Solution Of Fredholm Integral Equations Research Articles

Information Complexity of Multivariate Fredholm Integral Equations in Sobolev Classes

In this paper, the complexity of full solution of Fredholm integral equations of the second kind with data from the Sobolev classWr2is studied. The exact order of information complexity is derived. The lower bound is proved using a Gelfand number technique. The upper bound is shown by providing a concrete algorithm of optimal order, based on a specific hyperbolic cross approximation of the kernel function. Numerical experiments are included, comparing the optimal algorithm with the standard Galerkin method.

Complexity of Local Solution of Multivariate Integral Equations

In this paper the complexity of the local solution of Fredholm integral equations is studied. For certain Sobolev classes of multivariate periodic functions with dominating mixed derivative we prove matching lower and upper hounds. The lower bound is shown using relations to s-numbers. The upper hound is proved in a constructive way providing an implementable algorithm of optimal order based on Fourier coefficients and a hyperbolic cross approximation.

On the diffraction of high-frequency waves by a cone of arbitrary shape

The scalar problem of diffraction of a plane scalar wave by an arbitrary shape cone is considered. The boundary condition is a Dirichlet one. The spherical wave (the center of the sphere is the cone vertex) scattered by the cone vertex arises as a result of the diffraction process. Our subject is a numerical calculation of the wave amplitude. The calculations are based on V.P. Smyshlyaev's results. The problem required the numerical solution of Fredholm integral equations along the line of intersection of the cone and the unit sphere, center of which is the cone vertex, and calculations of integrals containing the solutions of the integral equations. The method is possible to use if some restrictions to the directions of the observations are held.

Determination of limiting radii of the contact area in axi-symmetric contact problems with frictional heat generation

Axi-symmetric contact problems for a half-space with frictional heat generation have been considered by Barber, [(1975) Thermoelastic contact of a rotating sphere and a half-space. Wear 35, 283–289; (1976) Some thermoelastic contact problems involving frictional heating. Q. J. Mech. Appl. Math. 29, 1–13], Generalov et al. [(1976) The contact problem of thermoelasticity for rotating bodies. Izvestija AN USSR. Mechanika Tverdogo Tela. 3, 46–52] and Grilitski and Kulchytsky-Zhyailo [(1991) The axi-symmetrical contact problem of thermoelasticity for the bodies of rotating. Physicochem. Mech. Mater. 3, 93–97]. The problems have been treated by different methods and their solutions included estimates of the critical values of the dimensionless contact area radius. The present paper gives closed form expressions for these limiting radii by the application of Atkinson's result [(1976) A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind. SIAM, Philadelphia (SIAM Monograph)] to the governing Fredholm integral equations, rather than using a full solution as was done by previous authors.

Accelerated degenerate-kernel methods for linear integral equations

A new approach to the theory of kernel approximations is developed for the numerical solution of Fredholm integral equations of the second kind. The objective is to determine the solution to any desired accuracy using a degenerate-kernel operator of fixed rank — that is, without increasing the number of terms in the approximate kernel. Consequently, high accuracy is achieved without incurring the computational costs associated with solving large algebraic systems. The procedure employs projection techniques to construct accelerated degenerate-kernel schemes that give rise to a bisequence of higher-order approximate solutions. Error expressions are obtained and a numerical example of solving a weakly singular equation is given.

Taylor polynomial solutions of Volterra integral equations

The method of Kanwal and Liu for the solution of Fredholm integral equations is applied to certain linear and nonlinear Volterra integral equations of the second kind. Some equations considered by other authors are solved in terms of Taylor polynomials and the results are compared.

Solution of Fredholm integral equations by conversion to known transforms

By manipulation of the kernel, it is possible to convert some types of Fredholm integral equations of the first kind to a well known transform or to a more tractable integral equation. In this paper manipulation of the kernel is used to replace the unknown function by a differential equation involving it, with the result that the integral equation has a simplified form. The technique is illustrated by a number of examples of conversion to Laplace and Mellin transform. Although the method requires only elementary manipulations, it highlights the interesting interplay between the boundary values of the unknown function and the solution of the integral equation.

Parallel solution of Fredholm integral equations of the second kind by accelerated projection methods

Conventional projection methods for the numerical solution of integral equations are serial in structure, and require substantial synchronization which seriously degrades their performance in a parallel environment. The present work focuses on the exploitation and implementation of a new class of computational techniques, known as accelerated projection methods, which are distinctly different from conventional methods, capable of high accuracy and which are particularly appropriate for concurrent computing. The methods are implemented in detail on a multiprocessor for a particular linear integral equation of interest. It is shown that accelerated projection methods possess a high degree of parallelism that leads to extremely efficient parallel algorithms.

An application of the generalized alternating polynomials to the numerical solution of Fredholm integral equations

Generalized alternating polynomials have been introduced by the author earlier. In the present paper their indirect analogue is constructed for numerical solution of the Fredholm linear integral equations. Although the proposed method is a particular case of the general projection scheme, its valuable feature is the presence of a sequence of parameters, which, for sufficiently smooth kernels and inhomogeneous terms, serves as an “indicator of the quality of approximation”.

Numerical tools for analysis and solution of Fredholm integral equations of the first kind

The author surveys several numerical tools that can be used for the analysis and solution of systems of linear algebraic equations derived from Fredholm integral equations of the first kind. These tools are based on the singular value decomposition (SVD) and the generalized SVD, and they allow the user to study many details of the integral equation. The tools also aid the user in choosing a good regularization parameter that balances the influence of regularization and perturbation errors.

