Fredholm Type Integral Equations Research Articles

Acoustic scattering by a circular semi-transparent conical surface

The scattering of a plane acoustic wave by a circular semi-transparent conical surface with impedance-type boundary conditions is studied. The analytic solution is constructed on the basis of the incomplete separation of variables and the reduction of the problem to a functional difference equation of the second order. Although the latter is equivalent to a Carleman boundary-value problem for analytic vectors, the solution is studied by means of the direct reduction method, that is, converting the functional difference equations to a Fredholm-type integral equation. Its unique solvability is then studied and the expression for the scattering amplitude of the spherical wave from the vertex is discussed. Some numerical results for axial incidence are also presented.

A homotopy perturbation algorithm to solve a system of Fredholm–Volterra type integral equations

In this paper, an application of He’s homotopy perturbation (HPM) method is applied to solve the system of Fredholm and Volterra type integral equations, the results revealing that the HPM is very effective and simple.

Extraction of Ocean Wave Spectra From Simulated Noisy Bistatic High-Frequency Radar Data

An algorithm is developed for the inversion of bistatic high-frequency (HF) radar sea echo to give the nondirectional wave spectrum. The bistatic HF radar second-order cross section of patch scattering, consisting of a combination of four Fredholm-type integral equations, contains a nonlinear product of ocean wave directional spectrum factors. The energy inside the first-order cross section is used to normalize this integrand. The unknown ocean wave spectrum is represented by a truncated Fourier series. The integral equation is then converted to a matrix equation and a singular value decomposition (SVD) method is invoked to pseudoinvert the kernel matrix. The new algorithm is verified with simulated radar Doppler spectrum for varying water depths, wind velocities, and radar operating frequencies. To make the simulation more realistic, zero-mean Gaussian noise from external sources is also taken into account

The Eshelby Tensors in a Finite Spherical Domain—Part I: Theoretical Formulations

This work is concerned with the precise characterization of the elastic fields due to a spherical inclusion embedded within a spherical representative volume element (RVE). The RVE is considered having finite size, with either a prescribed uniform displacement or a prescribed uniform traction boundary condition. Based on symmetry and group theoretic arguments, we identify that the Eshelby tensor for a spherical inclusion admits a unique decomposition, which we coin the “radial transversely isotropic tensor.” Based on this notion, a novel solution procedure is presented to solve the resulting Fredholm type integral equations. By using this technique, exact and closed form solutions have been obtained for the elastic disturbance fields. In the solution two new tensors appear, which are termed the Dirichlet–Eshelby tensor and the Neumann–Eshelby tensor. In contrast to the classical Eshelby tensor they both are position dependent and contain information about the boundary condition of the RVE as well as the volume fraction of the inclusion. The new finite Eshelby tensors have far-reaching consequences in applications such as nanotechnology, homogenization theory of composite materials, and defects mechanics.

Integral Equations of Fredholm Type with Rapidly Varying Kernels and Their Relationship to Dynamic Systems

The relationship between the eigenfunctions of a Fredholm-type integral equation with rapidly oscillating kernel and the dynamic mapping is analyzed. Differential operators commuting with the Fourier operator are constructed. These operators are closely related to nontrivial solutions of the unperturbed nonlinear functional equation related to the dynamic mapping. Bibliography: 6 titles.

Some families of Mathieu a-series and alternating Mathieu a-series

The main purpose of this paper is to present a number of potentially useful integral representations for the familiar Mathieu a -series as well as for its alternating version. These results are derived here from many different considerations and are shown to yield sharp bounding inequalities involving the Mathieu and alternating Mathieu a -series. Relationships of the Mathieu a -series with the Riemann Zeta function and the Dirichlet Eta function are also considered. Such special functions as the classical Bessel function J ν ( z ) and the confluent hypergeometric functions 0 F 1 and 1 F 2 are characterized by means of certain Fredholm type integral equations of the first kind, which are associated with some of these Mathieu type series. Several integrals containing Mathieu type series are also evaluated. Finally, some closely-related new questions and open problems are indicated with a view to motivating further investigations on the subject of this paper.

