Numerical Solution Of Integral Equations Research Articles

Light element depth distribution by the intensity ratio of incoherent and coherent scattering

A method is proposed for analyzing the distribution of elements with a small atomic number Z < 9 over the depth of the substrate. The method is based on measuring the ratio of incoherent and coherent scattering peaks intensities at different exit angles of X‐rays from the sample surface. Varying the beam exit angle at a constant scattering angle makes it possible to vary the depth of radiation penetration into the sample by almost two orders of magnitude and, correspondingly, to vary the information averaging zone. Mathematically, the method reduces to the numerical solution of integral equations of the first kind, in which the impurity distribution profile is given by the model function. Experimental measurements were performed on model samples of beryllium (Z = 4) and aluminum (Z = 13) foils and real coatings of boron (Z = 5) and titanium nitride. The adequacy of the received solutions to the certified characteristics is shown.

A Fourier Extension Based Numerical Integration Scheme for Fast and High-Order Approximation of Convolutions with Weakly Singular Kernels

Computationally efficient numerical methods for high-order approximations of convolution integrals involving weakly singular kernels find many practical applications including those in the development of fast quadrature methods for numerical solution of integral equations. Most fast techniques in this direction utilize uniform grid discretizations of the integral that facilitate the use of FFT for $O(n\log n)$ computations on a grid of size $n$. In general, however, the resulting error converges slowly with increasing $n$ when the integrand does not have a smooth periodic extension. Such extensions, in fact, are often discontinuous, and, therefore, their approximations by truncated Fourier series suffer from Gibbs oscillations. In this paper, we present and analyze an $O(n\log n)$ scheme, based on a Fourier extension approach for removing such unwanted oscillations, that not only converges with high order but is also relatively simple to implement. We include a theoretical error analysis as well as a wide variety of numerical experiments to demonstrate its efficacy.

A Novel Approach to Estimate Solution of Volterra Integral Equations

Approximating numerical solution of integral equations is considered to be very important as such equations have number of applications in various fields. In this paper we introduce novel approach to estimate numerical solutions of Volterra integral equations. In the proposed technique, Chebyshev polynomial is employed to approximate solution for an unknown function in the Volterra integral equation. It is observed that the proposed technique is highly suitable for such problems and have very encouraging results. We compare accuracy and efficiency of the method with existing techniques.

Local multiquadric scheme for solving two‐dimensional weakly singular Hammerstein integral equations

AbstractThe main purpose of this article is to present a numerical method for solving two‐dimensional Fredholm‐Hammerstein integral equations of the second kind with weakly singular kernels. The scheme utilizes locally supported (inverse) multiquadric functions constructed on scattered points as a basis in the discrete collocation method. The local (inverse) multiquadrics estimate a function in any dimensions via a small set of data instead of all points in the solution domain. The proposed method uses a special accurate quadrature formula based on the nonuniform Gauss‐Legendre integration rule for approximating singular integrals appeared in the scheme. In comparison with the globally supported (inverse) multiquadric for the numerical solution of integral equations, the proposed method is stable and uses much less computer memory. Moreover, the algorithm of the presented approach is attractive and easy to implement on computers. Since the scheme does not require any mesh generations on the domain, it can be identified as a meshless method. The error analysis of the method is provided. The convergence accuracy of the new technique is examined over several two‐dimensional Hammerstein integral equations and obtained results confirm the theoretical error estimates.

A new approach for numerical solution of two-dimensional nonlinear Fredholm integral equations in the most general kind of kernel, based on Bernstein polynomials and its convergence analysis

Regarding the efficient previous method for approximation of one-dimensional functions integrals through Bernstein polynomials in Amirfakhrian (2011), this paper presents a development of this method for approximation of two-dimensional functions integrals for the first time. Then, by combining this approximation with Bernstein collocation method for numerical solution of two-dimensional nonlinear Fredholm integral equations, the kernels double integrals of integral equations will be approximated. Combination of two-dimensional functions numerical integration method with numerical solution of integral equations method (in both methods, Bernstein polynomials were used) will result in increase of convergence speed and accuracy of the method. The convergence analysis of the method is completely presented. Finally, numerical examples are presented to illustrate the efficiency and superiority of our method in comparing it with other methods.

Analysis of cracked layers under transient temperature field

ABSTRACTThe uncoupled thermoelasticity problem of a homogenous isotropic layer containing a line of temperature discontinuity and a Volterra edge dislocation is solved. The analysis is performed for discontinuity lines with two different directions of parallel and perpendicular to the layer boundary. The solutions for heat flux and stress fields exhibit the familiar Cauchy type singularity. These results are utilized to derive integral equations for the intensity of heat flux and densities of dislocation in a layer weakened by interacting cracks which are parallel and/or perpendicular to the layer boundary. The numerical solution of integral equations is performed and stress intensity factors for cracks are obtained.

