Fredholm Type Research Articles

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.

Numerical piecewise approximate solution of Fredholm integro-differential equations by the Tau method

A general form of numerical piecewise approximate solution of linear integro-differential equations of Fredholm type is discussed. It is formulated for using the operational Tau method to convert the differential part of a given integro-differential equation, or IDE for short, to its matrix representation. This formulation of the Tau method can be useful for such problems over long intervals and also can be used as a good and simple alternative algorithm for other piecewise approximations such as splines or collocation. A Tau error estimator is also adapted for piecewise application of the Tau method. Some numerical examples are considered to demonstrate the implementation and general effect of application of this (segmented) piecewise Chebyshev Tau method.

An approximate solution for solving linear system of integral equation with application on "Stiff" problems

An approximate solution of the liner system of ntegral cquations fot both fredholm(SFIEs)and Volterra(SIES)types has been derived using taylor series expansion.The solusion is essentailly

An approximate solution for solving linear system of integral equation with application on "Stiff" problems

An approximate solution of the liner system of ntegral cquations fot both fredholm(SFIEs)and Volterra(SIES)types has been derived using taylor series expansion.The solusion is essentailly

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.

A Newton-Like Iterative Process for the Numerical Solution of Fredholm Nonlinear Integral Equations

In this paper, we give a semi-local convergence result for an iterative process of Newton-Kantorovichtype to solve nonlinear integral equations of Fredholm type and second kind. We also illustrate with several examples the technique for constructing a functional sequence that approaches solutions.

Buckling and postcritical behaviour of the elastic infinite plate strip resting on linear elastic foundation

In this paper the von Kármán model for thin, elastic, infinite plate strip resting on a linear elastic foundation of Winkler type is studied. The infinite plate strip is simply-supported and subjected to evenly distributed compressive loads. The critical values of bifurcation parameters and buckling modes for given frequency of longitudinal waves are found on the basis of investigation of linearized problem. The mathematical nonlinear model is reduced to operator equation with Fredholm type operator of index 0 depending on parameters defined in corresponding Hölder spaces. The Lyapunov–Schmidt reduction and the Crandall–Rabinowitz bifurcation theorem (gradient case) are used to examine the postcritical behaviour of the plate. It is proved that there exists maximal frequency of longitudinal waves depending on the compressive load and the stiffness modulus of foundation.

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.

Volterra integral equations solved in Fredholm form using Walsh functions

Recently Walsh function methods have been developed for the numerical solution of several classes of problems, mainly linear and nonlinear integral equations of both Volterra and Fredholm types. In addition, modifications of the basic approach have led to the solution of functional differential equations, variational problems and parameter estimation problems. Linear Volterra integral equations are re-written as Fredholm integral equations with appropriately modified kernels. In this Fredholm equation form, the Walsh function solution method is more efficient than directly solving the Volterra equation. Walsh function methods are spectral methods but they have a natural grid interpretation. Multigrid methods and a variation on the use of Richardson extrapolation are used on six well known Volterra test problems, re-written in Fredholm form, to illustrate that these methods provide effective and efficient solution methods.

Splitting of resonant and scattering frequencies under shape deformation

It is well known that the main difficulty in solving eigenvalue problems under shape deformation relates to the continuation of multiple eigenvalues of the unperturbed configuration. These eigenvalues may evolve, under shape deformation, as separated, distinct eigenvalues, and the splitting may only become apparent at high orders in their Taylor expansion. In this paper, we address the splitting problem in the evaluation of resonant and scattering frequencies of the two-dimensional Laplacian operator under boundary variations of the domain. By using surface potentials we show that the eigenvalues are the characteristic values of meromorphic operator-valued functions that are of Fredholm type with index 0. We then proceed from the generalized Rouché's theorem to investigate the splitting problem.

Asymptotic-preserving & well-balanced schemes for radiative transfer and the Rosseland approximation

We are concerned with efficient numerical simulation of the radiative transfer equations. To this end, we follow the Well-Balanced approach's canvas and reformulate the relaxation term as a nonconservative product regularized by steady-state curves while keeping the velocity variable continuous. These steady-state equations are of Fredholm type. The resulting upwind schemes are proved to be stable under a reasonable parabolic CFL condition of the type Δt≤O(Δx2) among other desirable properties. Some numerical results demonstrate the realizability and the efficiency of this process.

Root Vectors for Geometrically Simple Multiparameter Eigenvalues

A class of two-parameter eigenvalue problems involving generally non self-adjoint and unbounded operators is studied. A basis for the root subspace at a geometrically simple eigenvalue of Fredholm type is computed in terms of the underlying two-parameter system. Comparison with Faierman's work on two-parameter boundary value problems of Sturm-Liouville type is given as an application.

Root vectors for geometrically simple two-parameter eigenvalues

A class of two-parameter eigenvalue problems involving generally non self-adjoint and unbounded operators is studied. A basis for the root subspace at a geometrically simple eigenvalue of Fredholm type is computed in terms of the underlying two-parameter system. Comparison with Faierman's work on two-parameter boundary value problems of Sturm-Liouville type is given as an application.

A modification of the convergence conditions for Picard's iteration

The convergence of the method of successive approximations is usually studied by the fixed point theorem. An alternative to this theorem is given in this work, where a contraction mapping is not necessary. An application to nonlinear integral equations of Fredholm type and second kind is also presented.

Numerical solution of functional differential, integral and integro-differential equations

This paper describes a numerical method, based on Lagrange interpolation and Chebyshev interpolation, to treat functional integral equations of Volterra type and Fredholm type. Also, the method can be extended to functional differential and integro-differential equations. Various numerical examples are treated.

Investigation of a Basic Plane Interface Problem of Statics for Elastic Mixtures

Abstract The two-dimensional interface problem of statics of elastic mixtures is investigated when differences between displacement vectors and stress vectors are given on the contact curve. For this problem special potentials with complex densities are constructed and by means of them the contact problem is reduced to a system of integral equations of Fredholm type. Analysis of the equations obtained is given.

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.

Lagrange interpolation to compute the numerical solutions of differential, integral and integro-differential equations

This paper describes new two methods for the numerical solution of linear integral equations of Fredholm type and Volterra type. Also, the methods have been extended to linear integro-differential and differential equations. The methods are tested by different examples.

An Analog of the Green's Formula and Its Applications in Electrodynamics

A new formula similar to the Green's formula, which is related to the Helmholtz operator, is obtained. This formula enables one to state three boundary-value problems. They are reduced to a system of integral equations of the Fredholm type of the second kind, which are solved by the perturbation method. Bibliography: 6 titles.