Natural hp‐BEM for the electric field integral equation with singular solutions

We apply the hp ‐version of the boundary element method (BEM) for the numerical solution of the electric field integral equation (EFIE) on a Lipschitz polyhedral surface Γ. The underlying meshes are supposed to be quasi‐uniform triangulations of Γ, and the approximations are based on either Raviart‐Thomas or Brezzi‐Douglas‐Marini families of surface elements. Nonsmoothness of Γ leads to singularities in the solution of the EFIE, severely affecting convergence rates of the BEM. However, the singular behavior of the solution can be explicitly specified using a finite set of functions (vertex‐, edge‐, and vertex‐edge singularities), which are the products of power functions and poly‐logarithmic terms. In this article, we use this fact to perform an a priori error analysis of the hp ‐BEM on quasi‐uniform meshes. We prove precise error estimates in terms of the polynomial degree p, the mesh size h, and the singularity exponents. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012     

A fast numerical method for a natural boundary integral equation for the Helmholtz equation

A Neumann boundary value problem of the Helmholtz equation in the exterior circular domain is reduced into an equivalent natural boundary integral equation. Using our trigonometric wavelets and the Galerkin method, the obtained stiffness matrix is symmetrical and circulant, which lead us to a fast numerical method based on fast Fourier transform. Furthermore, we do not need to compute the entries of the stiffness matrix. Especially, our method is also efficient when the wave number k in the Helmholtz equation is very large.     

The boundary integral equation approach for numerical solution of the one‐dimensional Sine‐Gordon equation

This article describes a numerical method based on the boundary integral equation and dual reciprocity method for solving the one‐dimensional Sine‐Gordon (SG) equation. The time derivative is approximated by the time‐stepping method and a predictor–corrector scheme is employed to deal with the nonlinearity which appears in the problem. Numerical results are presented for some problems to demonstrate the usefulness and accuracy of this approach. In addition, the conservation of energy in SG equation is investigated. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

A fast numerical method for harmonic equation based on natural boundary integral

Based on the coupling of the natural boundary integral method and the finite elements method, we mainly investigate the numerical solution of Neumann problem of harmonic equation in an exterior elliptic. Using our trigonometric wavelets and Galerkin method, there obtained stiffness matrix is symmetrical and circulant, which lead us to a fast numerical method based on fast Fourier transform. Furthermore, we do not need to compute the entries of the stiffness matrix. On the other hand, we prove that the numerical solution possesses exponential convergence rate. Especially, examples state that our method still has good accuracy for small j when the solution u ₀(θ) is almost singular.     

Approximate solution of the system of nonlinear integral equation by Newton-Kantorovich method

The Newton-Kantorovich method is developed for solving the system of nonlinear integral equations. The existence and uniqueness of the solution are proved, and the rate of convergence of the approximate solution is established. Finally, numerical examples are provided to show the validity and the efficiency of the method presented.     

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.     

Convergence of Galerkin method solutions of the integral equation for thin wire antennas

In this paper we consider the Pocklington integro–differential equation for the current induced on a straight, thin wire by an incident harmonic electromagnetic field. We show that this problem is well posed in suitable fractional order Sobolev spaces and obtain a coercive or Gårding type inequality for the associated operator. Combining this coercive inequality with a standard abstract formulation of the Galerkin method we obtain rigorous convergence results for Galerkin type numerical solutions of Pocklington's equation, and we demonstrate that certain convergence rates hold for these methods.     

Locally one-dimensional difference schemes for the fractional order diffusion equation

Locally-one-dimensional difference schemes for the fractional diffusion equation in multidimensional domains are considered. Stability and convergence of locally one-dimensional schemes for this equation are proved.     

基于迭代正则化高斯-牛顿法的非线性Urysohn积分方程数值解

非线性Urysohn积分方程在许多领域中都有广泛的应用,但由于该方程具有不适定性的特点,数据的微小扰动可能导致解的巨大变化,给数值求解带来很大困难.为了获得稳定的、准确的数值解,本文利用迭代正则化高斯-牛顿法对此方程进行求解,给出了利用Sigmoid-型函数确定迭代正则化参数的方法.对一类重力测定问题进行了数值模拟,将得到的数值解和相应的精确解作比较.结果表明,本文提出的方法在求解非线性Urysohn积分方程时是可行的也是有效的.     

