Polynomial spline collocation method for solving weakly regular Volterra integral equations of the first kind

TL;DR

This paper introduces a polynomial spline collocation method with Gauss-type quadrature for solving weakly regular Volterra integral equations of the first kind, including accuracy estimates and stochastic arithmetic for optimization.

Contribution

The paper presents a novel polynomial spline collocation approach with stochastic arithmetic integration for improved solution accuracy of weakly regular Volterra equations.

Findings

Method achieves high accuracy in numerical examples.

Stochastic arithmetic helps optimize method parameters.

Efficient solution for weakly regular Volterra equations.

Abstract

The polynomial spline collocation method is proposed for solution of Volterra integral equations of the first kind with special piecewise continuous kernels. The Gauss-type quadrature formula is used to approximate integrals during the discretisation of the proposed projection method. The estimate of accuracy of approximate solution is obtained. Stochastic arithmetics is also used based on the Contr\^{o}le et Estimation Stochastique des Arrondis de Calculs (CESTAC) method and the Control of Accuracy and Debugging for Numerical Applications (CADNA) library. Applying this approach it is possible to find optimal parameters of the projective method. The numerical examples are included to illustrate the efficiency of proposed novel collocation method.