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Continuous and discrete one dimensional autonomous fractional ODEs

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  • In this paper, we study 1D autonomous fractional ODEs $D_c^{γ}u=f(u), 0< γ <1$, where $u: [0,∞) \to \mathbb{R}$ is the unknown function and $D_c^{γ}$ 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 ($γ \to 0$), 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 Grönwall inequalities for discretized equations, and perform some numerical simulations to confirm our analysis.

    Mathematics Subject Classification: Primary: 34A08.


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  • Figure 1.  Solution curves for $f(u)=Au^2$ with $u(0)=u_0$. (a). $A=1, u_0=0.12 ,\gamma=0.6$; (b). $A=1, u_0=1.2, \gamma=0.6$

    Figure 2.  Blow-up time versus $\gamma$. The red solid line shows the numerical results of the blow-up time. The blue dotted line is the estimated upper bound and the green dashed line is the lower bound, provided by Theorem 5.2. (a). $A=1, u_0=0.12$; (b). $A=1, u_0=1.2$

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