Solution Of Two-dimensional Integral Equations Research Articles

On the numerical solution of two-dimensional integral equations using a meshless local discrete Galerkin scheme with error analysis

This investigation presents a simple and effective method for numerically solving two-dimensional Fredholm integral equations of the second kind on non-rectangular domains. The general framework of the current scheme is based on the Galerkin method together with moving least squares (MLS) technique constructed on scattered points in which all integrals are approximated by a suitable quadrature formula. The MLS approach estimates an unknown function utilizing a locally weighted least square polynomial fitting. The method does not require any cell structures, so it is meshless and consequently independent of the geometry of the domain. The algorithm of the presented scheme is attractive and easy to implement on computers. Furthermore, the error bound and the convergence rate of the presented method are obtained. Illustrative examples clearly show the reliability and the efficiency of the new method and confirm the theoretical error estimates provided in the current paper.

The dynamics of the motion of a flat punch on the boundary of an elastic half-plane

The dynamic contact problem of the motion of a flat punch on the boundary of an elastic half-plane is considered. During motion, the punch deforms the elastic half-plane, penetrating it in such a manner that its base remains parallel to the boundary of the half-plane at each instant of time. In movable coordinates connected to the moving punch, the contact problem reduces to solving a two-dimensional integral equation, whose two-dimensional kernel depends on the difference between the arguments for each of the variables. An approximate solution of the integral equation of the problem is constructed in the form of a Neumann series, whose zeroth term is represented in the form of the superposition of the solutions of two-dimensional integral equations on the coordinate semiaxis minus the solution of the integral equation on the entire axis. This approach provides a way to construct the solution of the two-dimensional integral equation of the problem in four velocity ranges of motion of the punch, which cover the entire spectrum of its velocities, as well as to perform a detailed analysis of the special features of the contact stresses and vertical displacements of the free surface on the boundary of the contract area. An approximate method for solving the integral equation, which is based on a special approximation of the integrand of the kernel of the integral equation in the complex plane, is proposed for obtaining effective solutions of the problem that do not contain singular quadratures.

A new differential transformation approach for two-dimensional Volterra integral equations

The differential transformation method provides an iterative procedure to obtain the spectrum of analytic solutions. In this paper we extend the differential transformation approach for the solution of two-dimensional integral equations. We give some basic properties and a new differential transformation-type method for the solution of linear and nonlinear two-dimensional Volterra integral equations. By extension of the operations, a two-dimensional integral equation in the domain of interest can be transformed into an algebraic equation in the domain K, H. We show that, after transforming the original equation into an algebraic equation, the coefficients W(k, h) for k, h=1, 2, … are determined and then, by substituting the values of W(k, h) in the transformed equation, the closed form solution of the original equation can be obtained. The reliability and efficiency of the proposed scheme are demonstrated by numerical experiments.

Integral-equation solution using coherent optical feedback

Confocal feedback systems can perform analog optical solutions of two-dimensional integral equations. Methods for solution of Fredholm equations by space-variant feedback and of Volterra equations using multiple feedback have been found and experimental results obtained. Two-dimensional problems of this type are difficult and time consuming to solve digitally, but the optical solution is obtained at high speed.

Numerical Solution of Two-Dimensional Integral Equations Using Linear Elements

A general procedure is presented for numerically solving linear Fredholm integral equations of the first kind in two integration variables. The approximate solution is expressed as piecewise bilinear or linear functions on rectangles or triangles, respectively. The method involves collocation followed by the solution of an appropriately scaled stabilized matrix least squares problem. The stabilizing procedure consists of appending additional equations to the system that resulted from collocation. These additional equations arise from considering an appropriately weighted seminorm on the unknown solution. Three numerical examples are given.

Numerical algorithm for the solution of linear two-dimensional integral equations of the first kind

A numerical algorithm is proposed for the solution of two-dimensional integral equations of the first kind, to which some inverse problems of heat conduction are reduced.

Elementary Solutions of the Reduced Three-Dimensional Transport Equation

By treating one of the space dimensions exactly and approximating the other two by the exp (−iB·r) assumption, which is suggested by asymptotic transport theory, it is possible to reduce the three-dimensional transport equation to an equation that is of one-dimensional form and that still contains details of the complete three-dimensional angular distribution. In this paper we develop the method of elementary solutions for the reduced transport equation in the case of time-independent, monoenergetic neutron transport in homogeneous media with isotropic scattering. The spectrum of the transport operator consists of a pair of discrete points if B2 is sufficiently small and a continuum which occupies a two-dimensional region in the complex spectral plane. The eigenfunctions possess full-range and half-range orthogonality and completeness properties, which are proved via the solution of two-dimensional integral equations using the theory of boundary-value problems for generalized analytic functions. As applications we solve the Green's function for an infinite homogeneous prism and the albedo operator for a semi-infinite homogeneous prism. Also discussed are possible generalizations of the method to more complicated forms of the reduced transport equation.