Finite dimensional approximation to solutions of minimization problems in functional spaces

In this paper we consider minimization problems whose objectives are defined on functional spaces. The integral global optimization technique is applied to characterize a global minimum as the limit of a sequence of approximating solutions on finite dimensional subspaces. Necessary and sufficient optimality conditions are presented. A variable measure algorithm is proposed to find such approximating solutions. Examples are presented to illustrate the variable measure method     

A numerical approximation theory for second-order integral differential equations

This paper is a companion paper to “An Oscillation Theory for Second-Order Integral Differential Equations.” The underlying theme is that both topic (oscillation theory and numerical oscillation theory) follow as a corollary to an approximation theory of quadratic forms given previously by the author. In Section I we give the mathematical preliminaries; some of which are not included in the earlier paper. This includes the relationship between the fundamental quadratic form and its integral differential equation (the Euler-Lagrange equations). In Section II the approximating quadratic forms are defined on the approximating Hilbert space. In Section III we show that our approximating hypothesis are satisfied and give the fundamental inequality relationships [Eqs. (12) and (13)]. We also show that the mth oscillation point is a continuous function of our approximating parameter. Finally in Section IV we show how that the approximating indices may be easily obtained by computer algorithms.     

A numerical study of iterative refinement schemes for weakly singular integral equations

Three iterative refinement schemes are studied for approximating the solutions of linear weakly singular Fredholm integral equations of the second kind. The rates of convergence and computational costs of the three schemes are studied and compared with the classical approach by applying them respectively to: (i) a sparse linear system associated with an integral equation modelling a real life Astrophysics problem, and (ii) an integral equation whose associated linear problem is dense.     

On strongly elliptic singular integral operators with piecewise continuous coefficients

The concept of strongly elliptic operators is one of the main tools for approximating the solution of boundary integral equations by finite element methods (see [4-10]). In the present paper necessary and sufficient conditions for the strong ellipticity of singular integral operators with piecewise continuous matrix coefficients on a closed or open Ljapunov curve are obtained.     

The Beesack–Darst–Pollard inequalities and approximations of the Riemann–Stieltjes integral

Utilising the Beesack version of the Darst–Pollard inequality, some error bounds for approximating the Riemann–Stieltjes integral are given. Some applications related to the trapezoid and mid-point quadrature rules are provided.     

The solution of a problem in the theory of epidemics

In a 1971 paper, Hoppensteadt and Waltman consider a deterministic epidemic model that accounts for certain threshold phenomena occurring in the spread of infection. A system of nonlinear delay integral equations describe this model. We describe a method for constructing functions which approximate the solution of the system of integral equations. The approximating functions are shown to exist and to converge to the solution of the system.     

A meshless method for numerical solution of the one-dimensional wave equation with an integral condition using radial basis functions

The hyperbolic partial differential equation with an integral condition arises in many physical phenomena. In this paper, we propose a numerical scheme to solve the one-dimensional hyperbolic equation that combines classical and integral boundary conditions using collocation points and approximating the solution using radial basis functions (RBFs). The results of numerical experiments are presented, and are compared with analytical solution and finite difference method to confirm the validity and applicability of the presented scheme.     

Shifted rectangular quadrature rule approximations to Dawson's integral F(x)

This paper investigates the relationship between some rapidly convergent series of exponential functions for computing Dawson's integral. These series are the result of approximating a certain improper integral by a shifted rectangular quadrature rule. Dawson's original method and a more recent expansion due to Rybicki are shown to be special cases of our quadrature approach. An error bound is derived to compare the accuracy of the resulting approximations.     

An algorithm for the construction of best approximations based on Kolmogorov's criterion

A descent algorithm for approximating continuous functions having values in a unitary space by functions in a convex class is given. Applications include complex, multidimensional, monotone approximations and the approximation of kernels of integral equations. Numerous complex Chebyshev polynomials are computed numerically.     

