Solution Of Linear Integral Equations Research Articles

An iterative solution for the second kind integral equations using radial basis functions

The purpose of this paper is to present a new method based on radial basis functions (RBFs) and iterative procedure for the solution of linear integral equations of Fredholm and Volterra types of the second kind. To obtain a primary approximate solution, the collocation method with a small number of nodes, based on the RBFs and their first-order derivatives, can be adapted. Moreover, we present an iterative procedure from the integral equation to improve the accuracy of solution. Numerical examples show that our approach has the potentiality to become an efficient method.

Expansion method for linear integral equations by cardinal B-spline wavelet and Shannon wavelet as bases for obtain Galerkin system

In this paper, we apply the B-spline and Shannon wavelets bases for approximation the solution of linear integral equations of the second kind, then by this wavelet bases we construct Galerkin system, which is important the expansion methods in solving linear integral equations. At the end, For showing efficiency of this method, we use numerical examples.

Numerical solution of linear integral equations by using Sinc–collocation method

A collocation procedure is developed for linear Fredholm integral equations of the second kind, using Sinc basis functions. The convergence rate of the method is O ( e - k M ) . Numerical results are included to confirm the efficiency and accuracy of the method.

Numerical solution of integral equations by means of the Sinc collocation method based on the double exponential transformation

Numerical solution of linear integral equations by means of the Sinc collocation method based on the double exponential transformation, abbreviated the DE transformation, is considered. We first apply the method to the Volterra integral equation of the second kind and then to the Volterra equation of the first kind. This method is also applied to the Fredholm integral equation of the second kind. For the Volterra equations we employed a formula for numerical indefinite integration developed by Muhammad and Mori obtained by applying the DE transformation incorporated into the Sinc expansion of the integrand, while for the Fredholm equation we employed the conventional DE transformation for definite integrals. An error analysis of the method is given and in every case a convergence rate of O ( exp ( - cN / log N ) ) for the error is established where N is a parameter representing the number of terms of the Sinc expansion. Also, the condition of the matrix of the main system of linear equations is watched through an estimate of the condition number returned by the program. Numerical examples show the convergence rate mentioned above and confirm the high efficiency of the present method.

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.

Painlevé equations—nonlinear special functions

The six Painlevé equations (PI–PVI) were first discovered about a hundred years ago by Painlevé and his colleagues in an investigation of nonlinear second-order ordinary differential equations. Recently, there has been considerable interest in the Painlevé equations primarily due to the fact that they arise as reductions of the soliton equations which are solvable by inverse scattering. Consequently, the Painlevé equations can be regarded as completely integrable equations and possess solutions which can be expressed in terms of solutions of linear integral equations, despite being nonlinear equations. Although first discovered from strictly mathematical considerations, the Painlevé equations have arisen in a variety of important physical applications including statistical mechanics, plasma physics, nonlinear waves, quantum gravity, quantum field theory, general relativity, nonlinear optics and fibre optics.The Painlevé equations may be thought of a nonlinear analogues of the classical special functions. They possess hierarchies of rational solutions and one-parameter families of solutions expressible in terms of the classical special functions, for special values of the parameters. Further the Painlevé equations admit symmetries under affine Weyl groups which are related to the associated Bäcklund transformations.In this paper, I discuss some of the remarkable properties which the Painlevé equations possess including connection formulae, Bäcklund transformations associated discrete equations, and hierarchies of exact solutions. In particular, the second Painlevé equation PII is used to illustrate these properties and some of the applications of PII are also discussed.

Recovery of potential using module values of its gradient. 2

The method of the recovery of the potential on the module of its gradient under conditionof the similarity of the potential to that given as the limit of the succession of the solations of the boundary problems of Neumann for the Laplace equation that define the disturbing potential proposed in the first part is substantiated. The succession of the disturbing potential is generated by the succession of the solutions of linear integral equations of the second kind with compact operators having large cores. A correct solubility of the given kind of equations is established and the convergence of the succession of the solution of the Neumann's problems to the function unambiguously generating the desired potential is proven.

Consistency of a Method of Moments Estimator Based on Numerical Solutions to Asset Pricing Models

This paper considers the properties of estimators based on numerical solutions to a class of economic models. In particular, the numerical methods discussed are those applied in the solution of linear integral equations, specifically Fredholm equations of the second kind. These integral equations arise out of economic models in which endogenous variables appear linearly in the Euler equations, but for which easily characterized solutions do not exist. Tauchen and Hussey [24] have proposed the use of these methods in the solution of the consumption-based asset pricing model. In this paper, these methods are used to construct method of moments estimators where the population moments implied by a model are approximated by the population moments of numerical solutions. These estimators are shown to be consistent if the accuracy of the approximation is increased with the sample size. This result depends on the solution method having the property that the moments of the approximate solutions converge uniformly in the model parameters to the moments of the true solutions.

Integral equations and exact solutions for the fourth Painlevé equation

We consider a special case of the fourth Painlevé equation given by d 2 ƞ / dξ 2 = 3 ƞ 5 + 2ξ ƞ 3 + (1/4ξ 2 - v - 1/2 ) ƞ , (1) with v a parameter, and seek solutions ƞ (ξ; v ) satisfying the boundary condition ƞ (∞)=0. (2) Equation (1) arises as a symmetry reduction of the derivative nonlinear Schrödinger (DNLS) equation, which is a completely integrable soliton equation solvable by inverse scattering techniques. Solutions of equation (1), satisfying (2), are expressed in terms of the solutions of linear integral equations obtained from the inverse scattering formalism for the DNLS equation. We obtain exact ‘bound state’ solutions of equation (1) for v = n , a positive integer, using the integral equation representation, which decay exponentially as ξ→ ± ∞ and are the first example of such solutions for the Painlevé equations. Additionally, using Bäcklund transformations for the fourth Painlevé equation, we derive a nonlinear recurrence relation (commonly referred to as a Bäcklund transformation in the context of soliton equations) for equation (1) relating ƞ (ξ; v ) and ƞ (ξ; v + 1).

