Numerical evaluation of inverse integral transforms: Dynamic response of elastic materials

This study discusses the use of numerical integration in evaluating the improper integrals appearing as inverse integral transforms of non-analytic functions. These transforms appear while studying the response of various sources in an elastic medium through integral transform method. In these studies, the inverse Fourier transforms are solved numerically without bothering about the singularities and branch points in the corresponding integrands. References on numerical integration cited in relevant papers do not support such an evaluation but suggest contrary. Approximation of inverse Laplace transform integral into a series is used without following the essential restrictions and assumptions. Volume of the published papers using these dubious procedures has reached to an alarming level. The discussion presented aims to draw the attention of researchers as well as journals so as to stop this menace at the earliest possible.Keywords: Inverse Fourier transforms, inverse Laplace transforms, Romberg integration, improper integral, elastic waves

Solving Linear Volterra – Fredholm Integral Equation of the Second Type Using Linear Programming Method

In this paper, a new technique is offered for solving three types of linear integral equations of the 2nd kind including Volterra-Fredholm integral equations (LVFIE) (as a general case), Volterra integral equations (LVIE) and Fredholm integral equations (LFIE) (as special cases). The new technique depends on approximating the solution to a polynomial of degree and therefore reducing the problem to a linear programming problem(LPP), which will be solved to find the approximate solution of LVFIE. Moreover, quadrature methods including trapezoidal rule (TR), Simpson 1/3 rule (SR), Boole rule (BR), and Romberg integration formula (RI) are used to approximate the integrals that exist in LVFIE. Also, a comparison between those methods is produced. Finally, for more explanation, an algorithm is proposed and applied for testing examples to illustrate the effectiveness of the new technique.

A Valid Scheme to Evaluate Fuzzy Definite Integrals by Applying the CADNA Library

The aim of this paper is to estimate the value of a fuzzy integral and to find the optimal step size and nodes via the stochastic arithmetic. For this purpose, the fuzzy Romberg integration rule is considered as an integration rule, then the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method is applied which is a method to describe the discrete stochastic arithmetic. Also, in order to implement this method, the CADNA (Control of Accuracy and Debugging for Numerical Applications) is applied which is a library to perform the CESTAC method automatically. A theorem is proved to show the accuracy of the results by means of the concept of common significant digits. Then, an algorithm is given to perform the proposed idea on sample fuzzy integrals by computing the Hausdorff distance between two fuzzy sequential results which is considered to be an informatical zero in the termination criterion. Three sample fuzzy integrals are evaluated based on the proposed algorithm to find the optimal number of points and validate the results.