Continuous and discrete one dimensional autonomous fractional ODEs

TL;DR

This paper investigates one-dimensional autonomous fractional ODEs with Caputo derivatives, establishing solution monotonicity, comparison principles, asymptotic behaviors, and numerical validations, revealing how memory effects influence solution blow-up and decay.

Contribution

It provides new monotonicity, comparison principles, and detailed asymptotic analysis for fractional ODEs with generalized Caputo derivatives, including bounds on blow-up times and decay rates.

Findings

Blow-up time tends to zero as memory effect strengthens for large initial values.

Solutions decay more slowly than classical derivatives when A<0, p>1.

Numerical simulations confirm theoretical results.

Abstract

In this paper, we study 1D autonomous fractional ODEs $D_c^{\gamma}u=f(u), 0< \gamma <1$, where $u: [0,\infty)\mapsto\mathbb{R}$ is the unknown function and $D_c^{\gamma}$ is the generalized Caputo derivative introduced by Li and Liu ( arXiv:1612.05103). Based on the existence and uniqueness theorem and regularity results in previous work, we show the monotonicity of solutions to the autonomous fractional ODEs and several versions of comparison principles. We also perform a detailed discussion of the asymptotic behavior for $f(u)=Au^p$. In particular, based on an Osgood type blow-up criteria, we find relatively sharp bounds of the blow-up time in the case $A>0, p>1$. These bounds indicate that as the memory effect becomes stronger ($\gamma\to 0$), if the initial value is big, the blow-up time tends to zero while if the initial value is small, the blow-up time tends to infinity. In the case $A<0, p>1$, we show that the solution decays to zero more slowly compared with the usual derivative. Lastly, we show several comparison principles and Gr\"onwall inequalities for discretized equations, and perform some numerical simulations to confirm our analysis.