Short memory principle and a predictor-corrector approach for fractional differential equations.

*(English)*Zbl 1121.65128In recent years, fractional differential equations have proven to be a very useful tool for the modeling of various phenomena in physics, engineering, and other sciences. This is, at least in part, due to the fact that they can be used in a very natural way to model processes with memory. However, the fact that memory has to be handled has the disadvantage that the computational complexity of numerical algorithms for fractional differential equations is much higher than for their classical (non-fractional) counterparts. An approach that can be used to modify various numerical methods in such a way that the complexity is reduced substantially without the loss of accuracy has been proposed by N. J. Ford and A. C. Simpson [Numer. Algorithms 26, No. 4, 333–346 (2001; Zbl 0976.65062)].

In the paper under review, the concept of Ford and Simpson [loc. cit.] is combined with additional ideas to improve the performance, in particular for equations of order \(\alpha \in (1,2)\). The implementation of the combination of this principle with the predictor-corrector method of the reviewer, N. J. Ford and A. D. Freed [Numer. Algorithms 36, No. 1, 31–52 (2004; Zbl 1055.65098)] is described in detail.

In the paper under review, the concept of Ford and Simpson [loc. cit.] is combined with additional ideas to improve the performance, in particular for equations of order \(\alpha \in (1,2)\). The implementation of the combination of this principle with the predictor-corrector method of the reviewer, N. J. Ford and A. D. Freed [Numer. Algorithms 36, No. 1, 31–52 (2004; Zbl 1055.65098)] is described in detail.

Reviewer: Kai Diethelm (Braunschweig)

##### MSC:

65R20 | Numerical methods for integral equations |

26A33 | Fractional derivatives and integrals |

65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |

45J05 | Integro-ordinary differential equations |

##### Keywords:

fractional differential equation; Caputo derivative; short memory principle; computational complexity; predictor-corrector method
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\textit{W. Deng}, J. Comput. Appl. Math. 206, No. 1, 174--188 (2007; Zbl 1121.65128)

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##### References:

[1] | Agrawal, O.P.; Tenreiro Machado, J.A.; Sabatier, Jocelyn, Introduction, Nonlinear dynamics, 38, 1-2, (2004) |

[2] | Barkai, E.; Metzler, R.; Klafter, J., From continuous time random walks to the fractional fokker – planck equation, Phys. rev. E, 61, 132-138, (2000) |

[3] | Bonnet, C.; Partington, J.R., Coprime factorizations and stability of fractional differential systems, Systems control lett., 41, 167-174, (2000) · Zbl 0985.93048 |

[4] | Butzer, P.L.; Westphal, U., An introduction to fractional calculus, (2000), World Scientific Singapore · Zbl 0987.26005 |

[5] | Caputo, M., Linear models of dissipation whose Q is almost frequency independent-II, Geophys. J. roy. astron. soc., 13, (1967) |

[6] | Daftardar-Gejji, V.; Babakhani, A., Analysis of a system of fractional differential equations, J. math. anal. appl., 293, 511-522, (2004) · Zbl 1058.34002 |

[7] | Deng, W.H.; Li, C.P., Synchronization of chaotic fractional Chen system, J. phys. soc. jpn., 74, 1645-1648, (2005) · Zbl 1080.34537 |

[8] | Deng, W.H.; Li, C.P., Chaos synchronization of the fractional Lü system, Physica A, 353, 61-72, (2005) |

[9] | W.H. Deng, C.P. Li, J.H. Lü, Stability analysis of linear fractional differential system with multiple time-delays, Nonlinear Dynamics, accepted for publication. |

[10] | Deng, Z.; Singh, V.P.; Bengtsson, L., Numerical solution of fractional advection-dispersion equation, J. hydraul. eng., 130, 422-431, (2004) |

[11] | Diethelm, K.; Ford, N.J., Analysis of fractional differential equations, J. math. anal. appl., 265, 229-248, (2002) · Zbl 1014.34003 |

[12] | Diethelm, K.; Ford, N.J., Numerical solution of the bagley – torvik equation, Bit, 42, 490-507, (2002) · Zbl 1035.65067 |

