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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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References

Koeller, R. C., 1984, “Applications of Fractional Calculus to the Theory of Viscoelasticity,” ASME J. Appl. Mech., 51(2), pp. 299–307. [CrossRef]
Cao, J., Ma, C., Xie, H., and Jiang, Z., 2010, “Nonlinear Dynamics of the Duffing System With Fractional Order Damping,” ASME J. Comput. Nonlinear Dyn., 5, p. 041012. [CrossRef]
Ingman, D., and Suzdalnitsky, J., 2008, “Dynamic Viscoelastic Rod Stability Modeling by Fractional Differential Operator,” ASME J. Appl. Mech., 75, p. 014502. [CrossRef]
Liebst, B. S., and Torvik, P. J., 1996, “Asymptotic Approximations for Systems Incorporating Fractional Derivative Damping,” ASME J. Dyn. Syst., Meas., Control, 118(3), pp. 572–579. [CrossRef]
Sabatier, J., Agrawal, O. P., and Tenreiro Machado, J. A., 2007, Advances in Fractional Calculus-Theoretical Developments and Applications in Physics and Engineering, Springer, Berlin.
Schneider, W. R., and Wyss, W., 1989, “Fractional Diffusion and Wave Equations,” J. Math. Phys, 30, pp. 134–144. [CrossRef]
Cao, J., Cao, B., Zhang, X., and Wen, G., 2008, “Fractional Proportional Integral Control for Pneumatic Position Servo Systems,” Proceedings of the IEEE/ ASME International Conference on Mechatronic and Embedded Systems and Applications, Beijing, pp. 347–352.
Agrawal, O. P., 2004, “A General Formulation and Solution Scheme for Fractional Optimal Control Problems,” Nonlinear Dyn., 38, pp. 323–337. [CrossRef]
Podlubny, I., 1999, Fractional Differential Equations, Academic, San Diego.
Kilbas, A. A., Srivastava, H. M., and Trujillo, J. J., 2006, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam.
Tenreiro Machado, J. A., Kiryakova, V., and Mainardi, F., 2011, “Recent History of Fractional Calculus,” Commun. Nonlinear Sci. Numer. Simul., 3(16), pp. 1140–1153. [CrossRef]
Tenreiro Machado, J. A., and Galhano, A., 2008, “Fractional Dynamics: A Statistical Perspective,” ASME J. Comput. Nonlinear Dyn., 3, p. 021201. [CrossRef]
Bagley, R. L., and Torvik, P. J., 1984, “On the Appearance of the Fractional Derivative in the Behavior of Real Materials,” ASME J. Appl. Mech., 51(4), pp. 294–298. [CrossRef]
Eldred, L. B., Baker, W. P., and PalazottoA. N., 1995, “Kelvin-Voigt Versus Fractional Derivative Model as Constitutive Relations for Viscoelastic Materials,” AIAA J., 33, pp. 547–550. [CrossRef]
Chen, H. S., Hou, T. T., and Feng, Y. P., 2010, “Fractional Model for the Physical Aging of Polymers,” Sci. Sin. Phys. Mech. Astron., 40, pp. 1267–1274(in Chinese), Available at: http://phys.scichina.com:8083/Jwk_sciG_cn/CN/Y2010/V40/I10/1267.
Coronado, A., Trindade, M. A., and Sampaio, R., 2002, “Frequency-Dependent Viscoelastic Models for Passive Vibration Isolation Systems,” Shock Vib., 9, pp. 253–264, Available at:http://iospress.metapress.com/content/xh4255aq40db7eu7/.
Rossikhin, Y. A., and Shitikova, M. V., 2010, “Application of Fractional Calculus for Dynamic Problems of Solid Mechanics: Novel Trends and Recent Results,” Appl. Mech. Rev., 63, p. 010801. [CrossRef]
Miller, K. S., and Ross, B., 1993, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York.
Diethelm, K., Ford, N. J., and Freed, A. D., 2002, “A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations,” Nonlinear Dyn., 29, pp. 3–22. [CrossRef]
Deng, W. H., 2007, “Short Memory Principle and a Predictor-Corrector Approach for Fractional Differential Equations,” J. Comput. Appl. Math., 206, pp. 174–188. [CrossRef]
Matsuzaka, T., and Nakagawa, M., 2003, “Chaos Neuron Model With Fractional Differential Equation,” J. Phys. Soc. Jpn., 72, pp. 2678–2684. [CrossRef]
Diethelm, K., Ford, N. J., Freed, A. D., and Luchko, Yu., 2005, “Algorithms for the Fractional Calculus: A Selection of Numerical Methods,” Comput. Methods Appl. Mech. Eng., 194, pp. 743–773. [CrossRef]
