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Discrete and Continuous Dynamical Systems - Series B (DCDS-B)
 

Continuous and discrete one dimensional autonomous fractional ODEs
Page number are going to be assigned later 2017

doi:10.3934/dcdsb.2017210      Abstract        References        Full text (598.6K)      

Yuanyuan Feng - Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, United States (email)
Lei Li - Department of Mathematics, Duke University, Durham, NC 27708, United States (email)
Jian-Guo Liu - Department of Physics and Department of Mathematics, Duke University, Durham, NC 27708, United States (email)
Xiaoqian Xu - Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, United States (email)

1 M. Allen, L. Caffarelli and A. Vasseur, A parabolic problem with a fractional time derivative, Archive for Rational Mechanics and Analysis, 221 (2016), 603-630.       
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.       
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.       
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.       
7 K. Diethelm, The Analysis of Fractional Differential Equations: An Application-oriented Exposition Using Differential Operators of Caputo Type, Springer, 2010.       
8 K. Diethelm and N. J. Ford, Analysis of fractional differential equations, Journal of Mathematical Analysis and Applications, 265 (2002), 229-248.       
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.       
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.       
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. Lin, X. Li and C. Xu, Finite difference/spectral approximations for the fractional cable equation, Mathematics of Computation, 80 (2011), 1369-1396.       
16 Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, Journal of Computational Physics, 225 (2007), 1533-1552.       
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.       
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.       
20 J. D. Munkhammar, Riemann-Liouville fractional derivatives and the Taylor-Riemann series, UUDM project report, 7 (2004), 1-18.
21 C. A. Roberts, D. G. Lasseigne and W. E. Olmstead, Volterra equations which model explosion in a diffusive medium, J. Integral Equations Appl., 5 (1993), 531-546.       
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.       
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.       
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.       

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