October  2018, 23(8): 3109-3135. doi: 10.3934/dcdsb.2017210

Continuous and discrete one dimensional autonomous fractional ODEs

1. 

Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA

2. 

Department of Mathematics, Duke University, Durham, NC 27708, USA

Received  March 2017 Revised  July 2017 Published  October 2018 Early access  September 2017

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.

Citation: Yuanyuan Feng, Lei Li, Jian-Guo Liu, Xiaoqian Xu. Continuous and discrete one dimensional autonomous fractional ODEs. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3109-3135. doi: 10.3934/dcdsb.2017210
References:
[1]

M. AllenL. Caffarelli and A. Vasseur, A parabolic problem with a fractional time derivative, Archive for Rational Mechanics and Analysis, 221 (2016), 603-630.  doi: 10.1007/s00205-016-0969-z.

[2]

N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, Courier Corporation, 1975.

[3]

H. Brunner and Z. W. Yang, Blow-up behavior of Hammerstein-type Volterra integral equations, J. Integral Equations Appl, 24 (2012), 487-512.  doi: 10.1216/JIE-2012-24-4-487.

[4]

M. Caputo, Linear models of dissipation whose Q is almost frequency independent-Ⅱ, Geophysical Journal International, 13 (1967), 529-539. 

[5]

P. Clément and J. A. Nohel, Abstract linear and nonlinear Volterra equations preserving positivity, SIAM Journal on Mathematical Analysis, 10 (1979), 365-388.  doi: 10.1137/0510035.

[6]

P. Clément and J. A. Nohel, Asymptotic behavior of solutions of nonlinear Volterra equations with completely positive kernels, SIAM Journal on Mathematical Analysis, 12 (1981), 514-535.  doi: 10.1137/0512045.

[7]

K. Diethelm, The Analysis of Fractional Differential Equations: An Application-oriented Exposition Using Differential Operators of Caputo Type, Springer, 2010. doi: 10.1007/978-3-642-14574-2.

[8]

K. Diethelm and N. J. Ford, Analysis of fractional differential equations, Journal of Mathematical Analysis and Applications, 265 (2002), 229-248.  doi: 10.1006/jmaa.2000.7194.

[9]

J. Dixon and S. McKee, Weakly singular discrete Grönwall inequalities, ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 66 (1986), 535-544. 

[10]

C. S. Drapaca and S. Sivaloganathan, A fractional model of continuum mechanics, Journal of Elasticity, 107 (2012), 105-123.  doi: 10.1007/s10659-011-9346-1.

[11]

R. Gorenflo and F. Mainardi, Fractional Calculus, Springer, 1997.

[12]

H. J. Haubold, A. M. Mathai and R. K. Saxena, Mittag-Leffler functions and their applications, Journal of Applied Mathematics, 2011 (2001), Art. ID 298628, 51 pp. doi: 10.1155/2011/298628.

[13]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, 204 (2006), xvi+523 pp, URL http://www.sciencedirect.com/science/article/pii/S0304020806800010.

[14]

L. Li and J. -G. Liu, A generalized definition of Caputo derivatives and its application to fractional ODEs, arXiv preprint, arXiv: 1612. 05103v2.

[15]

Y. LinX. Li and C. Xu, Finite difference/spectral approximations for the fractional cable equation, Mathematics of Computation, 80 (2011), 1369-1396.  doi: 10.1090/S0025-5718-2010-02438-X.

[16]

Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, Journal of Computational Physics, 225 (2007), 1533-1552.  doi: 10.1016/j.jcp.2007.02.001.

[17]

F. Mainardi and R. Gorenflo, On Mittag-Leffler-type functions in fractional evolution processes, Journal of Computational and Applied Mathematics, 118 (2000), 283-299.  doi: 10.1016/S0377-0427(00)00294-6.

[18]

F. Mainardi, P. Paradisi and R. Gorenflo, Probability distributions generated by fractional diffusion equations, arXiv preprint, arXiv: 0704.0320.

[19]

R. K. Miller and A. Feldstein, Smoothness of solutions of Volterra integral equations with weakly singular kernels, SIAM Journal on Mathematical Analysis, 2 (1971), 242-258.  doi: 10.1137/0502022.

[20]

J. D. Munkhammar, Riemann-Liouville fractional derivatives and the Taylor-Riemann series, UUDM project report, 7 (2004), 1-18. 

[21]

C. A. RobertsD. G. Lasseigne and W. E. Olmstead, Volterra equations which model explosion in a diffusive medium, J. Integral Equations Appl., 5 (1993), 531-546.  doi: 10.1216/jiea/1181075776.

[22]

C. A. Roberts and W. E. Olmstead, Growth rates for blow-up solutions of nonlinear Volterra equations, Quarterly of Applied Mathematics, 54 (1996), 153-159.  doi: 10.1090/qam/1373844.

[23]

M. Taylor, Remarks on fractional diffusion equations, Preprint.

[24]

D. G. Weis, Asymptotic behavior of some nonlinear Volterra integral equations, Journal of Mathematical Analysis and Applications, 49 (1975), 59-87.  doi: 10.1016/0022-247X(75)90162-6.

