On optimal high accuracy linear multistep methods for first kind volterra integral equations

General linear multistep methods which include those of Holyhead et al. [9] and Gladwin et al. [8] are introduced. A new simple root condition for stability is deduced. A theorem relating the coefficients of this new associated stability polynomial to those of the method is proved. This permits a constructive approach for obtaining high accuracy convergent methods. Two such methods are derived and numerical results are presented.This work was done under the financial support of: Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), and Financiadora Nacional de Empreendimentos e Projetos (FINEP)—Ministério do Planejamento (Brazil).     

On linear Fredholm integro-differential equations of first kind

We study the question whether linear one-dimensional integro-differential equations with constant limits of integration (equations of Fredholm type) containing no free differential expression (equations of first kind) can be reduced to integral equations of first kind and to Fredholm integro-differential equations of second kind.Translated fromVychislitel'naya i Prikladnaya Matematika, Issue 71, 1990, pp. 20–27.     

On the solution of volterra integral equations of the first kind

Summary Two classes of high order finite difference methods for first kind Volterra integral equations are constructed. The methods are shown to be convergent and numerically stable.     

On parametric families of solutions of Volterra integral equations of the first kind with piecewise smooth kernel

We suggest a method for constructing asymptotic approximations to parametric families of solutions of Volterra integral equations of the first kind with piecewise smooth kernel and prove the existence of continuous solutions.     

An adaptive numerical method for solving linear Fredholm integral equations of the first kind

A general procedure is presented for numerically solving linear Fredholm integral equations of the first kind. The approximate solution is expressed as a continuous piecewise linear (spline) function. The method involves collocation followed by the solution of an appropriately scaled stabilized linear algebraic system. The procedure may be used iteratively to improve the accuracy of the approximate solution. Several numerical examples are given.Supported in part by the Office of Naval Research under Contract No. NR 044-457.Supported in part by the National Science Foundation under Grant No. GJ-31827.     

On the approximate solution of one-dimensional singular integral equations of first kind

We prove an estimate for the error in approximate solution of one-dimensional singular integral equations. The estimate is obtained by an approximation of the kernel. For a specific problem we give a comparison of numerical results. Translated fromMatematichni Metody i Fiziko-Mekhanichni Polya, Vol. 38, 1995.     

A convergence rate for approximate solutions of Fredholm integral equations of the first kind

We study the convergence rate for solving Fredholm integral equations of the first kind by using the well known collocation method. By constructing an approximate interpolation neural network, we deduce the convergence rate of the approximate solution by only using continuous functions as basis functions for the Fredholm integral equations of the first kind. This convergence rate is bounded in terms of a modulus of smoothness.     

Numerical solution of integral equations of the first kind

An approximate method for solving integral equations of the first kind is considered, with the approximate solution represented as a finite expansion in some basis. Solution examples for a number of model problems are given. The dependence of the approximation error on the accuracy of the initial data is analyzed numerically.Translated from Metody Matematicheskogo Modelirovaniya, Avtomatizatsiya Obrabotki Nablyudenii i Ikh Primeneniya, pp. 72–79, 1986.     

Asymptotic regularization for integral equations of the first kind

The paper concerns solving a certain class of i11-posed problems including integral equationsof the first kind. The proposed regularization consists in replacing the considered i11-posedproblem by an ass ociated dynamical system. Well posedness of introduced system and an asymptotic connection of its solution with the solution we look for are proved.     

Multigrid solvers and preconditioners for first kind integral equations

We discuss multigrid methods and multilevel preconditioners for first kind boundary integral equations with weakly and hypersingular kernels. We find that the number of iterations needed is bounded or grows no worse than logarithmically in the numbers of unknowns. We also discuss the complexity for parallel implementations.     

Oscillating solutions of integral equations and linear recursions

Answering a question of Z. Daróczy we show that there are positive sequences _n satisfying the recursion

Approximate solutions of Fredholm integral equations of the second kind

This note is concerned with the problem of determining approximate solutions of Fredholm integral equations of the second kind. Approximating the solution of a given integral equation by means of a polynomial, an over-determined system of linear algebraic equations is obtained involving the unknown coefficients, which is finally solved by using the least-squares method. Several examples are examined in detail.     

On an iterative method for a class of integral equations of the first kind

In this paper, we investigate an iterative method which has been proposed [1] for the numerical solution of a special class of integral equations of the first kind, where one of the essential assumptions is the positivity of the kernel and the given right-hand side. Integral equations of this special type occur in experimental physics, astronomy, medical tomography and other fields where density functions cannot be measured directly, but are related to observable functions via integral equations. In order to take into account the non-negativity of density functions, the proposed iterative scheme was defined in such a way that only non-negative solutions can be approximated. The first part of the paper presents a motivation for the iterative method and discusses its convergence. In particular, it is shown that there is a connection between the iterative scheme and a certain concave functional associated with integral equations of this type. This functional can be interpreted as a generalization of the log-likelihood function of a model from emission tomography. The second part of the paper investigates the convergence properties of the discrete analogue of the iterative method associated with the discretized equation. Sufficient conditions for local convergence are given; and it is shown that, in general, convergence is very slow. Two numerical examples are presented.     

On volterra equations of the first kind

The existence of a solution β of the equation $$\int_0^t {a(t - s)d\beta (s) = 1, t > 0} $$ is studied under fairly general assumptions on the function a. Sufficient conditions for the measure β to be absolutely continuous or satisfy some additional regularity properties are given. An extension to nonconvolution kernels is also considered.     

Polynomial Volterra integral equations of the first kind and the Lambert function

The role of the Lambert function in the theory of polynomial Volterra equations of the first kind is considered. New results are presented in addition to the known ones. In particular, the stability of a continuous solution of the first-kind polynomial Volterra equation of degree N is investigated. Based on the techniques of majorant equations, sufficient stability conditions are obtained.