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Fractional high order methods for the nonlinear fractional ordinary differential equation. (English) Zbl 1118.65079

The paper begins by referring to applications of fractional order equations, along with a brief summary of the main results achieved for this type of equation in the last decade.
The authors consider the nonlinear fractional-order order differential equation (NFOODE), \(_0D_t^\alpha y(t)=f(y,t), (t>0), n-1<\alpha \leq n, y^{(i)}(0)=y_0^{(i)}, i=0, 1, 2,\dots, n-1\) where \(f(y,t)\) satisfies the condition \(| f(y_1,t)-f(y_2,t)| \leq L| y_1-y_2| \) in \(t\in [0,T]\).
Existence and uniqueness theorems for the NFOODE, by K. Diethelm and N. J. Ford [J. Math. Anal. Appl. 265, No. 2, 229–248 (2002; Zbl 1014.34003)], are stated. High order fractional linear multi-step methods (\(p\)-HOFLMSM) are introduced. Definitions pertaining to their consistency and stability are stated. New results relating to the consistence, convergence and stability of these methods are presented and proved. The paper concludes with numerical examples which demonstrate the computational efficiency of the \(p\)-HOFLMSM.
Reviewer: Pat Lumb (Chester)

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
26A33 Fractional derivatives and integrals
34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)
34A34 Nonlinear ordinary differential equations and systems
65R20 Numerical methods for integral equations
45J05 Integro-ordinary differential equations
45G10 Other nonlinear integral equations

Citations:

Zbl 1014.34003
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References:

[1] Bagley, R. L.; Calico, R. A., Fractional order state equations for the control of viscoelastic strucures, J. Guid. Control Dyn., 14, 2 (1999)
[2] Benson, D. A.; Wheatcraft, S. W.; Meerschaert, M. M., Application of a fractional advection-despersion equation, Water Resour. Res., 36, 6, 1403-1412 (2000)
[3] Benson, D. A.; Wheatcraft, S. W.; Meerschaert, M. M., The fractional-order governing equation of Levy motion, Water Resour. Res., 36, 6, 1413-1423 (2000)
[4] Diethelm, K.; Ford, N. J., Analysis of fractional differential equations, J. Math. Anal. Appl., 265, 220-248 (2002) · Zbl 1014.34003
[5] Diethelm, K.; Ford, N. J.; Freed, A. D., Details error analysis for a fractional Adams method, Numer. Algorithms, 36, 31-52 (2004) · Zbl 1055.65098
[6] Djrbashian, M. M.; Nersesian, A. B., Fractional derivatives and the cauchy problem for differential equations of fractional order, Izv. Acad. Nauk Armjanskoy SSR, 3, 1, 3-29 (1968), (in Russian)
[7] Huang, F.; Liu, F., The time fractional diffusion and advection-dispersion equation, ANZIAM J., 46, 317-330 (2005) · Zbl 1072.35218
[8] Ichise, M.; Nagayanagi, Y.; Kojima, T., An analog simulation of non-integer order transfer functions for analysis of electrode processes, J. Electroanal. Chem. Interfacial Electrochem., 33, 253-265 (1971)
[9] Kilbas, A. A.; Saigo, M.; Saxena, R. K., Solutions of Volterra integro-differential equations with generalised Mittag-Leffler function in the kernels, J. Integral Equations Appl., 14, 4, 377-396 (2002) · Zbl 1041.45011
[10] Koeller, R. C., Polynomial operators, Stieltjes convolution, and fractional colculus in hereditary methabics, Acta Mech., 58, 299-307 (1984) · Zbl 0544.73052
[11] Koeller, R. C., Application of fractional calculus to the theory of viscoelasticity, J. Appl. Mech., 51, 299-307 (1984) · Zbl 0544.73052
[12] Li, R., Numerical Solution of Differential Equation (1996), High Education Press: High Education Press Beijing, China
[13] Lin, R.; Liu, F., High order approximations for the fractional ordinary differential equation with initial value problem, J. Xiamen Univ. Natur. Sci., 43, 1, 25-30 (2004)
[14] Lin, R.; Liu, F., Analysis of fractional-order numerical method for the fractional relaxation equation, Comput. Mech. (CD-ROM), ID-362 (2004)
[15] Liu, F.; Anh, V.; Turner, I.; Zhuang, P., Time fractional advection-dispersion equation, J. Appl. Math. Comput., 233-246 (2003) · Zbl 1068.26006
[16] Liu, F.; Anh, V.; Turner, I., Numerical solution of space fractional Fokker-Planck equation, J. Comput. Appl. Math., 166, 209-219 (2004) · Zbl 1036.82019
[17] Liu, F.; Anh, V.; Turner, I.; Zhuang, P., Numerical simulation for solute transport in fractal porous media, ANZIAM J., 45, E, 461-473 (2004) · Zbl 1123.76363
[18] Liu, F.; Shen, S.; Anh, V.; Turner, I., Analysis of a discrete non-Markovian random walk approximation for the time fractional diffusion equation, ANZIAM J., 46, E, 488-504 (2005) · Zbl 1082.60511
[19] Lubich, Ch., Discretized fractional calculus, SIAM J. Math. Anal., 17, 3, 704-719 (1986) · Zbl 0624.65015
[20] Mandelbrot, B., Some noises with \(1 / f\) spectrum, a bridgre between direct current and white noise, IEEE Trans. Inform. Theory, 13, 2, 289-298 (1967) · Zbl 0148.40507
[21] Mark, R. J.; Hall, M. W., Differentegral interpolation from a bandlimited signal’s samples, IEEE Trans. Acoust. Speech Signal Process., 29, 872-877 (1981) · Zbl 0525.65005
[22] Miller, K. S.; Boss, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), Wiley: Wiley New Tork, USA
[23] Oldham, K. B.; Spanier, J. C., The Fractional Calculus (1974), Academic Press: Academic Press San Diego, CA, USA
[24] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press San Diego, CA, USA · Zbl 0918.34010
[25] Shen, S.; Liu, F., A computational effective method for fractional order Bagley-Torvik equation, J. Xiamen Univ. Natur. Sci., 45, 3, 306-311 (2004) · Zbl 1122.65368
[26] Shen, S.; Liu, F., Error analysis of an explicit finite difference approximation for the space fractional diffusion equation with insulated ends, ANZIAM J., 46, E, 871-887 (2005)
[27] Skaar, S. B.; Michel, A. N.; Miller, R. K., Stability of viscoelastic control systems, IEEE Trans. Automat. Control, 3, 4, 348-357 (1988) · Zbl 0641.93051
[28] Sun, H. H.; Onaral, N.; Tsao, Y., Application of position reality principle to metal electrode linear polarization phenomena, IEEE Trans. Ciomed. Eng., 31, 10, 664-674 (1984)
[29] Sun, H. H.; Abdelwahab, A. A.; Onaral, B., Linear approximation of transfer function with a pole of fractional order, IEEE Trans. Automat. Control, 29, 5, 441-444 (1984) · Zbl 0532.93025
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