Class Of Linear Integral Equations Research Articles

On the numerical solution of integral equations of the second kind over infinite intervals

In this paper, we discuss the numerical solution of a class of linear integral equations of the second kind over an infinite interval. The method of solution is based on the reduction of the problem to a finite interval by means of a suitable family of mappings so that the resulting singular equation can be accurately solved using spectral collocation at the Jacobi-Gauss points. Several selected numerical examples are presented and discussed to illustrate the application and effectiveness of the proposed approach.

A class of linear integral equations of the first kind with two independent variables

A class of linear integral equations of the first kind with two independent variables

A transfer integral technique for solving a class of linear integral equations: Convergence and applications to DNA

An eigenvalue problem, the convergence difficulties that arise and a mathematical solution are considered. The eigenvalue problem is motivated by simplified models for the dissociation equilibrium between double-stranded and single-stranded DNA chains induced by temperature (thermal denaturation), and by the application of the so-called transfer integral technique. Namely, we extend the Peyrard–Bishop model for DNA melting from the original one-dimensional model to a three-dimensional one, which gives rise to an eigenvalue problem defined by a linear integral equation whose kernel is not in L2. For the one-dimensional model, the corresponding kernel is not in L2 either, which is related to certain convergence difficulties noticed by previous researchers. Inspired by methods from quantum scattering theory, we transform the three-dimensional eigenvalue problem, obtaining a new L2 kernel which has improved convergence properties.

A Class of Linear Integral Equations and Systems with Sum and Difference Kernel

By means of Fourier transform and Cauchy integral techniques a complete investigation of a class of linear integral equations and corresponding systems of equations of cross-correlation type in the Lebesgue spaces L^1 and L^2 is performed. Integral equations of first and second kind are reduced to explicitly solvable Riemann-Hilbert problems for a holomorphic function in the upper half-plane and the system of equations to conjugacy problems for a sectionally holomorphic function, where in the case of a finite interval also the analytic continuation of the solutions to the lower half-plane can be carried out in explicit way. Further, a resolvent representation of the solution to the integral equation and its adjoint equation is derived.

The Thermodynamic Bethe Ansatz and a Connection with Painlevé Equations

We summarize some recent connections between a class of nonlinear integral equations related to the thermodynamic Bethe Ansatz and a class of linear integral equations related to the Painlevé equations.

On the Asymptotic Solution to a Class of Linear Integral Equations

We consider Fredholm integral equations of the first kind whose kernels are a function of the difference between two points times a large parameter. Conditions on the kernel are stated in terms of a function corresponding to a Wiener–Hopf factorization of the Fourier transform of the kernel. We give the complete asymptotic expansions of the solution to the integral equations.As applications of our results, we consider the steady-state, acoustical scattering of a plane wave by both a hard strip and a soft strip. Our results are uniform with respect to the direction of incidence.

Bivariational Methods for Linear Integral Equations with Nonsymmetric Kernels

Bivariational methods are used to develop upper and lower bounding functionals for arbitrary inner products $\langle {g,\phi } \rangle $ associated with the solutions $\phi $ of a class of linear integral equations with nonsymmetric kernels. The underlying structure of the variational principles is seen to relate to a linear combination of the integral equation and an adjoint equation. The new functionals derived are more accurate than previous bounds obtained from a direct approach. Their efficiency is illustrated by using them to calculate actual pointwise bounds on some solutions of integral equations. In two test cases, the accuracy achieved was superior to that arising from a standard 160-step Simpson quadrature calculation.

General algebraic theory of identical particle scattering

We consider the nonrelativistic N-body scattering problem for a system of particles in which some subsets of the particles are identical. We demonstrate how the particle identity can be included in a general class of linear integral equations for scattering operators or components of scattering operators. The Yakubovskii, Yakubovskii–Narodestkii, Rosenberg, and Bencze–Redish–Sloan equations are included in this class. Algebraic methods are used which rely on the properties of the symmetry group of the system. Operators depending only on physically distinguishable labels are introduced and linear integral equations for them are derived. This procedure maximally reduces the number of coupled equations while retaining the connectivity properties of the original equations.

Regularization using Fejer operators

THE USE of Fejer integral operators to construct regularizing operators (see [1], p. 46) for a class of linear integral equations of the 1st kind of convolution type, and for stable summation of Fourier series with approximate coefficients, is examined.

Functional analysis of integral transport equations in linear rarefied gas dynamics

A study is proposed on the functional properties of the solutions of an interesting class of linear integral equations governing linear problems in rarefied gas dynamics. The analysis is carried out through a systematic study of the integral operator generated by the kernel of the equations themselves.

Imbedding a Class of Linear Integral Equations Through the First Critical Point

In this paper, we investigate the continuation of an imbedded solution $x(t) = x(t,\alpha )$ of \[x(t) = f(t) + \int_0^\alpha {k(t,s)x(s)ds = f(t) + (K_\alpha x)(t)} \] through its first “critical point” $\alpha = c$. Under the assumption that the Fredholm resolvent $\Gamma _\alpha = (I - K_\alpha )^{ - 1} - I$ has a simple pole in its meromorphic expansion about $\alpha = c$ we obtain a simple eigenspace corresponding to$\lambda = 1$ for the operator $K_c $ ; and in accordance with the redholm alter-native, we have an imbedded solution for $\alpha > c$ for forcing functions orthogonal to the one-dimensional eigenspace of the adjoint operator $K_c^ * $. The principal technique is the explicit solving of the Bartle-Schmidt bifurcation equation.

Solution by the regularization method of a convolution type equation with an inaccurately defined kernel

THE solution by the regularization method of a class of linear integral equations of the first kind of the convolution type with an inaccurately defined right side and kernel is considered.

A method for the determination of the eigenvalues and eigenfunctions of a certain class of linear integral equations

This article presents a method for the determination of the eigenvalues and eigenfunctions of Fredholm's linear integral equation with a positive definite root, which is the correlation function of a function of a random variable, and is related to the phenomenon of white noise by a linear differential equation. This method is a modification of the method given in article [1] for the solution of an integral equation of the first kind, occuring in the statistical theory of optimal systems.