얼랑분포의 축차확률비검정과 관련된 적분 방정식의 해

본 연구에서는 얼랑(Erlang)분포의 규모모수에 대 한 축차확률비검정(SPRT)과 관련된 적분방정식의 정학한 해를 구하는 법을 살펴보기로 한다. 축차확률비검정에서 그 평균 표본 개수, 그리고 1종 오류 확률과 2종 오류 확률은 프레돔 형태의 적분 방정식으로 나타나게 된다. 이러한 적분 방정식은 보통 가우시안 쿼드러쳐(qudrature)를 이용하여 근사적으로 그 해를 구하는 것이 일반적이다. 얼랑분포의 경우 이러한 적분방정식의 해가 정확하게 구할 수 있음이 알려져 있다. 본 연구에서는 얼랑분포에서 그 해를 구하는 구체적 방법을 살펴보기로 한다. In this paper, we propose a method to evaluate the solutions of the renewal equations related to SPRT for Erlang distribution. In SPRT, the Average Sample Number(ASN) and type I or type II error probabilities are shown in Fredholm type integral equations. The integral equations are generally solved by the approximation method using Gaussian quadrature. For Erlang distribution, it has been known that the exact solutions of the equations exist. We propose the algorithm to solve the equations.

ITG-CHAMP01: a CHAMP gravity field model from short kinematic arcs over a one-year observation period

Global gravity field models have been determined based on kinematic orbits covering an observation period of one year beginning from March 2002. Three different models have been derived up to a maximum degree of n=90 of a spherical harmonic expansion of the gravitational potential. One version, ITG-CHAMP01E, has been regularized beginning from degree n=40 upwards, based on the potential coefficients of the gravity field model EGM96. A second model, ITG-CHAMP01K, has been determined based on Kaula’s rule of thumb, also beginning from degree n=40. A third version, ITG-CHAMP01S, has been determined without any regularization. The physical model of the gravity field recovery technique is based on Newton’s equation of motion, formulated as a boundary value problem in the form of a Fredholm-type integral equation. The observation equations are formulated in the space domain by dividing the one-year orbit into short sections of approximately 30-minute arcs. For every short arc, a variance factor has been determined by an iterative computation procedure. The three gravity field models have been validated based on various criteria, and demonstrate the quality of not only the gravity field recovery technique but also the kinematically determined orbits.

Unified Solutions of Integral Equations of SPRT for Exponential Random Variables

The main characteristics of the sequential probability ratio test (SPRT) are described by acceptance probability (AP) and average sample number (ASN). The properties of the cumulative sum control procedure, CUSUM in short, which is a variant of the SPRT, are mainly determined by the average run length (ARL). The characteristics are obtained by solving Fredholm type integral equations. For the sake of mathematical simplicity, the models for the exponential random variables and the sum of the exponential random variables have been considered to test the effects of the design parameters of the SPRT and CUSUM. The solution of the integral equation for each case obtained by various methods in previous studies. In this article, we show that all the integral equations related to the characteristics of AP, ASN and ARL for the models of the exponential random variable and the sum of exponential random variables are defined in a framework, and their solutions are obtained precisely in a unified method. The suggested solutions for exponential distribution do not require any other numerical steps to obtain the unknown constants used in the solutions.

Ruin Theory in a Discrete Time Risk Model with Interest Income

ABSTRACTIn this paper we consider a discrete time insurance risk model with interest income. Using the recursive calculation method of De Vylder & Goovaerts (1988), recursive equations for the finite time ruin probabilities and the distribution of the time of ruin are derived. Fredholm type integral equations for the ultimate ruin probability, the distribution of the severity of ruin, the joint distribution of surplus before and after ruin, and the probability of absolute ruin are obtained. Numerical results are included.

Radiative transfer in finite cylindrical media using transformed integral equations

A modified direct integration method is presented to solve three-dimensional radiative transfer in emitting, absorbing and linear-anisotropic scattering finite cylindrical media. This scheme effectively avoids an integral singularity in the coupled Fredholm type integral equations of radiative transfer. The scheme leads to faster and more accurate results, which are needed in combined mode and non-gray problems. The calculated incident radiation and heat fluxes agree well with published results by discrete ordinates method. Using the transformed integral equations, the effects of boundary emission and reflection can also be easily handled.

A New Method of Solving the Basic Plane Boundary Value Problems of Statics of the Elastic Mixture Theory

The basic plane boundary value problems of statics of the elastic mixture theory are considered when on the boundary are given: a displacement vector (the first problem), a stress vector (the second problem); differences of partial displacements and the sum of stress vector components (the third problem). A simple method of deriving Fredholm type integral equations of second order for these problems is given. The properties of the new operators are established. Using these operators and generalized Green formulas we investigate the above-mentioned integral equations and prove the existence and uniqueness of a solution of all the boundary value problems in a finite and an infinite domain.

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.

