Modified integral current methods in electrodynamics of nonhomogeneous media

The main difficulty in numerical solution of integral equations of electrodynamics is associated with the need to solve a high-order system of linear equations with a dense matrix. It is therefore relevant to develop numerical methods that lead to linear equation systems of lower order at the cost of more complex evaluation of the coefficients. In this article we propose a method for solving linear equations of electrodynamics which is a modification of the integral current method. The main distinctive feature of the proposed method is double integration of the electric Green’s tensor in the process of algebraization of the original integral equation. The solutions of the system of linear equations are thus integral means of the electric field inside the anomaly constructed by the proposed transformation formula. We prove convergence and derive error bounds for both the solution of the integral equation and the electromagnetic field components evaluated from approximate transformation formulas.     

Integral equation methods for the Yukawa-Beltrami equation on the sphere

An integral equation method for solving the Yukawa-Beltrami equation on a multiply-connected sub-manifold of the unit sphere is presented. A fundamental solution for the Yukawa-Beltrami operator is constructed. This fundamental solution can be represented by conical functions. Using a suitable representation formula, a Fredholm equation of the second kind with a compact integral operator needs to be solved. The discretization of this integral equation leads to a linear system whose condition number is bounded independent of the size of the system. Several numerical examples exploring the properties of this integral equation are presented.     

On the quadrature methods for the numerical solution of singular integral equations

A Cauchy type singular integral equation can be numerically solved by the use of an appropriate numerical integration rule and the reduction of this equation to a system of linear algebraic equations, either directly or after the reduction of the Cauchy type singular integral equation to an equivalent Fredholm integral equation of the second kind. In this paper two fundamental theorems on the equivalence (under appropriate conditions) of the aforementioned methods of numerical solution of Cauchy type singular integral equations are proved in sufficiently general cases of Cauchy type singular integral equations of the second kind.     

Numerical method for solving linear Fredholm fuzzy integral equations of the second kind

In this paper we use parametric form of fuzzy number and convert a linear fuzzy Fredholm integral equation to two linear system of integral equation of the second kind in crisp case. We can use one of the numerical method such as Nystrom and find the approximation solution of the system and hence obtain an approximation for fuzzy solution of the linear fuzzy Fredholm integral equations of the second kind. The proposed method is illustrated by solving some numerical examples.     

Analytical discussion for the mixed integral equations

This paper presents a numerical method for the solution of a Volterra–Fredholm integral equation in a Banach space. Banachs fixed point theorem is used to prove the existence and uniqueness of the solution. To find the numerical solution, the integral equation is reduced to a system of linear Fredholm integral equations, which is then solved numerically using the degenerate kernel method. Normality and continuity of the integral operator are also discussed. The numerical examples in Sect. 5 illustrate the applicability of the theoretical results.     

On approximation and numerical solution of Fredholm integral equations of second kind using quasi-interpolation

In this article a method is presented, which can be used for the numerical treatment of integral equations. Considered is the Fredholm integral equation of second kind with continuous kernel, since this type of integral equation appears in many applications, for example when treating potential problems with integral equation methods.The method is based on the approximation of the integral operator by quasi-interpolating the density function using Gaussian kernels. We show that the approximation of the integral equation, gained with this method, for an appropriate choice of a certain parameter leads to the same numerical results as Nyström’s method with the trapezoidal rule. For this, a convergence analysis is carried out.     

椭圆外区域上的自然边界元法

１．引言 二十年来,自然边界元法已在椭圆问题求解方面取得了许多研究成果。它可以直接用来解决圆内（外）区域、扇形区域、球内（外）区域及半平面区域等特殊区域上的椭圆边值问题［１,２,５］,也可以结合有限元法求解一般区域上的椭圆边值问题,例如基于自然边界归化的耦合算法及区域分解算法就是处理断裂区域问题及外问题的一种有效手段［２－４,６］。 人们在设计求解外问题的耦合算法或者区域分解算法时,通常选取圆周或球面作人工边界。但对具有长条型内边界的外问题,以圆周或球面作人工边界显然并非最佳选择,它将会导致大量的…     

A Nyström method for a class of Fredholm integral equations of the third kind on unbounded domains

The author proposes a numerical procedure in order to approximate the solution of a class of Fredholm integral equations of the third kind on unbounded domains. The given equation is transformed in a Fredholm integral equation of the second kind. Hence, according to the integration interval, the equation is regularized by means of a suitable one-to-one map or is transformed in a system of two Fredholm integral equations that are subsequently regularized. In both cases a Nyström method is applied, the convergence and the stability of which are proved in spaces of weighted continuous functions. Error estimates and numerical tests are also included.     

