{2^{k-1},0}^{(1)}(f) and E_{2^{k-1},M}^{(1)}(f) of characteristic function and an estimator E_{2^{k-1},M}^{(2)}(f) of function of Hddot{text {o}}lder’s class H^{alpha } [0,1) of order 0<alpha leqslant 1 have been established using Bernoulli wavelets. A new technique has been applied for solving Volterra integral equation of second kind using Bernoulli wavelet operational matrix of integration as well as product operational matrix. These matrices have been utilized to reduce the Volterra integral equation into a system of algebraic equations, which are easily solvable. Some examples are illustrated to show the validity and efficiency of proposed technique of this research paper.