Error bounds for the solution of Volterra and delay equations

The construction of efficient and reliable numerical software requires a theory for two distinct issues: a convergence analysis and a way of bounding the error in the computed results. While the first of these questions has received much attention, less is known about the second one. Here we describe a method for bounding the error in the solution of Fredholm integral equations and show how the approach can be extended to Volterra and delay equations.

Approximate solution of fredholm integral equations and piecewise cubic interpolatory polynomials

Numerical solution of Fredholm integral equations is considered while using the class of Piecewise Cubic Interpolatory Polynomials, X-splines. We include the case where the kernel has a logarithmic singularity. The homogenous equation is shown reduced to a standard Eigen value problem. Examples, with applications, are considered and the results of the proposed method are compared with other existing methods. We indicate the convergence of the numerical solution for the approximate function belonging to the space of continuous as well as differentiable function.

Parallel information-based complexity

We develop the theory of information-based complexity from a parallel point of view. For a model of computation with p processors, each being capable of arithmetic operations and decisions, we analyze the complexity of function approximation, numerical integration, and solution of Fredholm integral equations. We obtain tight bounds on the complexity, considered as a function of three variables simultaneously: the number of processors, the required precision, and (in the case of approximation and integral equations) the number of points, in which the approximate solution is to be determined.

Analysis of positron annihilation lifetime data by numerical Laplace inversion: Corrections for source terms and zero-time shift errors

Abstract We have recently described modifications to the program CONTIN [S.W. Provencher, Comput. Phys. Commun. 27 (1982) 229] for the solution of Fredholm integral equations with convoluted kernels of the type that occur in the analysis of positron annihilation lifetime data [R.B. Gregory and Yongkang Zhu, Nucl. Instr. and Meth. A290 (1990) 172]. In this article, modifications to the program to correct for source terms in the sample and reference decay curves and for shifts in the position of the zero-time channel of the sample and reference data are described. Unwanted source components, expressed as a discrete sum of exponentials, may be removed from both the sample and reference data by modification of the sample data alone, without the need for direct knowledge of the instrument resolution function. Shifts in the position of the zero-time channel of up to half the channel width of the multichannel analyzer can be corrected. Analyses of computer-simulated test data indicate that the quality of the reconstructed annihilation rate probability density functions is improved by employing a reference material with a short lifetime and indicate that reference materials which generate free positrons by quenching positronium formation (i.e. strong oxidizing agents) have lifetimes that are too long (400–450 ps) to provide reliable estimates of the lifetime parameters for the shortlived components with the methods described here. Well-annealed single crystals of metals with lifetimes less than 200 ps, such as molybdenum (123 ps) and aluminum (166 ps) do not introduce significant errors in estimates of the lifetime parameters and are to be preferred as reference materials. The performance of our modified version of CONTIN is illustrated by application to positron annihilation in polytetrafluoroethylene.

A new method for the numerical solution of Fredholm integral equations of the first kind

This paper describes a new approach for the numerical solution of Fredholm integral equations of the first kind (FIEFK). Examples of where this kind of problem may occur in chemical engineering include determination site-energy distributions, determination of pore-size distributions and using chemically reacting tracers to measure temperature profiles in plug-flow systems. FIEFK also occur frequently in other fields of engineering and science. Several numerical examples are used to illustrate the new technique.

Hyperbolic cross and the complexity of the approximate solution of Fredholm integral equations of the second kind with differentiable kernels

Hyperbolic cross and the complexity of the approximate solution of Fredholm integral equations of the second kind with differentiable kernels

The solution of faddeev integral equations for three-body scattering by means of B-splines

A method for the solution of Fredholm integral equations of the second kind with sigularities both in the kernel and in the solution is developed, based on the approximation of the solution by B-splines. The main problem of this method, the distribution of knots, is extensively investigated. The Faddeev equations describing the physical problem of pion deuteron elastic scattering are solved as an application of this method.

Parallel solution of Fredholm integral equations

We consider here parallel variants of the Nyström and Fast Galerkin methods for the solution of Fredholm integral equations of the second kind. Numerical examples, and timings for an Ada implementation on a multiprocessor Sequent Balance, are given for both smooth and non-smooth kernels.

Parallel solution of Fredholm integral equations of the second kind by orthogonal polynomial expansions

Given a Fredholm integral equation of the second kind, the Nyström method based on an n-point Gaussian quadrature rule and the perturbed Galerkin method in which inner products and integral transforms are approximated using the Gaussian rule, are equivalent in the sense that the Nyström interpolant and the orthogonal polynomial expansion of degree n − 1> agree at the n quadrature nodes. The perturbed collocation method, using the n quadrature nodes as collocation points, in which integral transforms are approximated using the Gaussian rule, is also equivalent. A practical implication is that one can set up and solve the simple n × n linear system corresponding to the Nyström method, and then easily transform the result to that of the perturbed Galerkin method, in order to economically gain the advantages of the orthogonal expansion. This approach appears particularly useful for the numerical solution on multiprocessor systems of the integral formulation of certain two-point boundary value problems. The paper analyzes the relationship among numerical methods, it establishes the weighted least-squares convergence of the orthogonal expansions, and it describes corresponding parallel algorithms.

The numerical solution of Fredholm integral equations on parallel computers

In this paper algorithmic strategies for the Parallel solution of Fredholm Integral Equations of the first kind are presented. The solution which is obtained by the minimal least squares method uses the singular value decomposition procedure. The parallel computation of the singular value decomposition (SVD) algorithm is derived in 3 phases, where each phase is implemented in a suitable form to achieve a high level of parallelism. The algorithm was implemented on the Loughborough NEPTUNE system for 1, 2, 3 and 4 processors and the performance results analysed.