A numerical solution of one class of Volterra integral equations of the first kind in terms of the machine arithmetic features

TL;DR

This paper presents a numerical approach for solving a specific class of Volterra integral equations of the first kind, emphasizing the impact of machine arithmetic features on solution accuracy.

Contribution

It introduces a computational method that accounts for floating point significand limitations in solving Volterra equations derived from heat conduction problems.

Findings

Kernel calculation at fixed significand length

Implementation of product integration and midpoint rule methods

Demonstration of systematic error effects in numerical solutions

Abstract

The research is devoted to a numerical solution of the Volterra equations of the first kind that were obtained using the Laplace integral transforms for solving the equation of heat conduction. The paper consists of an introduction and two sections. The first section deals with the calculation of kernels from the respective integral equations at a fixed length of the significand in the floating point representation of a real number. The PASCAL language was used to develop the software for the calculation of kernels, which implements the function of tracking the valid digits of the significand. The test examples illustrate the typical cases of systematic error accumulation. The second section presents the results obtained from the computational algorithms which are based on the product integration method and the midpoint rule. The results of test calculations are presented to demonstrate the performance of the difference methods.