New enclosure algorithms for the verified solutions of nonlinear Volterra integral equations

Abstract

a b s t r a c tThis paper is concerned with new algorithms which provide the sharp bounds that are guaranteed to contain the exact solutions of nonlinear Volterra integral equations. We develop new enclosure algorithms based on the interval methods which was first introduced by Moore in [24] together with the Taylor polynomials to improve the accuracy of the scheme by reducing the width of interval solutions. The modified methods calculate a priori bound automatically in parallel with the computation of solutions of integral equations. We will show that the accuracy of the proposed algorithms is dependent on the number of interval subdivisions. Some numerical experiments are also included to demonstrate the validity and applicability of the scheme and showing a marked improvement in comparison with the recent existing numerical results.

Piecewise constant bounds for the solution of nonlinear Volterra-Fredholm integral equations

“…Recently, Murashige and Oishi [21] presented numerical verification of solutions of periodic integral equations with a singular kernel and Nekrasov's integral equation. In comparison to Fredholm type, there are a few enclosure methods on solving Volterra integral equations especially in the nonlinear form [8,15].…”