A Method for Solving Fredholm Integral Equations of the First Kind Based on Chebyshev Wavelets

In this paper, we suggest a method for solving Fredholm integral equation of the first kind based on wavelet basis. The continuous Legendre and Chebyshev wavelets of the first, second, third and fourth kind on [0,1] are used and are utilized as a basis in Galerkin method to approximate the solution of integral equations. Then, in some examples the mentioned wavelets are compared with each other.

EXTRAPOLATION FOR COLLOCATION METHOD OF THE FIRST KIND VOLTERRA INTEGRAL EQUATIONS

1. Introduction It is known that the following Cauchy problem for a parabolic partial differential equation (where the values at the right boundary, u.(1, t)=v(t) are unknown and sought for) is ill-posed: the solution (v) does not depend continuously on the data (g). In order to treat the ill-posedness and develop the numerical method, one reformulates the problem as a Volterra integral equation of the first kind wish a convolution type kernel (see Sneddon [1], Carslaw and Jaeger [2])

ON THE REGULARIZATION METHOD OF THE FIRST KIND OFFREDHOLM INTEGRAL EQUATION WITH A COMPLEX KERNEL AND ITS APPLICATION

The regularized integrodifferential equation for the first kind of Fredholm, integral equation with a complex kernel is derived by generalizing the Tikhonov regularization method and the convergence of approximate regularized solutions is discussed. As an application of the method, an inverse problem in the two-dimensional wave-making problem of a flat plate is solved numerically, and a practical approach of choosing optimal regularization parameter is given.

The Collocation Method and the Splitting Extrapolation for the First Kind of Boundary Integral Equations on Polygonal Regions

In this paper,the collocation methods are used to solve the boundary integral equations of the first kind on the polygon.By means of Sidi’s periodic transformation and domain decomposition,the errors are proved to possess the multi-parameter asymptotic expansion at the interior point with the powers h^(3)/_(i)(i=1,...,d),which means that the approximations of higher accuracy and a posteriori estimation of the errors can be obtained by splitting extrapolations.Numerical experiments are carried out to show that the methods are very efficient.

DIRECT IMPLEMENTATION OF TIKHONOV REGULARIZATION FOR THE FIRST KIND INTEGRAL EQUATION

A common way to handle the Tikhonov regularization method for the first kind Fredholm integral equations,is first to discretize and then to work with the final linear system.This unavoidably inflicts discretization errors which may lead to disastrous results,especially when a quadrature rule is used.We propose to regularize directly the integral equation resulting in a continuous Tikhonov problem.The Tikhonov problem is reduced to a simple least squares problem by applying the Golub-Kahan bidiagonalization(GKB)directly to the integral operator.The regularization parameter and the iteration index are determined by the discrepancy principle approach.Moreover,we study the discrete version of the proposed method resulted from numerical evaluating the needed integrals.Focusing on the nodal values of the solution results in a weighted version of GKB-Tikhonov method for linear systems arisen from the Nystr¨om discretization.Finally,we use numerical experiments on a few test problems to illustrate the performance of our algorithms.

Inversion and III-Posed Problem Solutions in Atmospheric Remote Sensing

With the swift advances in earth observation,satellite remote sensing and application of atmospheric radiation theory have been developed in the past decades,atmospheric sensing inversion with its algorithms is getting more and more importance.It is known that since a remote sensing equation falls into an integral equation of the first kind,thus leading to the fact that it is ill-posed and particularly the solution is unsteady,tremendous difficulties arise from the retrieval.This paper will present a simple review on the inversion techniques with some necessary remarks,before introducing the successful efforts with respect to such equations and the encouraging solutions achieved in recent decades by researchers of the world.

SPLITTING EXTRAPOLATIONS FOR SOLVING BOUNDARY INTEGRAL EQUATIONS OF LINEAR ELASTICITY DIRICHLET PROBLEMS ON POLYGONS BY MECHANICAL QUADRATURE METHODS

Taking hm as the mesh width of a curved edge Гm （m = 1, ..., d ） of polygons and using quadrature rules for weakly singular integrals, this paper presents mechanical quadrature methods for solving BIES of the first kind of plane elasticity Dirichlet problems on curved polygons, which possess high accuracy O（h0^3） and low computing complexities. Since multivariate asymptotic expansions of approximate errors with power hi^3 （i = 1, 2, ..., d） are shown, by means of the splitting extrapolations high precision approximations and a posteriori estimate are obtained.

On Integral Equation and Least Squares Methods for Scattering by Diffraction Gratings

In this paper we consider the scattering of a plane acoustic or electromagnetic wave by a one-dimensional,periodic rough surface.We restrict the discussion to the case when the boundary is sound soft in the acoustic case,perfectly reflecting with TE polarization in the EM case,so that the total field vanishes on the boundary.We propose a uniquely solvable first kind integral equation formulation of the problem,which amounts to a requirement that the normal derivative of the Green’s representation formula for the total field vanish on a horizontal line below the scattering surface.We then discuss the numerical solution by Galerkin’s method of this(ill-posed)integral equation.We point out that,with two particular choices of the trial and test spaces,we recover the so-called SC(spectral-coordinate)and SS(spectral-spectral)numerical schemes of DeSanto et al.,Waves Random Media,8,315-414,1998.We next propose a new Galerkin scheme,a modification of the SS method that we term the SSmethod,which is an instance of the well-known dual least squares Galerkin method.We show that the SSmethod is always well-defined and is optimally convergent as the size of the approximation space increases.Moreover,we make a connection with the classical least squares method,in which the coefficients in the Rayleigh expansion of the solution are determined by enforcing the boundary condition in a least squares sense,pointing out that the linear system to be solved in the SSmethod is identical to that in the least squares method.Using this connection we show that(reflecting the ill-posed nature of the integral equation solved)the condition number of the linear system in the SSand least squares methods approaches infinity as the approximation space increases in size.We also provide theoretical error bounds on the condition number and on the errors induced in the numerical solution computed as a result of ill-conditioning.Numerical results confirm the convergence of the SSmethod and illustrate the ill-conditioning that arises.