The Finite Difference Methods for Fractional Ordinary Differential Equations

REFERENCES

• K. Miller and B. Ross ( 1993 ). An Introduction to the Fractional Calculus and Fractional Differential Equations . Wiley , New York .
   Google Scholar
• I. Podlubny ( 1999 ). Fractional Differential Equations . Acdemic Press , San Dieg .
   Google Scholar
• G. M. Zaslavsky ( 2002 ). Chaos, fractional kinetics, and anomalous transport . Phys. Rep. 371 : 461 – 580 .
   Web of Science ®Google Scholar
• K. B. Oldham and J. Spanier ( 2006 ). The Fractional Calculus . Academic Press , New York .
   Google Scholar
• K. Diethelm and N. J. Ford ( 2002 ). Analysis of fractional differential equations . J. Math. Anal. Appl. 265 : 229 – 248 .
   Web of Science ®Google Scholar
• K. Diethelm , N. J. Ford , and A. D. Freed ( 2004 ). Detailed error analysis for a fractional Adams method . Numer. Algor. 36 : 31 – 52 .
   Web of Science ®Google Scholar
• K. Diethelm , N. J. Ford , A. D. Freed , and Yu. Luchko ( 2005 ). Algorithms for the fractional calculus: a selection of numerical methods . Comput. Methods Appl. Mech. Engrg. 194 : 743 – 773 .
   Web of Science ®Google Scholar
• C. P. Li and C. X. Tao ( 2009 ). On the fractional Adams method . Comput. Math. Appl. 58 : 1573 – 1588 .
   Web of Science ®Google Scholar
• C. P. Li , A. Chen , and J. J. Ye ( 2011 ). Numerical approaches to fractional calculus and fractional ordinary differential equation . J. Comput. Phys. 230 : 3352 – 3368 .
   Web of Science ®Google Scholar
• Z. Odibat and S. Momani ( 2008 ). An algorithm for the numerical solution of differential equations of fractional order . J. Appl. Math. Informatics 26 : 15 – 27 .
   Google Scholar
• C. Yang and F. Liu ( 2006 ). A computationally effective Predictor-Corrector method for simulating fractional-order dynamical control system . ANZIAM J. 47 : C137 – C153 .
   Google Scholar
• C. Yin , F. Liu , and V. Anh ( 2007 ). Numerical simulation of the nonlinear fractional dynamical systems with fractional damping for the extensible and inextensible pendulum . J. Algorithm Comput. Tech. 1 : 427 – 447 .
   Google Scholar
• W. H. Deng ( 2007 ). Numerical algorithm for the time fractional Fokker–Planck equation . J. Comput. Phys. 227 : 1510 – 1522 .
   Web of Science ®Google Scholar
• F. Liu and K. Burrage ( 2011 ). Novel techniques in parameter estimation for fractional dynamical models arising from biological systems . Comput. Math. Appl. 62 : 822 – 833 .
   Web of Science ®Google Scholar
• I. Podlubny , A. Chechkin , T. Skovranek , Y. Q. Chen , and B. Vinagre ( 2009 ). Matrix approach to discrete fractional calculus II: partial fractional differential equations . J. Comput. Phys. 228 : 3137 – 3153 .
   Web of Science ®Google Scholar
• R. Lin and F. Liu ( 2007 ). Fractional high order methods for the nonlinear fractional ordinary differential equation . Nonlinear Analysis 66 : 856 – 869 .
   Web of Science ®Google Scholar
• C. Lubich ( 1986 ). Discretized fractional calculus . SIAM J. Math. Anal. 17 : 704 – 719 .
   Web of Science ®Google Scholar
• T. Tang ( 1993 ). A finite difference scheme for partial integro-differential equations with weakly singular kernel . Appl. Numer. Math. 11 : 309 – 319 .
   Web of Science ®Google Scholar
• Q. Sheng and T. Tang ( 1995 ). Optimal convergence of an Euler and finite difference method for nonlinear partial integro-differential equations . Math. Comput. Model. 21 : 1 – 11 .
   Web of Science ®Google Scholar
• C. T. H. Baker ( 2002 ). A perspective on the numerical treatment of Volterra equations . J. Comput. Appl. Math. 125 : 217 – 249 .
   Web of Science ®Google Scholar
• K. Diethelm , N. J. Ford , A. D. Freed , and M. Weilbeer ( 2006 ). Pitfalls in fast numerical solvers for fractional differential equations . J. Comput. Appl. Math. 186 : 482 – 503 .
   Web of Science ®Google Scholar
• L. Galeone and R. Garrappa ( 2009 ). Explicit methods for fractional differential equations and their stability properties . J. Comput. Appl. Math. 228 : 548 – 560 .
   Web of Science ®Google Scholar
• R. Garrappa ( 2009 ). On some explicit Adams multistep methods for fractional differential equations . J. Comput. Appl. Math. 229 : 392 – 399 .
   Web of Science ®Google Scholar
• C. Lubich ( 1986 ). A stability analysis of convolution quadratures for Abel-Volterra integral equations . IMA Journal of Numerical Analysis 6 : 87 – 101 .
   Web of Science ®Google Scholar
• R. Garrappa (2010). On linear stability of predictor-corrector algorithms for fractional differential equations. Int. J. Comput. Math. 87:2281–2290.
   Web of Science ®Google Scholar
• Y. Huang and T. Läu ( 2003 ). A generalization of discrete Gronwall inequality and its application to weakly singular Volterra integral equation of the second kind . J. Math. Anal. Appl. 282 : 56 – 62 .
   Web of Science ®Google Scholar
• J. Dixon ( 1985 ). On the order of the error in discretization methods for weakly singular second kind non-smooth solutions . BIT 25 : 623 – 634 .
   Google Scholar
• H. P. Ma and W. W. Sun ( 2001 ). Optimal error estimates of the Legendre-Petrov-Galerkin method for the Korteweg-de Vries equation . SIAM J. Numer. Anal. 39 : 1380 – 1394 .
   Web of Science ®Google Scholar