Error analysis of a spectrally accurate Volterra-transformation method for solving 1-D Fredholm integro-differential equations

Abstract

This is a repository copy of Error analysis of a spectrally accurate Volterra-transformation method for solving 1-D Fredholm integro-differential equations.

A priori Nyström-method error bounds in approximate solutions of 1-D Fredholm integro-differential equations

“…The primary aim of the present paper is to develop and to implement a computational method that yields explicitly computable and spectrally accurate a priori error predictions for the numerical solution of (1) using proves a convergence theorem for the iterative method, but does not explicitly analyse errors. By contrast, the authors' recent companion paper [13] extends the IDE-to-FIE approach of [18] to obtain the first-ever spectrally accurate a priori error estimates of numerical solutions of (1).…”

“…Accordingly, it is hoped that the present work will augment the existing literature in a novel and useful way by providing a method for computing error bounds for the (Nyström) numerical solution of (1) not only a priori, but also, as in [13], explicitly in terms of only the numerical solution itself. As it transpires, the predicted bounds are spectrally accurate for a diverse class of problems, adding to the merit of the approach.…”

“…and recall that the authors' related method [13] can be applied only when m = 1. Using (7) and (8), the approximate solution u N (x) of (1) satisfies the discretised equation…”

“…In the terminology of [13], (62) is deemed to be a "smooth" problem: both computed and predicted errors for more challenging so-called "Runge", "steep" and "oscillatory" problems [13, Table 1 and Figure 5] are less accurate than those for the "smooth" problem. However, since such kernels are implemented and comprehensively analysed in [13], they are not pursued herein.…”

Solving Fractional Fredholm Integro-Differential Equations by Laguerre Polynomials

“…Integro-differential equations play an important role in the modeling of numerous of physical phenomena from science and engineering. Hence, searching the exact and approximate solutions of integro-differential equations have attracted appreciable attention for scientists and applied mathematicians (Dzhumabaev 2018;Fairbairn & Kelmanson 2018;Hendi & Al-Qarni 2017;Kürkçü et al 2017;Rahimkhani et al 2017;Rohaninasab et al 2018;Yüzbaşı & Karaçayır 2017). The fractional calculus represents a powerful tool in applied mathematics to study a myriad of problems from different fields of science and engineering, with many break-through results found in mathematical physics, finance, hydrology, biophysics, thermodynamics, control theory, statistical mechanics, astrophysics, cosmology and bioengineering (Abbas et al 2015).…”