Multistep collocation methods for weakly singular Volterra integral equations with application to fractional differential equations

TL;DR

This paper develops and analyzes multistep collocation methods for solving weakly singular Volterra integral equations, demonstrating their convergence, superconvergence, and application to fractional differential equations through numerical examples.

Contribution

It introduces new multistep collocation methods tailored for weakly singular kernels and provides convergence analysis and superconvergence results.

Findings

Methods achieve specific convergence orders.

Superconvergence phenomena are identified.

Numerical examples validate the methods' effectiveness.

Abstract

We discuss the application of multistep collocation methods to Volterra integral equations which contain a weakly singular kernel $(t-\tau)^{\alpha-1}$ with $0 <\alpha <1.$ Convergence orders of the methods are determined and their superconvergence is also analyzed. The paper closes with numerical examples and application to fractional differential equations.