Numerical solution of some classes of integral equations using Bernstein polynomials

This paper is concerned with obtaining approximate numerical solutions of some classes of integral equations by using Bernstein polynomials as basis. The integral equations considered are Fredholm integral equations of second kind, a simple hypersingular integral equation and a hypersingular integral equation of second kind. The method is explained with illustrative examples. Also, the convergence of the method is established rigorously for each class of integral equations considered here.     

一类对偶积分方程组正则化为Cauchy奇异积分方程组解法

本文基于Mellin变换法求解复杂更一般形式的对偶积分方程组．通过积分变换，由实数域化成复数域上的方程组，引入未知函数的积分变换，移动积分路径，应用Cauchy积分定理，实现退耦正则化为Cauchy奇异积分方程组，由此给出一般性解，并严格证明了对偶积分方程组退耦正则化为Cauchy奇异积分方程组与原对偶积分方程组等价性，以及对偶积分方程组解的存在性和唯一性．给出的解法和理论解，作为求解复杂对偶积分方程组一种有效解法，可供求解复杂的数学、物理、力学中的混合边值问题应用．     

含三角函数的一般形式复杂对偶积分方程组的理论解

本文基于Gopson法，进行研究，改进，推广，应用于一般形式，复杂的对偶积分方程组的求解，首先引入函数进行方程组变换，其次引入未知函数的积分变换实现退耦，应用Abel反演变换，使方程组正则化为Fredholm第二类积分方程组，并由此给出对偶积分方程组的一般性解，本文给出的解法和理论解，可供求解复杂的数学，物理，力学中的混合边值问题参考，选用．同时也提供求解复杂的对偶积分方程组另一种有效的解法。     

Boundary integral equation method in micropolar elasticity

This paper presents the integral representations of the displacement and rotation fields of the micropolar continuum, and the regularized integral representations of the traction and couple vector. Thus the boundary integral equations and the boundary integro-differential equations are derived. In addition, the formulation for crack problems is given by both the boundary integral equations and the boundary integro-differential equations.     

A schauder decomposition and weakly singular integral equations in Lp

In this paper, a Schauder decomposition in L_p is used to obtain numerical solutions for the Fredholm integral equations of the second kind. Considerations are also given to weakly singular integral equations and two dimensional weakly singular integral equations.     

A boundary integral equation method for three-dimensional crack problems in elasticity

This paper presents a solution procedure for three-dimensional crack problems via first kind boundary integral equations on the crack surface. The Dirichlet (Neumann) problem is reduced to a system of integral equations for the jump of the traction (of the field) across the crack surface. The calculus of pseudodifferential operators is used to derive existence and regularity of the solutions of the integral equations. With the concept of the principal symbol and the Wiener-Hopf technique we derive the explicit behavior of the densities of the integral equations near the edge of the crack surface. Based on the detailed regularity results we show how to improve the boundary element Galerkin method for our integral equations. Quasi-optimal asymptotic estimates for the Galerkin error are given.     

The Far-Field Equations in Linear Elasticity for Disconnected Rigid Bodies and Cavities

In this paper the far-field equations in linear elasticity for scattering from disjoint rigid bodies and cavities are considered. The direct scattering problem is formulated in differential and integral form. The boundary integral equations are constructed using a combination of single- and double-layer potentials. Using a Fredholm type theory it is proved that these boundary integral equations are uniquely solvable. Assuming that the incident field is produced by a superposition of plane incident waves in all directions of propagation and polarization it is established that the scattered field is also expressed as the superposition of the corresponding scattered fields. A pair of integral equations of the first kind which hold independently of the boundary conditions are constructed for the far-field region. The properties of the Herglotz functions are used to derive solvability conditions for the far-field equations. It is also proved that the far-field operators, in terms of which we can express the far-field equations, are injective and have dense range. An analytical example for spheres illuminates the theoretical results.     

ON INFINITE DIMENSIONAL RICCATI INTEGRALEQUATIONS WITH ALL COEFFICIENTSINDEFINITE

1IntroductionXiccatiequationshavebeenplayinganimportantroleinmoderncontroltheoryduetotheircloserelationtolinearquadraticoptimalcontrol,optimalfiltering,Hoooptimalcontrolanddifferentialgameproblems.ThereareajreadyagreatdealofworkdevotedtothestudyofRiccatiequationsofbothfinitedimensional([11,etc.)andinfinitedimensiollal([2],etc.).Letusbrieflyreviewsomeknownresults.LetY,U,VbeHilbertspacest0相似文献   

An Investigation of a Pair of Integral Equations for the Biharmonic Problem

The paper contains an elementary investigation of the questionof uniqueness of a pair of integral equations connected withthe plane biharmonic problem. It is shown that for two particularexceptional geometries of the boundary curve the pair of integralequations does not have a unique solution. This defect can beremoved by adding two supplementary integral conditions whichthe solution of the integral equations must satisfy. As an illustrationthe integral equations are solved numerically with and withoutthese extra conditions.     