On the convergence of a numerical method for a hypersingular integral equation on a closed surface

We consider a linear integral equation with a hypersingular integral treated in the sense of the Hadamard finite value. This equation arises in the solution of the Neumann boundary value problem for the Laplace equation with a representation of a solution in the form of a double-layer potential. We consider the case in which the interior or exterior boundary value problem is solved in a domain; whose boundary is a smooth closed surface, and an integral equation is written out on that surface. For the integral operator in that equation, we suggest quadrature formulas like the method of vortical frames with a regularization, which provides its approximation on the entire surface for the use of a nonstructured partition. We construct a numerical scheme for the integral equation on the basis of suggested quadrature formulas, prove an estimate for the norm of the inverse matrix of the related system of linear equations and the uniform convergence of numerical solutions to the exact solution of the hypersingular integral equation on the grid.     

An improved boundary element formulation for the motion of spherical bubbles

We suggest a modified boundary element method for modeling the potential flow caused by the motion of many spherical bubbles. The problem for the fluid velocity potential is reduced to a Fredholm integral equation of the second kind. The kernel of that integral equation has no singularity; as a result, its numerical solution does not encounter any difficulties. The matrix representation of the integral equation is diagonally dominant and is well suited to handle multiple bubble system.     

On the solution of a mixed nonlinear integral equation

In this paper, we consider a mixed nonlinear integral equation of the second kind in position and time. The existence of a unique solution of this equation is discussed and proved. A numerical method is used to obtain a system of Harmmerstein integral equations of the second kind in position. Then the modified Toeplitz matrix method, as a numerical method, is used to obtain a nonlinear algebraic system. Many important theorems related to the existence and uniqueness solution to the produced nonlinear algebraic system are derived. The rate of convergence of the total error is discussed. Finally, numerical examples when the kernel of position takes a logarithmic and Carleman forms, are presented and the error estimate, in each case, is calculated.     

A boundary integral equation for the transmission eigenvalue problem for Maxwell equation

We propose a new integral equation formulation to characterize and compute transmission eigenvalues in electromagnetic scattering. As opposed to the approach that was recently developed by Cakoni, Haddar and Meng (2015) which relies on a two‐by‐two system of boundary integral equations, our analysis is based on only one integral equation in terms of the electric‐to‐magnetic boundary trace operator that results in a simplification of the theory and in a considerable reduction of computational costs. We establish Fredholm properties of the integral operators and their analytic dependence on the wave number. Further, we use the numerical algorithm for analytic nonlinear eigenvalue problems that was recently proposed by Beyn (2012) for the numerical computation of the transmission eigenvalues via this new integral equation.     

Numerical solution of nonlinear stochastic Itô‐Volterra integral equations driven by fractional Brownian motion

In this paper, an efficient numerical technique is applied to provide the approximate solution of nonlinear stochastic Itô‐Volterra integral equations driven by fractional Brownian motion with Hurst parameter . The proposed method is based on the operational matrices of modification of hat functions (MHFs) and the collocation method. In this approach, by approximating functions that appear in the integral equation by MHFs and using Newton's‐Cotes points, nonlinear integral equation is transformed to nonlinear system of algebraic equations. This nonlinear system is solved by using Newton's numerical method, and the approximate solution of integral equation is achieved. Some theorems related to error estimate and convergence analysis of the suggested scheme are also established. Finally, 2 illustrative examples are included to confirm applicability, efficiency, and accuracy of the proposed method. It should be noted that this scheme can be used to solve other appropriate problems, but some modifications are required.     