Plane Strain Compression of Rigid-Perfectly Plastic Strip Between Parallel Dies With Slipping Friction

This paper presents the results of computer calculations of a class of slipline solutions for compression between parallel dies with slipping friction at the die-metal interface such that the frictional shear traction is a constant proportion of the yield stress. The slipline fields considered here have previously only been suggested qualitatively. The fields are of “indirect type”, requiring the solution of linear integral equations. They have been analyzed and computed here using the recently developed matrix operator procedure. The numerical results obtained are compared with those obtained from approximate upper bound and other “technological” theories.

On moment-discretization and least-squares solutions of linear integral equations of the first kind

Let K( s, t) be a continuous function on [0, 1] × [0, 1], and let K be the linear integral operator induced by the kernel K( s, t) on the space L 2[0, 1]. This note is concerned with moment-discretization of the problem of minimizing ‖ K x− y ‖ in the L 2-norm, where y is a given continuous function. This is contrasted with the problem of least-squares solutions of the moment-discretized equation: ∝ 0 1 K( s i , t) x( t) dt = y( s i ), i = 1, 2, h., n. A simple commutativity result between the operations of “moment-discretization” and “least-squares” is established. This suggests a procedure for approximating K†y (where K † is the generalized inverse of K ), without recourse to the normal equation K∗Kx = K∗y, that may be used in conjunction with simple numerical quadrature formulas plus collocation, or related numerical and regularization methods for least-squares solutions of linear integral equations of the first kind.

Iterative methods for best approximate solutions of linear integral equations of the first and second kinds

Least-squares solutions of Fredholm and Volterra equations of the first and second kinds are studied using generalized inverses. The method of successive approximations, the steepest descent and the conjugate gradient methods are shown to converge to a least-squares solution or to the least-squares solution of minimal norm, both for integral equations of the first and second kinds. An iterative method for matrices due to Cimmino is generalized to integral equations of the first kind and its convergence to the least-squares solution of minimal norm is established.

Solution of linear integral equations by Gregory's method

Two techniques for using Gregory's method to solve Fredholm integral equations of the second kind are described. Since the kernel function is allowed to be mildly discontinuous, Volterra integral equations of the second kind can be solved in the same manner. Some numerical examples are given.

An embedding technique for the solution of linear integral equations

Starting from the classical Fredholm theory, it is shown that the solution of a linear integral equation at an eigenvalue λ = λ 0 (say) can be expressed as the limit function (when this exists) of the unique solution of a standard non-homogeneous linear integral equation with λ ≠ λ 0, as λ → λ 0. Developing this idea, most of the standard properties are readily derived, and also, some which are believed to be new. Although, for simplicity, the present theory is restricted to Riemann integrals and kernels of only limited discontinuity, the fundamental principle here explained is obviously capable of a far wider generality in application, and it is hoped to so extend it in later papers.

The Numerical Solution of Integral Equations on the Half-Line

The Numerical Solution of Integral Equations on the Half-Line

Chebyshev Solution of Differential, Integral and Integro-Differential Equations

This paper describes a new method for the numerical solution of linear integral equations of Fredholm type and of Volterra type. The method has been extended to the linear integro-differential equations and ordinary differential equations. It can also be applied to non-linear problems. In each case numerical examples are treated and the method compares quite favourably with other known methods.

Solution of Linear Integral Equations by the Use of Invariant Functions

A simple method is presented for the solution of linear integral equations. The relationship of this method to the replacement of an integral equation by an infinite system of equations and to the solution of integral equations by the use of characteristic functions (eigenfunctions) is shown, and its application is illustrated by two simple examples.

Concerning a method of solution of linear integral equations of the second kind

Concerning a method of solution of linear integral equations of the second kind

Nonlinear prediction of a class of random processes

This paper is concerned with the minimum mean-squared error (MMSE) nonlinear prediction of a class of random processes. A class of random processes is defined by the property that its MMSE zero-memory predictor is represented by a finite sum of separable terms. Sufficient conditions for the existence of such processes are also considered. The nonlinear predictor is restricted to be composed of a linear filter in parallel with a zero-memory nonlinearity (ZNL) preceded by a variable delay, The optimum predictor is shown to be the solution of linear integral equations with the same kernel as for the optimum linear predictor. The first step of the derivation also yields a simpler scheme which ,only requires the addition of a ZNL to the optimum linear predictor. The improvements in the MMSE of the two nonlinear systems over the linear case are compared and illustrated by a numerical example.

Numerical solution of linear integral equations of the first kind. Calculation of molecular weight distributions from sedimentation equilibrium data.

A new method is described for the numerical solution of g(x)= ∫ abK(x, t)f(t)dtfor f(t) when g(x) is experimental data of limited accuracy. Under these conditions, the well-known least-squares and algebraic methods often give poor solutions. The new method is an optimum combination of these two. It has the stability of the least-squares method to errors in g(x), but still does not assume a functional form for f(t). The problem of the determination of the molecular weight distributions (MWD) of ideal polydisperse solutes from sedimentation equilibrium data is discussed in detail. Results of the analyses of sets of computer-simulated experimental data containing pseudorandom errors show that the method can give good estimates of the MWD from data accurate to only two significant figures. This accuracy should be readily attainable experimentally. The method should be useful in the analysis of data from many other types of experiments, particularly scattering, where the presently used Fourier methods are often unsatisfactory because they require the extrapolation of data in order to evaluate the Fourier transforms.