On the solution of a class of integral equations II

A class of integral equations is investigated, particular examples of which occur in the consideration of certain three- and four-part mixed boundary-value problems in applied mathematics. A constructive method is given for reformulating the integral equations as Fredholm integral equations of the second kind and three examples are examined in detail to illustrate the general methods developed in the paper.     

Comparison of two approaches to integration on separated topological spaces

A class of integral equations is investigated, particular examples of which occur in the consideration of certain three- and four-part mixed boundary-value problems in applied mathematics. A constructive method is given for reformulating the integral equations as Fredholm integral equations of the second kind and three examples are examined in detail to illustrate the general methods developed in the paper.     

动力学方程的积分型直接积分法

提出了一个求解动力学问题的新方法(DIM-IM).将动力学方程化成积分方程的形式,借助于该方程构造出了具有显式预测-校正的单步、自起动和四阶精度的积分型直接积分算法.理论分析和算例指出,这一方法较中心差分法、Houbolt法、Newmark法和Wilson-θ法都有较高的精度.本方法适用于强非线性,非保守系统.     

Integral Equations in Measure Spaces

In this paper, we consider a class of integral equations in measure spaces, and the corresponding integral inequalities. Special cases are Volterra type integral equations and Gronwall type integral inequalities. We give different necessary and su.cient, and only su.cient conditions which together with the Lipschitz condition imply the existence and the uniqueness of solutions of the considered integral equations. We study the successive approximations for the considered integral equations. We derive estimates for the solutions of the studied integral equations and integral inequalities. Submitted: June 20, 2000?Revised: July 10, 2001     

Numerical solutions of two-dimensional nonlinear integral analysis

In this paper, the approximate solutions for two different type of two-dimensional nonlinear integral equations: two-dimensional nonlinear Volterra-Fredholm integral equations and the nonlinear mixed Volterra-Fredholm integral equations are obtained using the Laguerre wavelet method. To do this, these two-dimensional nonlinear integral equations are transformed into a system of nonlinear algebraic equations in matrix form. By solving these systems, unknown coefficients are obtained. Also, some theorems are proved for convergence analysis.Some numerical examples are presented and results are compared with the analytical solution to demonstrate the validity and applicability of the proposed method.     

A COLLOCATION METHOD FOR THE CONDUCTIVITY PROBLEM WITH DISCONTINUOUS COEFFICIENT

In this paper, a new collocation BEM for the Robin boundary value problem of the conductivity equation ▽(γ▽u) = 0 is discussed, where the 7 is a piecewise constant function. By the integral representation formula of the solution of the conductivity equation on the boundary and interface, the boundary integral equations are obtained. We discuss the properties of these integral equations and propose a collocation method for solving these boundary integral equations. Both the theoretical analysis and the error analysis are presented and a numerical example is given.     

A remark on the single scattering preconditioner applied to boundary integral equations

This article deals with boundary integral equation preconditioning for the multiple scattering problem. The focus is put on the single scattering preconditioner, corresponding to the diagonal part of the integral operator, for which two results are proved. Indeed, after applying this geometric preconditioner, it appears that, firstly, every direct integral equations become identical to each other, and secondly, that the indirect integral equation of Brakhage–Werner becomes equal to the direct integral equations, up to a change of basis. These properties imply in particular that the convergence rate of a Krylov subspaces solver will be exactly the same for every preconditioned integral equations. To illustrate this, some numerical simulations are provided at the end of the paper.     

An embedding theorem for integral equations

In this paper, an embedding theorem is established for a system of nonlinear integral equations of the Volterra type. The main result is basic in the development of a maximum principle for an optimal control problem in which the state variables are determined as solutions to integral equations.     

SINGULAR INTEGRAL EQUATIONS ALONG AN OPEN ARC WITH SOLUTIONS HAVING SINGULARITIES OF HIGHER ORDER

In this paper, the difficulties on calculation in solving singular integral equations are overcome when the restriction of curve of integration to be a closed contour is cancelled. When the curve is an open arc and the solutions for singular integral equations possess singularities of higher order, the solution and the solvable condition for characteristic equations as well as the generalized Noether theorem for complete equations are given.     

Laplace transforms and American options

Laplace transform methods are used to study the valuation of American call and put options with constant dividend yield, and to derive integral equations giving the location of the optimal exercise boundary. In each case studied, the main result of this paper is a nonlinear Fredholm-type integral equation for the location of the free boundary. The equations differ depending on whether the dividend yield is less than or exceeds the risk-free rate. These integral equations contain a transform variable, so the solution of the equations would involve finding the free boundary that satisfies the equations for all values of this transform variable. Expressions are also given for the transform of the value of the option in terms of this free boundary.     

Rationalized Haar functions method for solving Fredholm and Volterra integral equations

This paper presents a computational technique for Fredholm integral equation of the second kind and Volterra integral equation of the second kind. The method is based upon Haar functions approximation. Properties of Rationalized Haar functions are first presented, the operational matrix of integration together with product operational matrix and Newton–Cotes nodes are utilized to reduce the computation of integral equations into some algebraic equations. The method is computationally attractive and applications are demonstrated through illustrative examples.