Convergence of the piecewise linear approximation and collocation method for a hypersingular integral equation on a closed surface

We study the numerical solution of a linear hypersingular integral equation arising when solving the Neumann boundary value problem for the Laplace equation by the boundary integral equation method with the solution represented in the form of a double layer potential. The integral in this equation is understood in the sense of Hadamard finite value. We construct quadrature formulas for the integral occurring in this equation based on a triangulation of the surface and an application of the linear approximation to the unknown function on each of the triangles approximating the surface. We prove the uniform convergence of the quadrature formulas at the interpolation nodes as the triangulation size tends to zero. A numerical solution scheme for this integral equation based on the suggested quadrature formulas and the collocation method is constructed. Under additional assumptions about the shape of the surface, we prove a uniform estimate for the error in the numerical solution at the interpolation nodes.     

关于二维双曲型初边值问题的自然积分方程

本文将自然边界元方法应用于求解一类双曲型初边值问题。给出自然积分方程及Poisson积分公式,研究了自然积分算子的性质,并详细讨论了自然积分方程的数值解法,最后给出数值例子。     

A boundary integral equation method for the transmission eigenvalue problem

We propose a new integral equation formulation to characterize and compute transmission eigenvalues for constant refractive index that play an important role in inverse scattering problems for penetrable media. As opposed to the recently developed approach by Cossonnière and Haddar [1,2] which relies on a two by two system of boundary integral equations our analysis is based on only one integral equation in terms of Dirichlet-to-Neumann or Robin-to-Dirichlet operators which results in a noticeable reduction of computational costs. We establish Fredholm properties of the integral operators and their analytic dependence on the wave number. Further we employ the numerical algorithm for analytic non-linear eigenvalue problems that was recently proposed by Beyn [3] for the numerical computation of transmission eigenvalues via this new integral equation.     

New predictor-corrector approach for nonlinear fractional differential equations: error analysis and stability

In this paper, the predictor-corrector approach is used to propose two algorithms for the numerical solution of linear and non-linear fractional differential equations (FDE). The fractional order derivative is taken to be in the sense of Caputo and its properties are used to transform FDE into a Volterra-type integral equation. Simpson''s 3/8 rule is used to develop new numerical schemes to obtain the approximate solution of the integral equation associated with the given FDE. The error and stability analysis for the two methods are presented. The proposed methods are compared with the ones available in the literature. Numerical simulation is performed to demonstrate the validity and applicability of both the proposed techniques. As an application, the problem of dynamics of the new fractional order non-linear chaotic system introduced by Bhalekar and Daftardar-Gejji is investigated by means of the obtained numerical algorithms.     

Product integration methods for solving a system of nonlinear Volterra integral equations

In this paper the technique of subtracting out singularities is used to derive explicit and implicit product Euler schemes with order one convergence and a product trapezoidal scheme with order two convergence for a system of Volterra integral equations with a weakly singular kernel. The convergence proofs of the numerical schemes are presented; these are nonstandard since the nonlinear function involved in the integral equation system does not satisfy a global Lipschitz condition.     

抛物型初边值问题的自然积分方程及其数值解法

１．引言数值求解无界区域的偏微分方程,自然的处理方式是削去区域的无界部分,即引入一条适当的人工边界ｒ。,将原问题的求解限制在一个适当的有界区域Ｄ内,这样必须在人工边界上引入适当边界的条件．于是很自然地导致这样一个问题：'是否存在一个人工边界条件,使得在这边界条件下,原问题在区域Ｄ内所求得的数值解与原无界区域的解在Ｄ上的限制是完全一致的？"这里我们的着眼点是寻求与原无界区域问题等价的数学形式,以便于数值求解．因为边界元方法可以将区域内的问题转化到区域的边界上去处理,经典的边界元方法常被应用．七十年代…     

Splinefunktionen bei der Lösung von Integralgleichungen

Summary The paper presents a numerical method to solve integral equations of the second kind. An interpolating spline is substituted for the unknown function, so that the integral can be evaluated. Similar applications of this so-called productintegration are given in [11, 22]. Satisfying the equation in the interpolation-points one obtains a system of linear algebraic equations for the values of the unknown function in these points. An estimation of the numerical error is given.

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Gekürzte Fassung der Dissertation des Verfassers.