Daftardar-Geji, V., and Jafari, H., 2005, “Adomian Decomposition: A Tool for Solving a System of Fractional Differential Equations,” J. Math. Anal. Appl., 301, pp. 508–518. [CrossRef]
Saha, R. S., and Bera, R. K., 2005, “Analytical Solution of the Bagley-Torvik Equation by Adomian Decomposition Method,” Appl. Math. Comput., 168, pp. 398–410. [CrossRef]
Saha, R. S., 2009, “Analytical Solution for the Space Fractional Diffusion Equation by Two-Step Adomian Decomposition Method,” Commun. Nonlinear Sci. Numer. Simul., 14, pp. 1295–1306. [CrossRef]
Cafagna, D., and Grassi, G., 2008, “Fractional-Order Chua’s Circuit: Time-Domain Analysis, Bifurcation, Chaotic Behavior and Test for Chaos,” Int. J. Bifurcation Chaos Appl. Sci. Eng., 3(18), pp. 615–639. [CrossRef]
Saha, R. S., Poddar, B. P., and Bera, R. K., 2005, “Analytical Approximate Solution of a Dynamic System Containing Fractional Derivative of Order One-Half by Adomian Decomposition Method,” ASME J. Appl. Mech., 72, pp. 290–295. [CrossRef]
Momani, S., and Odibat, Z., 2007, “Numerical Approach to Differential Equations of Fractional Order,” J. Comput. Appl. Math., 207, pp. 96–110. [CrossRef]
Drăgănescu, G. E., 2006, “Application of a Variational Iteration Method to Linear and Nonlinear Viscoelastic Models With Fractional Derivatives,” J. Math. Phys., 47, p. 082902. [CrossRef]
Cang, J., Tan, Y., Xu, H., and Liao, S. J., 2009, “Series Solutions of Non-Linear Riccati Differential Equations With Fractional Order,” Chaos Solitons Fractals, 40, pp. 1–9. [CrossRef]
Li, C. P., and Wang, Y. H., 2009, “Numerical Algorithm Based on Adomian Decomposition for Fractional Differential Equations,” Comput. Math. Appl., 57, pp. 1672–1681. [CrossRef]
Pálfalvi, A., 2010, “Efficient Solution of a Vibration Equation Involving Fractional Derivatives,” Int. J. Non-Linear Mech., 45, pp. 169–175. [CrossRef]
Zhou, J. K., 1986, Differential Transformation and its Applications for Electrical Circuits, Huazhong University Press, Wuhan (in Chinese).
Odibat, Z. M., Bertelle, C., Aziz-Alaoui, M. A., and Duchamp, G. H. E., 2010, “A Multi-Step Differential Transform Method and Application to Non-Chaotic or Chaotic Systems,” J. Comput. Math. Appl., 59, pp. 1462–1472. [CrossRef]
Ertük, V. S., Odibat, Z. M., and Momani, S., 2011, “An Approximate Solution of a Fractional Order Differential Equation Model of Human T-Cell Lymphotropic Virus I (HTLV-I) Infection of CD4+ T-Cells,” Comput. Math. Appl., 3(62), pp. 996–1002. [CrossRef]
Odibat, Z. M., Momani, S., and Ertürk, V. S., 2008, “Generalzied Differential Transform Method: Application to Differential Equations of Fractional Order,” J. Appl. Math. Comput., 197, pp. 467–477. [CrossRef]
Ford, N. J., and Simpson, A. C., 2001, “The Numerical Solution of Fractional Differential Equations: Speed Versus Accuracy,” Numer. Algorithms, 26, pp. 333–346. [CrossRef]
Momani, S., and Shawagfeh, N., 2006, “Decomposition Method for Solving Fractional Riccati Differential Equations,” Appl. Math. Comput., 182, pp. 1083–1092. [CrossRef]
Hosseinnia, S. H., Ranjbar, A., and Momani, S., 2008, “Using an Enhanced Homotopy Perturbation Method in Fractional Differential Equations Via Deforming the Linear Part,” Comput. Math. Appl., 56, pp. 3138–3149. [CrossRef]
Odibat, Z. M., 2011, “On Legendre Polynomial Approximation With the VIM or HAM for Numerical Treatment of Nonlinear Fractional Differential Equations,” J. Comput. Appl. Math., 235, pp. 2956–2968. [CrossRef]
Sun, K. H., Wang, X., and Sprott, J. C., 2010, “Bifurcations and Chaos in Fractional-Order Simplified Lorenz System,” Int. J. Bifurcation Chaos, 4(20), pp. 1209–1219. [CrossRef]

Figures

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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