Long-time behavior of numerical solutions to nonlinear fractional ODEs

TL;DR

This paper investigates the long-term behavior of numerical solutions to fractional ODEs, improving existing rates and developing new analytical tools to ensure contractivity and dissipativity are preserved in numerical methods.

Contribution

The paper introduces novel analytical techniques for analyzing fractional backward differential formulas and demonstrates their effectiveness in preserving qualitative properties of solutions.

Findings

Numerical solutions preserve the contractivity and dissipativity rates of continuous solutions.

High order F-BDFs also exhibit contractivity and dissipativity under certain conditions.

Numerical experiments show structure-preserving methods outperform traditional approaches.

Abstract

In this work, we study the long time behaviors, including asymptotic contractivity and dissipativity, of the solutions to several numerical methods for fractional ordinary differential equations (F-ODEs). The existing algebraic contractivity and dissipativity rates of the solutions to the scalar F-ODEs are first improved. In order to study the long time behavior of numerical solutions to fractional backward differential formulas (F-BDFs), two crucial analytical techniques are developed, with the first one for the discrete version of the fractional generalization of the traditional Leibniz rule, and the other for the algebraic decay rate of the solution to a linear Volterra difference equation. By mens of these auxiliary tools and some natural conditions, the solutions to F-BDFs are shown to be contractive and dissipative, and also preserve the exact contractivity rate of the continuous solutions. Two typical F-BDFs, based on the Grunwald-Letnikov formula and L1 method respectively, are studied. For high order F-BDFs, including some second order F-BDFs and $3-\alpha$ order method, their numerical contractivity and dissipativity are also developed under some slightly stronger conditions. Numerical experiments are presented to validate the long time qualitative characteristics of the solutions to F-BDFs, revealing very different decay rates of the numerical solutions in terms of the the initial values between F-ODEs and integer ODEs and demonstrating the superiority of the structure-preserving numerical methods.