Continuous and discrete one dimensional autonomous fractional ODEs

Abstract

In this paper, we study 1D autonomous fractional ODEs D γ c u = f (u), 0 < γ < 1, where u : [0, ∞) → R is the unknown function and D γ c 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 (γ → 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 Grönwall inequalities for discretized equations, and perform some numerical simulations to confirm our analysis.2010 Mathematics Subject Classification. Primary 34A08.

A Discretization of Caputo Derivatives with Application to Time Fractional SDEs and Gradient Flows

“…However, for H = 0.8 or α = 0.4, the mean square distance has a rapid drop at the early stage, but then the memory lingers for long time so that the convergence is very slow. This is also the case for normal fractional ODE [37]. In fact, in Fig.…”

“…By the theory in [37], u(·) is nondecreasing and thus R n ≤ 0. Consequently, applying Theorem 3.1 (4), we have u(t n ) ≤ u n .…”

“…If one can show that ξ n is bounded, then one can improve the orders. Lastly, we give a quick glimpse of the case H = R d , φ ∈ C 1 (R d ) (instead of requiring ∇φ to be Lipschitz as in [5,37]) so that (1.5) becomes the FODE: D α c u = −∇φ(u).…”

“…In [35,36], some spectral methods have been developed for fractional differential equations. Moreover, in [37], some comparison principles for the discrete fractional equations have been established. Unfortunately, using these discretizations to study the fractional SDE and time fractional gradient flows is not appropriate because the time continuous problems are not well understood yet.…”

On asymptotic properties of solutions to fractional differential equations

“…Proof. Assume first without loss of generality that one of the inequalities in (17) and (18) is strict, say C D α 0+ m 2 (t) < L(m 2 (t)) and m 2 (0) < m 0 ≤ m 1 (0). We will show that m 2 (t) < m 1 (t) for all t ∈ [0, T ].…”