[25]

D. V. Widder, Laplace Transform (PMS-6), Princeton University Press, 2015.

[26]

G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Physics Reports, 371 (2002), 461-580.  doi: 10.1016/S0370-1573(02)00331-9.

show all references

References:
[1]

M. AllenL. Caffarelli and A. Vasseur, A parabolic problem with a fractional time derivative, Archive for Rational Mechanics and Analysis, 221 (2016), 603-630.  doi: 10.1007/s00205-016-0969-z.

[2]

N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, Courier Corporation, 1975.

[3]

H. Brunner and Z. W. Yang, Blow-up behavior of Hammerstein-type Volterra integral equations, J. Integral Equations Appl, 24 (2012), 487-512.  doi: 10.1216/JIE-2012-24-4-487.

[4]

M. Caputo, Linear models of dissipation whose Q is almost frequency independent-Ⅱ, Geophysical Journal International, 13 (1967), 529-539. 

[5]

P. Clément and J. A. Nohel, Abstract linear and nonlinear Volterra equations preserving positivity, SIAM Journal on Mathematical Analysis, 10 (1979), 365-388.  doi: 10.1137/0510035.

[6]

P. Clément and J. A. Nohel, Asymptotic behavior of solutions of nonlinear Volterra equations with completely positive kernels, SIAM Journal on Mathematical Analysis, 12 (1981), 514-535.  doi: 10.1137/0512045.

[7]

K. Diethelm, The Analysis of Fractional Differential Equations: An Application-oriented Exposition Using Differential Operators of Caputo Type, Springer, 2010. doi: 10.1007/978-3-642-14574-2.

[8]

K. Diethelm and N. J. Ford, Analysis of fractional differential equations, Journal of Mathematical Analysis and Applications, 265 (2002), 229-248.  doi: 10.1006/jmaa.2000.7194.

[9]

J. Dixon and S. McKee, Weakly singular discrete Grönwall inequalities, ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 66 (1986), 535-544. 

[10]

C. S. Drapaca and S. Sivaloganathan, A fractional model of continuum mechanics, Journal of Elasticity, 107 (2012), 105-123.  doi: 10.1007/s10659-011-9346-1.

[11]

R. Gorenflo and F. Mainardi, Fractional Calculus, Springer, 1997.

[12]

H. J. Haubold, A. M. Mathai and R. K. Saxena, Mittag-Leffler functions and their applications, Journal of Applied Mathematics, 2011 (2001), Art. ID 298628, 51 pp. doi: 10.1155/2011/298628.

[13]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, 204 (2006), xvi+523 pp, URL http://www.sciencedirect.com/science/article/pii/S0304020806800010.

[14]

L. Li and J. -G. Liu, A generalized definition of Caputo derivatives and its application to fractional ODEs, arXiv preprint, arXiv: 1612. 05103v2.

[15]

Y. LinX. Li and C. Xu, Finite difference/spectral approximations for the fractional cable equation, Mathematics of Computation, 80 (2011), 1369-1396.  doi: 10.1090/S0025-5718-2010-02438-X.

[16]

Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, Journal of Computational Physics, 225 (2007), 1533-1552.  doi: 10.1016/j.jcp.2007.02.001.

[17]

F. Mainardi and R. Gorenflo, On Mittag-Leffler-type functions in fractional evolution processes, Journal of Computational and Applied Mathematics, 118 (2000), 283-299.  doi: 10.1016/S0377-0427(00)00294-6.

[18]

F. Mainardi, P. Paradisi and R. Gorenflo, Probability distributions generated by fractional diffusion equations, arXiv preprint, arXiv: 0704.0320.

[19]

R. K. Miller and A. Feldstein, Smoothness of solutions of Volterra integral equations with weakly singular kernels, SIAM Journal on Mathematical Analysis, 2 (1971), 242-258.  doi: 10.1137/0502022.

[20]

J. D. Munkhammar, Riemann-Liouville fractional derivatives and the Taylor-Riemann series, UUDM project report, 7 (2004), 1-18. 

[21]

C. A. RobertsD. G. Lasseigne and W. E. Olmstead, Volterra equations which model explosion in a diffusive medium, J. Integral Equations Appl., 5 (1993), 531-546.  doi: 10.1216/jiea/1181075776.

[22]

C. A. Roberts and W. E. Olmstead, Growth rates for blow-up solutions of nonlinear Volterra equations, Quarterly of Applied Mathematics, 54 (1996), 153-159.  doi: 10.1090/qam/1373844.

[23]

M. Taylor, Remarks on fractional diffusion equations, Preprint.

[24]

D. G. Weis, Asymptotic behavior of some nonlinear Volterra integral equations, Journal of Mathematical Analysis and Applications, 49 (1975), 59-87.  doi: 10.1016/0022-247X(75)90162-6.

[25]

D. V. Widder, Laplace Transform (PMS-6), Princeton University Press, 2015.

[26]

G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Physics Reports, 371 (2002), 461-580.  doi: 10.1016/S0370-1573(02)00331-9.

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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