Exponential quadrature rules for problems with time-dependent fractional source

TL;DR

This paper introduces new exponential quadrature rules tailored for stiff linear differential equations with time-dependent fractional sources, achieving higher order accuracy and demonstrating effectiveness through numerical experiments.

Contribution

The paper develops a novel class of exponential quadrature rules for fractional sources, improving convergence order and expressing solutions via fractional Mittag-Leffler functions.

Findings

New quadrature rules can reach order 1 + νr with proper collocation points.

Methods are expressed using fractional Mittag-Leffler functions.

Numerical experiments confirm theoretical convergence and effectiveness.

Abstract

In this manuscript, we propose newly-derived exponential quadrature rules for stiff linear differential equations with time-dependent fractional sources in the form $h(t^r)$, with $0<r<1$ and $h$ a sufficiently smooth function. To construct the methods, the source term is interpolated at $\nu$ collocation points by a suitable non-polynomial function, yielding to time marching schemes that we call Exponential Quadrature Rules for Fractional sources (EQRF$\nu$). The error analysis is done in the framework of strongly continuous semigroups. Compared to classical exponential quadrature rules, which in our case of interest converge with order $1+r$ at most, we prove that the new methods may reach order $1+\nu r$ for proper choices of the collocation points. We also show that the proposed integrators can be written in terms of special instances of the Mittag--Leffler functions that we call fractional $\varphi$ functions. Several numerical experiments demonstrate the theoretical findings and highlight the effectiveness of the approach.