[13] | Diethelm, K.; Ford, N.J., Multi-order fractional differential equations and their numerical solution, Appl. math. comput., 154, 621-640, (2004) · Zbl 1060.65070 |

[14] | Diethelm, K.; Ford, N.J.; Freed, A.D., A predictor – corrector approach for the numerical solution of fractional differential equations, Nonlinear dynamics, 29, 3-22, (2002) · Zbl 1009.65049 |

[15] | Diethelm, K.; Ford, N.J.; Freed, A.D., Detailed error analysis for a fractional Adams method, Numer. algorithms, 36, 31-52, (2004) · Zbl 1055.65098 |

[16] | Edwards, J.T.; Ford, N.J.; Simpson, A.C., The numerical solution of linear multi-term fractional differential equations: systems of equations, J. math. anal. appl., 148, 401-418, (2002) · Zbl 1019.65048 |

[17] | El-Raheem, Z.F.A., Modification of the application of a contraction mapping method on a class of fractional differential equation, Appl. math. comput., 137, 371-374, (2003) · Zbl 1034.34070 |

[18] | Ford, N.J.; Simpson, A.C., The numerical solution of fractional differential equations: speed versus accuracy, Numer. algorithms, 26, 336-346, (2001) · Zbl 0976.65062 |

[19] | Heaviside, O., Electromagnetic theory, (1971), Chelsea New York · JFM 25.1774.02 |

[20] | Heymans, N.; Podlubny, I., Physical interpretation of initial conditions for fractional differential equations with riemann – liouville fractional derivatives, Rheol. acta, 37, 1-7, (2005) |

[21] | Ichise, M.; Nagayanagi, Y.; Kojima, T., An analog simulation of noninteger order transfer functions for analysis of electrode processes, J. electroanal. chem., 33, 253-265, (1971) |

[22] | Kenneth, S.M.; Bertram, R., An introduction to the fractional calculus and fractional differential equations, (1993), Wiley-Interscience Publication US · Zbl 0789.26002 |

[23] | R.C. Koeller, Application of fractional calculus to the theory of viscoelasticity, J. Appl. Mech. (1984) 229-307. · Zbl 0544.73052 |

[24] | Kusnezov, D.; Bulgac, A.; Dang, G.D., Quantum levy processes and fractional kinetics, Phys. rev. lett., 82, 1136-1139, (1999) |

[25] | Li, C.P.; Deng, W.H.; Xu, D., Chaos synchronization of the Chua system with a fractional order, Physica A, 360, 171-185, (2006) |

[26] | Mainardi, F., Fractional relaxation – oscillation and fractional diffusion-wave phenomena, Chaos solitons fractals, 7, 1461-1477, (1996) · Zbl 1080.26505 |

[27] | Mandelbrot, B., Some noises with \(1 / f\) spectrum, a bridge between direct current and white noise, IEEE trans. inform. theory, 13, 289-298, (1967) · Zbl 0148.40507 |

[28] | D. Matignon, Stability results for fractional differential equations with applications to control processing, in: Computational Engineering in Systems and Application Multiconference, IMACS, IEEE-SMC, Lille, France, vol. 2, 1996, pp. 963-968. |

[29] | Matignon, D., Observer-based controllers for fractional differential equations, (), 4967-4972 |

[30] | Podlubny, I., Fractional differential equations, (1999), Academic Press New York · Zbl 0918.34010 |

[31] | Podlubny, I., Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. calculus appl. anal., 5, 367-386, (2002) · Zbl 1042.26003 |

[32] | Sugimoto, N., Burgers equation with a fractional derivative: hereditary effects on nonlinear acoustic waves, J. fluid mech., 225, 631-653, (1991) · Zbl 0721.76011 |

[33] | Sun, H.H.; Abdelwahab, A.A.; Onaral, B., Linear approximation of transfer function with a pole of fractional order, IEEE trans. automat. control, 29, 441-444, (1984) · Zbl 0532.93025 |

[34] | Zaslavsky, G.M., Chaos, fractional kinetics, and anomalous transport, Phys. rep., 371, 461-580, (2002) · Zbl 0999.82053 |

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