Meshless methods for one-dimensional oscillatory Fredholm integral equations

In this paper, efficient and simple algorithms based on Levin’s quadrature theory and our earlier work involving local radial basis function (RBF) and Chebyshev differentiation matrices, are adopted for numerical solution of one-dimensional highly oscillatory Fredholm integral equations. This work is focused on the comparative performance of local RBF meshless and pseudospectral procedures. We have tested the proposed methods on phase functions with and without stationary phase point(s), both on uniform and Chebyshev grid points. The proposed procedures are shown accurate and efficient, and therefore provide a reliable platform for the numerical solution of integral equations. From the numerical results, we draw some conclusions about accuracy, efficiency and robustness of the proposed approaches.

TESTING OF NUMERICAL SOLUTION OF THE PROBLEM OF DETERMINING SOURCES OF MAGNETOSTATIC FIELD IN MAGNETIZED MEDIUM

Purpose. Testing of numerical solution algorithm for integral equation for calculation of plane meridian magnetostatic field source distribution at interfaces of piecewise homogeneous magnetized medium by means of electrostatic analogy. Methodology. The piecewise homogeneous medium consists of three regions with different magnetic permeabilities: the shell of arbitrary meridian section, external unlimited medium outside the shell, and the medium inside the shell. For testing external homogeneous magnetic field effect on spherical shell is considered. The analytical solution of this problem on the basis of electrostatic analogy from the solution of the problem uniform electrostatic field effect on dielectric shell is obtained. We have compared results of numerical solution of integral equation with the data obtained by means of analytical solution at the variation of magnetic permeabilities of regions of medium. Results. Integral equation and the algorithm of its numerical solution for calculation of source field distribution at the boundaries of piecewise homogeneous medium is validated. Testing of integral equations correctness for calculation of fictitious magnetic charges distribution on axisymmetric boundaries of piecewise homogeneous magnetized medium and algorithms of their numerical solutions can be carried out by means of analytical solutions of problems of homogeneous electrostatic field effect analysis on piecewise homogeneous dielectric medium with central symmetry of boundaries – single-layer and multilayer spherical shells. In the case of spherical shell in wide range of values of the parameter λk, including close to ± 1, numerical solution of integral equation is stable, and relative error in calculating of fictitious magnetic charges surface density and magnetic field intensity inside the shell is from tenths of a percent up to several percent except for the cases of very small values of these quantities. Originality. The use analytical solutions for problems of calculation of external electrostatic field effect on piecewise homogeneous dielectric bodies for testing integral equations of magnetostatics and algorithms for their numerical solutions. Practical value. The described method of testing integral equations of magnetostatics and their numerical solutions can be used for calculation of magnetic fields of spacecraft control system electromagnets.

Erratum to: Quasi-interpolation based on the ZP-element for the numerical solution of integral equations on surfaces in $$\mathbb {R}^3$$ R 3

The aim of this paper is to present spline methods for the numerical solution of integral equations on surfaces of $$\mathbb {R}^3$$ , by using optimal superconvergent quasi-interpolants defined on type-2 triangulations and based on the Zwart–Powell quadratic box spline. In particular we propose a modified version of the classical collocation method and two spline collocation methods with high order of convergence. We also deal with the problem of approximating the surface. Finally, we study the approximation error of the above methods together with their iterated versions and we provide some numerical tests.

A note on improvement by iteration of approximate solutions of operator equations

Let T be a bounded linear operator on a Banach space X = C[0, 1]. Consider the operator equation $$\begin{aligned} u-Tu=f,\;u,f\in X. \end{aligned}$$ Collocation methods are a class of important methods to solve the above equation approximately We consider a modified projection method and an iterated modified projection for the solution of the above operator equation. Let \(\pi _n\) be an interpolatory projection onto the space \(X_n\) of piecewise polynomials of degree \(\le r-1.\) If the collocation points are the Gauss points, then the iterated collocation and iterated modified projection solution is an improvement on the collocation and the modified projection solution. Now let \(\pi _n: X\rightarrow X_n\) be the interpolatory projection with collocation points as the partition points of a uniform partition of [0, 1]. Then iteration fails to improve the order of convergence. But when the approximating space \(X_n\) is the space of continuous functions that are piecewise polynomials of even degree with respect to a uniform partition, then Atkinson (The numerical solutions of integral equations of the second kind, 1997) has shown that iterated collocation solution is an improvement on the collocation solution whereas we show that the iteration improves the order of convergence in the case of modified projection method.