Results on Newton methods. Part 1: A unified approach for constructing perturbed Newton-like methods in Banach space and their applications

Many applied problems can be formulated to fit the model of a nonlinear equation. Undoubtedly, Newton methods are the most popular methods for solving such equations. Here we report the results of our recent investigations in a series of two papers. We unified Newton methods and provided convergence results to solve nonlinear equations. We used Newton–Kantorovich type hypotheses and the majorant method. Our results depend on the existence of a Lipschitz function with which we replace the Lipschitz constant usually appearing in the hypotheses of these investigations. We find our methods useful for many reasons. From the computational point of view we have allowed our iterations to be corrected at every step and still converge to a solution of the equation at hand. In most earlier results the iterates are assumed to be computed exactly to achieve convergence. Moreover we showed that under weaker hypotheses our iterations converge faster than others already in the literature. Finally we used our results to solve Uryson and Fredholm-type integral equations appearing in radiative transfer.

Axi-symmetric contact problem with frictional heating for thermally nonlinear sliders

The axi-symmetric contact problem for a sliding elastic sphere on a rigid foundation is considered. It is assumed that the work done against shearing tractions is the source of heat generation and all the heat is absorbed in one moving sphere. The effect of the temperature-dependent properties of the sphere material is investigated. The problem is reduced to the solution of nonlinear Fredholm type integral equations. The approximate solution of these equations is obtained. For a sphere of carbon graphite material it is established that the influence of the nonlinear thermal properties on the contact area, the contact pressure and temperature is essential.

Regular Gaussian Random Operatorsand Strook--Varadhan Theorem for Symmetric StochasticFredholm-Type Equations

Regular Gaussian functionals are introduced and studied; the results obtained are applied to the investigation of properties of the distribution of solutions of equations with Gaussian random operators. In particular, an analogue of the Strook--Varadhan theorem for linear stochastic Fredholm-type integral equations is proved.

Thermoelastic contact problems with frictional heating and convective cooling

The contact problems with frictional heat generation are usually investigated with the idealised thermal boundary condition of perfect insulation in the separation region. This paper formulates axisymmetric and plane static thermoelasticity problems for a half-space when one boundary condition corresponds to convective heat transfer outside the contact area. The problems are reduced to the solution of Fredholm type integral equations. In each case the approximate solutions of those equations are obtained. It is established, when the influence of the heat exchange with the surrounding medium on the contact area, the contact pressure and temperature is essential and, also, when this effect is negligible.

Results on Newton methods: Part I. A unified approach for constructing perturbed Newton-like methods in Banach space and their applications

Many applied problems can be formulated to fit the model of a nonlinear equation. Newton methods are undoubtedly the most popular methods for solving such equations. Here we report the results of our recent investigations in a series of two papers. We unified Newton methods and provided convergence results to solve nonlinear equations. We used Newton-Kantorovich type hypotheses and the majorant method. Our results depend on the existence of a Lipschitz function with which we replace the Lipschitz constant usually appearing in the hypotheses of these investigations. We find our methods useful for many reasons. From the computational point of view we have allowed our iterations to be corrected at every step and still converge to a solution of the equation at hand. In most earlier results the iterates are assumed to be computed exactly to achieve convergence. Moreover we showed that under weaker hypotheses our iterations converge faster than others already in the literature. Finally we used our results to solve Uryson and Fredholm-type integral equations appearing in radiative transfer.

Solution of the basic equation of stellar statistics

The basic equation of stellar statistics connects the probability density function of a measurable quantity with the probability density of two variables, which cannot be observed directly, by the Bayes theorem of conditional probabilities. The resulting relation is a Fredholm-type integral equation of the first kind. If the two background variables are statistically independent we recover the convolution equation. The analytical solution based on the Fourier transformation is very sensitive to high-frequency noise. Eddington's solution attempts to find the unknown function in form of a series Sigma gamma jh(i)(z). Malmquist's method computes the conditional probability of the unknown variable assuming that the observed variable is given. The statistical aspect of the problem is expressed if one uses Lucy's algorithm which is a particular form of the more general EM algorithm. Dolan's matrix method solves numerically the matrix equation which approximates the integral equation. Methods are superior which retain the true statistical nature of the problem.

Application of harmonic perturbations to the calculation of periodically corrugated waveguides

Harmonic perturbation of the profile of an infinite axisymmetric periodically-corrugated waveguide is used for the calculation of its electrodynamic characteristics. The analysis is based on the method of Fredholm-type integral equations of the second kind. The problem is solved analytically for the perturbation of a cylindrical waveguide. These analytical results serve as the basis for the solution of the inverse problem of recovery of the waveguide profile from the given dispersion curve.