Research Papers

A Modified Multi-Step Differential Transform Method for Solving Fractional Dynamic Systems

[+] Author and Article Information
Min Shi

e-mail: 15996301586@163.com

Zaihua Wang

e-mail: zhwang@nuaa.edu.cn
State Key Laboratory of Mechanics and Control of Mechanical Structures,
Nanjing University of Aeronautics and Astronautics,
210016 Nanjing, China

Maolin Du

Institute of Science,
PLA University of Science and Technology,
211101 Nanjing, China
e-mail: dumaolindml@yahoo.com.cn

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received October 23, 2011; final manuscript received March 28, 2012; published online June 14, 2012. Assoc. Editor: J. A. Tenreiro Machado.

J. Comput. Nonlinear Dynam 8(1), 011008 (Jun 14, 2012) (8 pages) Paper No: CND-11-1178; doi: 10.1115/1.4006572 History: Received October 23, 2011; Revised March 28, 2012

This paper presents a semianalytical method, the modified multistep differential transform method (MMSDTM), for solving linear and nonlinear fractional-order differential equations with the order between 0 and 2. This method can be considered as a variant of the predictor-corrector method (PCM). The multistep differential transform method (MSDTM), which does not take the memory effect into account and yields unsatisfactory solution very rapidly, is first used to find an estimation as the predictor of the solution. In the corrector procedure, the memory term associated with the fractional-order derivative is decomposed by the subtraction of two integrals; one is abnormal with singularity and the other is normal without singularity and the two integrals are calculated by using the MSDTM and a simple numerical scheme, respectively. Four illustrative examples are given to show that the MMSDTM requires much less computational cost and retains high computational accuracy, compared with the widely used PCM.

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Grahic Jump Location
Fig. 1

The solution of the fractional-order biology system [35] with α = 0.85, h  0.05 obtained by using different methods. Solid line: fractional-order MSDTM [35], and dots: PCM [19].

Grahic Jump Location
Fig. 2

The solution of the fractional Riccati differential equation with α = 0.5. Solid line: MMSDTM, and dots: PCM.

Grahic Jump Location
Fig. 3

The solution of the nonlinear fractional differential equation with α = 1.5. Solid line: MMSDTM, and dots: PCM.

Grahic Jump Location
Fig. 4

The solution of the nonlinear fractional differential equation with α = 1.8. Solid line: MMSDTM, and dots: PCM.

Grahic Jump Location
Fig. 5

The phase portrait of the fractional-order Lorenz system with α  0.92 and T  20, by using MMSDTM

Grahic Jump Location
Fig. 6

The phase portrait of the fractional-order Lorenz system with α  0.94 and T  20, by using MMSDTM

Grahic Jump Location
Fig. 7

The solution of x(t) of the fractional-order Lorenz system with α  0.92 and T  20. Solid line: MMSDTM, and dots: PCM.

Grahic Jump Location
Fig. 8

The solution of x(t) of the fractional-order Lorenz system with α  0.94 and T  20. Solid line: MMSDTM, and dots: PCM.

Grahic Jump Location
Fig. 9

The solution of the fractional-order Baglay-Torvik equation with T=20. Solid line: MMSDTM, and dots: PCM.




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