Control of accuracy on Taylor-collocation method to solve the weakly regular Volterra integral equations of the first kind by using the CESTAC method

TL;DR

This paper introduces a stochastic arithmetic approach using the CESTAC method to enhance the accuracy control of the Taylor-collocation method for solving weakly regular Volterra integral equations of the first kind, replacing traditional error measures.

Contribution

It develops a new accuracy control technique based on stochastic arithmetic and proves a convergence theorem for the Taylor-collocation method using CESTAC, with practical numerical examples.

Findings

The CESTAC method effectively controls accuracy in the Taylor-collocation method.

The new termination criterion improves convergence assessment.

Numerical results demonstrate the method's efficiency and accuracy.

Abstract

Finding the optimal parameters and functions of iterative methods is among the main problems of the Numerical Analysis. For this aim, a technique of the stochastic arithmetic (SA) is used to control of accuracy on Taylor-collocation method for solving first kind weakly regular integral equations (IEs). Thus, the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method is applied and instead of usual mathematical softwares the CADNA (Control of Accuracy and Debugging for Numerical Applications) library is used. Also, the convergence theorem of presented method is illustrated. In order to apply the CESTAC method we will prove a theorem that it will be our licence to use the new termination criterion instead of traditional absolute error. By using this theorem we can show that number of common significant digits (NCSDs) between two successive approximations are almost equal to NCSDs between exact and numerical solution. Finally, some examples are solved by using the Taylor-collocation method based on the CESTAC method. Several tables of numerical solutions based on the both arithmetics are presented. Comparison between number of iterations are demonstrated by using the floating point arithmetic (FPA) for different values of $\varepsilon$.