The efficient solution of direct medium problems by using translation techniques

We consider a Fredholm integral equation arising from a time-harmonic electromagnetic scattering problem for inhomogeneous media. The discretization of this equation usually produces a large dense linear system that must be solved by iterative methods. To speed up these methods we propose an efficient computation of the action of the corresponding coefficient matrix on a generic vector. This computation is mainly based on the well known addition formula for the Hankel functions and a simple translation argument. We present some numerical examples to show the efficiency of the proposed method.     

The discrete collocation method for Fredholm–Hammerstein integral equations based on moving least squares method

The purpose of this paper is to investigate the discrete collocation method based on moving least squares (MLS) approximation for Fredholm–Hammerstein integral equations. The scheme utilizes the shape functions of the MLS approximation constructed on scattered points as a basis in the discrete collocation method. The proposed method is meshless, since it does not require any background mesh or domain elements. Error analysis of this method is also investigated. Some numerical examples are provided to illustrate the accuracy and computational efficiency of the method.     

Robust projection method for Fredholm integral equation of the second kind

The projection method based on the Schauder basis expansion for solving the Fredholm integral equation of the second kind is investigated. (1) It is shown that any given finite linearly independent function set in L ^p [0, 1] can be extended as a Schauder basis. The truncation of the dual Schauder bases is then used to form the projection in approximating the kernel. (2) With the finite rank approximation, the use of the Sherman–Morrison–Woodbury formula achieves efficient implementation. The computation is successive when dimension of projection space increases. Numerical experiments demonstrate the robustness of the Schauder basis expansion.     

A weighted H1 seminorm regularization method for Fredholm integral equations of the first kind

Many problems in mathematics and engineering lead to Fredholm integral equations of the first kind, e.g. signal and image processing. These kinds of equations are difficult to solve numerically since they are ill-posed. Therefore, regularization is required to obtain a reasonable approximate solution. This paper presents a new regularization method based on a weighted H¹ seminorm. Details of numerical implementation are given. Numerical examples, including one-dimensional and two-dimensional integral equations, are presented to illustrate the efficiency of the proposed approach. Numerical results show that the proposed regularization method can restore edges as well as details.     

Existence of Positive Periodic Solutions for Two Species Population Growth Model

Existence and nonexistence criteria are established for the positive periodic solutions of two species pop-ulation growth with periodic delay by applying continuation theorem of coincidence degree theory.     

商巴拿赫空间上的Fredholm复形理论

本文研究了严格Ｆｒｅｄｈｏｌｍ复形在自然的小和紧扰动下的性质和指标不变性，作为特例给出了Ｆｒｅｄｈｏｌｍ算子组的指标不变性的一个结果。     

Rigid body mode and spurious mode in the dual boundary element formulation for the Laplace problems

In this paper, the general formulation for the static stiffness is analytically derived using the dual integral formulations. It is found that the same stiffness matrix is derived by using the integral equation no matter what the rigid body mode and the complementary solutions are superimposed in the fundamental solution. For the Laplace problem with a circular domain, the circulant was employed to derive the stiffness analytically in the discrete system. In deriving the static stiffness, the degenerate scale problem occurs when the singular influence matrix can not be inverted. The Fredholm alternative theorem and the SVD updating technique are employed to study the degenerate scale problem mathematically and numerically. The direct treatment in the matrix level is achieved to deal with the degenerate scale problems instead of using a modified fundamental solution. The addition of a rigid body term in the fundamental solution is found to shift the zero singular value for the singular matrix without disturbing the stiffness.     

Solution for the general integral‐type nonlocal plasticity model with Tikhonov regularization

A new solution approach, based on Tikhonov regularization on the Fredholm integral equations of the first kind, is proposed to find the approximate solutions of the strain softening problems. In this approach, the consistency condition is regularized with the Tikhonov stabilizers along with a regularization parameter, and the internal variable increments are solved from the resulting Euler's equations. It is shown that, as the regularization parameter is increased, the solutions converge to a unique one. A nonlocal yield condition and a nonlocal return mapping algorithm are proposed to carry out the integration of constitutive equations in the time and spatial domains. A global plastic dissipation principle is proposed to relax the classical local plastic dissipation postulate. Numerical examples show that the proposed approach leads to objective, mesh‐independent solutions of the softening‐induced localization problems. A comparison of the results from the proposed approach with those from the gradient‐dependent plasticity model shows that the two models give close solutions.     

Optimal control of the order of asymptotics for elliptic equations with fast oscillating coefficients. I. Formal constructions

Formal algorithms for complete asymptotic decompositions of solutions to optimal globally bounded control problems for elliptic equations with fast oscillating coefficients are constructed. If the kernel of a differential operator of the initial problem is nonzero, asymptotics of different orders may exist in the control system.Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 84–95, September–October 2004.     

不同荷载速度下弹性土体中排桩的隔振效果分析

利用Muki和Sternberg的虚拟桩法,分析了不同荷载速度下弹性土体地基中排桩对移动荷载引起振动的被动隔振效果。首先由积分变换推导弹性土体表面受移动荷载的自由波场解、内部受圆形均布简谐荷载作用下的基本解,再根据基本解与桩土变形协调条件,建立了频域内土—桩的第二类Fredholm积分方程。最后通过求解积分方程与相应的积分逆变换得到时间域内评价隔振效果的振幅比。通过与已知文献结果比较,验证本文方法的正确性。数值结果表明:荷载速度对排桩的隔振效果有一定影响:当移动荷载小于土体的剪切波速时,随荷载速度增加排桩的隔振效果会降低,然而当荷载速度大于土体的剪切波速后,部分区域的隔振效果较好。