Method of boundary integral equations as applied to the numerical solution of the three-dimensional Dirichlet problem for the laplace equation in a piecewise homogeneous medium

A Dirichlet problem is considered in a three-dimensional domain filled with a piecewise homogeneous medium. The uniqueness of its solution is proved. A system of Fredholm boundary integral equations of the second kind is constructed using the method of surface potentials, and a system of boundary integral equations of the first kind is derived directly from Green’s identity. A technique for the numerical solution of integral equations is proposed, and results of numerical experiments are presented.     

Convergence of the Galerkin methods for equations with elliptic operators on subspaces and solving the electric field equation

Application of the Galerkin methods to the numerical analysis of the integro-differential electric field equation is justified. The convergence of the Galerkin methods is established for a class of equations with nonelliptic operators comprising the electric field equation. Theorems concerning the approximation of the elements belonging to a special Sobolev space by the basis Rao-Wilton-Glisson functions are proved. The rate of convergence is estimated.     

A numerical method to determine mode content in circular waveguide based on an integral identity equation

According to the orthogonality of circular waveguide modes, an orthogonal integral identity equation is proposed among plane-wave spectrum of different modes, under the same azimuthal angle index. By using this identity equation, a simple method, which can determine components of different modes, is proposed. In the end, some numerical experiment results are also reported, which show that the proposed method is efficient.     

Finite difference schemes for the two‐dimensional semilinear wave equation

In this article, two finite difference schemes for solving the semilinear wave equation are proposed. The unique solvability and the stability are discussed. The second‐order accuracy convergence in both time and space in the discrete H¹‐norm for the two proposed difference schemes is proved. Numerical experiments are performed to support our theoretical results.     

On the numerical solution of a boundary integral equation for the exterior Neumann problem on domains with corners

The authors propose a “modified” Nyström method to approximate the solution of a boundary integral equation connected with the exterior Neumann problem for Laplace's equation on planar domains with corners. They prove the convergence and the stability of the method and show some numerical tests.     

A multiscale Galerkin method for the hypersingular integral equation reduced by the harmonic equation

The aim of this paper is to investigate the numerical solution of the hypersingular integral equation reduced by the harmonic equation. First, we transform the hypersingular integral equation into 2π-periodic hypersingular integral equation with the map x=cot(θ/2). Second, we initiate the study of the multiscale Galerkin method for the 2π-periodic hypersingular integral equation. The trigonometric wavelets are used as trial functions. Consequently, the 2j+1 × 2j+1 stiffness matrix Kj can be partitioned j×j block matrices. Furthermore, these block matrices are zeros except main diagonal block matrices. These main diagonal block matrices are symmetrical and circulant matrices, and hence the solution of the associated linear algebraic system can be solved with the fast Fourier transform and the inverse fast Fourier transform instead of the inverse matrix. Finally, we provide several numerical examples to demonstrate our method has good accuracy even though the exact solutions are multi-peak and almost singular.     

A random integral quadrature method for numerical analysis of the second kind of Volterra integral equations

In this paper, a novel meshless technique termed the random integral quadrature (RIQ) method is developed for the numerical solution of the second kind of the Volterra integral equations. The RIQ method is based on the generalized integral quadrature (GIQ) technique, and associated with the Kriging interpolation function, such that it is regarded as an extension of the GIQ technique. In the GIQ method, the regular computational domain is required, in which the field nodes are scattered along straight lines. In the RIQ method however, the field nodes can be distributed either uniformly or randomly. This is achieved by discretizing the governing integral equation with the GIQ method over a set of virtual nodes that lies along straight lines, and then interpolating the function values at the virtual nodes over all the field nodes which are scattered either randomly or uniformly. In such a way, the governing integral equation is converted approximately into a system of linear algebraic equations, which can be easily solved.     

On existence and local attractivity of solutions of a quadratic Volterra integral equation of fractional order

In the paper we study the existence of solutions of a nonlinear quadratic Volterra integral equation of fractional order. This equation is considered in the Banach space of real functions defined, continuous and bounded on an unbounded interval. Moreover, we show that solutions of this integral equation are locally attractive.     

The finite integral transform method for a parabolic differential equation on a graph

The finite integral transform method is set forth and justified for solving a mixed problem for a parabolic differential equation posed on a graph.