The propagation of error in linear problems

The Numerical Solution of Integral Equations on the Half-Line

The basic integro-differential equation is subjected to a one-sided finite Laplace transform to obtain linear integral equations of angular distribution of bounding faces. These linear integral equations have been transformed into linear singular integral equations which have been solved exactly to get the emergent distributions from the bounding faces by the theory of linear singular operators. Some solutions of linear singular integral equations have also been derived for future use in radiative transfer problems.

Chapter 5 Padé Approximants and Linear Integral Equations

In the paper we develop a class of iterative methods for approximating solutions of linear operator equations in a Banach space. The main techniques involve a product integral representation of solutions to linear Stieltjes integral equations, a variation of parameters formula, and the asymptotic convergence of solutions to the homogeneous integral equation.

This article studies the strong conical hull intersection property (strong CHIP) for a general (possibly infinite) system of convex sets in normed linear spaces. The first part of the article presents necessary and sufficient conditions for strong CHIP in terms of tangent cones and feasible direction cones in the primal space, and reveals some differences between strong CHIP for finite and infinite systems. The second part of the paper proves that strong CHIP has the segment extension property, i.e. if strong CHIP is satisfied at some ‘base points’ then it can be extended from the base points along line segments to the entire set. Thus, the verification effort of strong CHIP can be greatly reduced in many cases. For example, if the intersection set is a translated cone, then strong CHIP at the vertex ensures strong CHIP for the entire cone. In the special case of Euclidean space, strong CHIP at the extreme points of the intersection set ensures strong CHIP for the entire intersection.

The main aim of this paper is to numerically solve the first kind linear Fredholm and Volterra integral equations by using Modified Bernstein–Kantorovich operators. The unknown function in the first kind integral equation is approximated by using the Modified Bernstein–Kantorovich operators. Hence, by using discretization, the obtained linear equations are transformed into systems of algebraic linear equations. Due to the sensitivity of the solutions on the input data, significant difficulties may be encountered, leading to instabilities in the results during actualization. Consequently, to improve on the stability of the solutions which imply the accuracy of the desired results, regularization features are built into the proposed numerical approach. More stable approximations to the solutions of the Fredholm and Volterra integral equations are obtained especially when high order approximations are used by the Modified Bernstein–Kantorovich operators. Test problems are constructed to show the computational efficiency, applicability and the accuracy of the method. Furthermore, the method is also applied to second kind Volterra integral equations.

Solving linear and nonlinear Abel fuzzy integral equations by homotopy analysis method

Simulation and evaluation of interval-valued fuzzy linear Fredholm integral equations with interval-valued fuzzy neural network

Impact interaction of cylindrical body with a surface of cavity during supercavitation motion in compressible fluid

On some linear integral equations generating solutions of nonlinear partial differential equations

Unless otherwise specified, E will be a real or complex normed linear space. A complete normed linear space is called a Banach space. E’ is the dual space, i.e., the Banach space of continuous linear functionals, equipped with the norm \(\left\| l \right\| = \mathop {\sup }\limits_{\left\| x \right\| = 1} \left| {l\left( x \right)} \right|\).KeywordsNormed Linear SpaceSmooth Banach SpaceStrong UniquenessCharacterization TheoremGaussian Quadrature FormulaThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper we consider periodic functional differential and integral equations. Massera proved that the existence of a bounded solution implies the existence of a periodic solution for linear (inhomogeneous) ordinary differential equations. Chow proved a similar result for linear functional differential equations with finite delay. We give a new proof for Chow's result that works for linear functional differential equations with infinite delay and also for integral equations. We also give examples that do not satisfy uniform boundedness or uniform ultimate boundedness, the conditions frequently used to prove that there is a periodic solution.

The authors’ new method of iterative refinement has been successfully applied to the linear fractional Fredholm integro-differential equation. In this work, we adapt our method to obtain approximate analytical solutions for linear Volterra equations of both the first and the second kind of fractional type. A detailed convergence analysis is presented for each kind. The authors also apply their method to the challenging first kind linear Fredholm integral equation. Three different cases are considered to test the efficacy of our method. These include mixed systems of various forms of linear integral and fractional integro-differential equations. We compare our method with six well-established methods: Picard’s successive approximations, an accelerated form of the latter, the Adomian decomposition, the variational iteration, the homotopy perturbation and the homotopy analysis. The results show the versatility of the new method in solving a wide variety of equations, its high convergence speed and accuracy and its capability of directly dealing with equations of the first kind. We also prove that Picard’s method for linear equations is a special case of our method.

The exact solution of a linear integral equation with weakly singular kernel

It is studied a regularization of systems of linear Volterra integral equations of the third kind with continuous kernel degenerated at points of the given segment on the diagonal. The determinant of the matrix outside the integral disappears at the starting point or at the end of the segment. Regularizing Lavrent’ev type operator is constructed that preserves the property of Volterra equation. It is proved the uniform convergence of the regularized solution to the exact solution, and defined the conditions of uniqueness of the solution in the Holder space. Also it is considered the case of systems of third kind Volterra linear integral equations with the approximated right hand side of equation. It is established conditions under which a regularized solution of the equation with approximately given right-hand side can serve as an approximate solution to the original equation.