An approximating algorithm for the solution of an integral equation from epidemics

The following delay integral equation

$ x(t)=\int\limits_{t-\tau}^{t}f(s,x(s))ds,\quad t\in \mathbb{R}, $

has been proposed by Cooke and Kaplan to describe the spread of certain infectious diseases with periodic contact rate that varies seasonally. This mathematical model can also be interpreted as an evolution equation of a single species population. The purpose of this paper is to present an approximating algorithm for the continuous positive solution of this integral equation from the theory of epidemics. This algorithm is obtained by applying the successive approximations method and the rectangle formula, used for the calculation of the approximate value of integrals which appear in the right-hand-side of the terms of the sequence of successive approximations. In order to establish this approximating algorithm, we will suppose that this integral equation has a unique solution. The main result contains also the error of approximation of the solution obtained by applying this approximating algorithm.

     

一类非线性积分,微分方程的逼近解

在不主紧型条件和耗散型条件的情形下，得到了Ｂａｎａｃｈ空间中一类非线性积分，微分方程的逼近解。     

Positive solutions to singular fractional differential system with coupled boundary conditions

By using the fixed point theory in cone and constructing some available integral operators together with approximating technique, the existence of positive solution for a singular nonlinear semipositone fractional differential system with coupled boundary conditions is established. Two examples are then given to demonstrate the validity of our main results.     

Numerical solution of nonlinear two-dimensional Fredholm integral equations of the second kind using Gauss product quadrature rules

The Gauss product quadrature rules and collocation method are applied to reduce the second-kind nonlinear two-dimensional Fredholm integral equations (FIE) to a nonlinear system of equations. The convergence of the proposed numerical method is proved under certain conditions on the kernel of the integral equation. An iterative method for approximating the solution of the obtained nonlinear system is provided and its convergence is proved. Also, some numerical examples are presented to show the efficiency and accuracy of the proposed method.     

Integral and nonnegativity preserving approximations of functions

In this paper we consider the problem of approximating a function by continuous piecewise linear functions that preserve the integral and nonnegativity of the original function.     

A constructive method for solving the displacement boundary-value problem of elastostatics by use of global basis systems

In this paper we are concerned with the three-dimensional homogeneous isotropic elastostatic boundary-value problem. Global basis vector fields are shown to be constructive structures for approximating the solution. The theoretical tool is a regularity condition developed from the classical integral equation approach. The paper ends with some reflections on the practical computability and numerical applicability.     

Error bounds in the approximation of eigenvalues of differential and integral operators

Various methods of approximating the eigenvalues and invariant subspaces of nonself-adjoint differential and integral operators are unified in a general theory. Error bounds are given, from which most of the error bounds in the literature can be derived. Computable error bounds are given for simple eigenvalues, and trace formulae are used to improve the accuracy of the computed eigenvalues.     

Best approximations of classes of functions defined by means of the modulus of continuity

This paper is devoted to an exact solution of problems of best approximation in the uniform and integral metrics of classes of periodic functions representable as a convolution of a kernel not increasing the oscillation with functions having a given convex upwards majorant of the modulus of continuity. The approximating sets are taken to be the trigonometric polynomials in the case of the uniform and integral metrics, and convolutions of the kernel defining the class with polynomial splines in the case of the integral metric.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 5, pp. 579–589, May, 1992.     

On some iterative numerical methods for a Volterra functional integral equation of the second kind

In this paper, we study an iterative numerical method for approximating solutions of a certain type of Volterra functional integral equations of the second kind (Volterra integral equations where both limits of integration are variables). The method uses the contraction principle and a suitable quadrature formula. Under certain conditions, we prove the existence and uniqueness of the solution and give error estimates for our approximations. We also included a numerical example which illustrates the fast approximations.     

Time-discretization schema for an integrodifferential Sobolev type equation with integral conditions

The appropriate discretization in time schema is considered for approximating the solution of a nonlinear integro-differential Sobolev type equation with integral conditions. We construct a discrete numerical solution of the discretized problem. Then, some a priori estimates for the approximations are derived, while the convergence of the method and the well posedness of the problem under study are established.