Conductive transport through a mixed-matrix membrane

Conductive transport through an infinite homogeneous medium across a layer of finite thickness, a planar array of infinite cylinders, or a planar array of three-dimensional particles with arbitrary conductivity is considered as a model of mixed-matrix membrane separation. The boundary distribution of the transported scalar field on the interior side of the layer, cylinders, or particles is proportional to that on the exterior side according to a linear sorption/desorption kinetics law, while the conductive flux is continuous across the interface. In the case of cylinders and particles, the solution of Laplace’s equation for the transported field is represented by an interfacial distribution of point sources expressed in terms of the periodic Green’s function of Laplace’s equation in two dimensions or the doubly periodic Green’s function of Laplace’s equation in three dimensions. Analytical solutions for small circular cylinders and small spherical particles are derived based on the integral representation, and numerical solution of integral equations arising from the interfacial conditions are computed by boundary-element methods. The results document the displacement of the linear profile of the transported field far from the interfacial layer or array with respect to that prevailing in the absence of membrane. Expressions for the effective diffusivity of the mixed-matrix membrane are derived.

Investigation of the Numerical Solution of Integral Equation with Kernels Involving Logarithmic Functions

The main aim of this paper is to investigate the numerical solution of first kind integral equation of logarithmic kernel when using spectral method. Our approach consists of limiting the boundary to the unit interval and specify a logarithmic kernel. The behavior of the solution on the unit interval was analyzed and the advantages and disadvantages of this approach was shown.

Collocation Technique for Numerical Solution of Integral Equations with Certain Orthogonal Basis Function in Interval [0, 1

This paper is concerned with the construction of a class of polynomial orthogonal with respect to the weight function w(x)=1-x2 over the interval [0,1]. The zeros of these polynomials were employed as points of collocation for the orthogonal collocation technique in the solution of integral equations. The method is illustrated with some numerical examples and the results obtained show that the method is effective.

Quasi-interpolation based on the ZP-element for the numerical solution of integral equations on surfaces in $$\mathbb {R}^3$$ R 3

Quasi-interpolation based on the ZP-element for the numerical solution of integral equations on surfaces in $$\mathbb {R}^3$$ R 3

Numerical Solution of Integral Equations with Fractional and Partial Integrals and Variable Integration Limits

The method of mechanical quadratures is applied to linear Volterra integral equations with partial integrals among which there is an integral with an unbounded kernel. We construct numerical algorithms based on replacing integrals with quadrature formulas and prove the convergence.

EXTRACTING SURFACE MACRO BASIS FUNCTIONS FROM LOW-RANK SCATTERING OPERATORS WITH THE ACA ALGORITHM

The Adaptive Cross Approximation (ACA) algorithm has been used to compress the rank- deficient sub-blocks of the matrices that arise in the numerical solution of integral equations (IEs) with the Method of Moments. In the context of the linear embedding via Green's operator (LEGO) method — a domain decomposition technique based on IEs — an electromagnetic problem is modelled by combining bricks in turn described by scattering operators which, in many situations, are singular. As a result, macro basis functions defined on the boundary of a brick can be generated by applying the ACA to a scattering operator. Said functions allow compressing the weak form of the LEGO functional equations which then use up less computer memory and are faster to invert.

Numerical Solution of Integral Equations by using Discrete GHM Multiwavelet

Numerical Solution of Integral Equations by using Discrete GHM Multiwavelet

On the Separation of Singularities in the Numerical Solution of Integral Equations of the Potential Theory

We propose new procedures for the separation of singularities in the kernel and in the potential density for weakly singular Fredholm integral equations for the simple-layer potential in the case where the boundary surface has edges, ribs, and corner points. These procedures are based on the projection methods for the solution of these equations and finite-element approximations of the required potential density.

The Estimation of the Error at Richardson’s Extrapolation and the Numerical Solution of Integral Equations of the Second Kind

The mode of definition of the error at polynomial Richardson’s extrapolation is described. Along with the table of extrapolations the new magnitudes reflecting expediency and efficiency of extrapolation are entered. On concrete examples it is shown that application of Richardson’s extrapolation to a solution of integral equations has appeared rather effective and gives a solution with a high exactitude. Application of formulas of interpolation leads to a solution in the analytical aspect.

Numerical solution of integral equations for a scalar diffraction problem

Numerical solution of integral equations for a scalar diffraction problem