- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Article Processing Charges ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Recently Accepted Articles ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Advances in Mathematical Physics
Volume 2013 (2013), Article ID 869484, 6 pages
The Periodic Solution of Fractional Oscillation Equation with Periodic Input
1School of Sciences, Shanghai Institute of Technology, Shanghai 201418, China
2School of Mathematics and Information Sciences, Zhaoqing University, Zhaoqing, Guangdong 526061, China
Received 5 July 2013; Revised 14 August 2013; Accepted 15 August 2013
Academic Editor: Ming Li
Copyright © 2013 Jun-Sheng Duan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
- A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands, 2006.
- F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, UK, 2010.
- D. Băleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus Models and Numerical Methods, vol. 3 of Series on Complexity, Nonlinearity and Chaos, World Scientific Publishing, Hackensack, NJ, USA, 2012.
- J. T. Machado, V. Kiryakova, and F. Mainardi, “Recent history of fractional calculus,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1140–1153, 2011.
- L. Song, S. Xu, and J. Yang, “Dynamical models of happiness with fractional order,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 3, pp. 616–628, 2010.
- R. Gu and Y. Xu, “Chaos in a fractional-order dynamical model of love and its control,” in Nonlinear Mathematics for Uncertainty and Its Applications, S. Li, X. Wang, Y. Okazaki, J. Kawabe, T. Murofushi, and L. Guan, Eds., vol. 100 of Advances in Intelligent and Soft Computing, pp. 349–356, Springer, Berlin, Germany, 2011.
- Y. A. Rossikhin and M. V. Shitikova, “Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids,” Applied Mechanics Reviews, vol. 50, no. 1, pp. 15–67, 1997.
- D. Băleanu, O. G. Mustafa, and R. P. Agarwal, “An existence result for a superlinear fractional differential equation,” Applied Mathematics Letters, vol. 23, no. 9, pp. 1129–1132, 2010.
- D. Băleanu and O. G. Mustafa, “On the global existence of solutions to a class of fractional differential equations,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1835–1841, 2010.
- J.-S. Duan, “Time- and space-fractional partial differential equations,” Journal of Mathematical Physics, vol. 46, no. 1, Article ID 013504, pp. 13504–13511, 2005.
- V. Daftardar-Gejji and H. Jafari, “An iterative method for solving nonlinear functional equations,” Journal of Mathematical Analysis and Applications, vol. 316, no. 2, pp. 753–763, 2006.
- G. Wu, “A fractional characteristic method for solving fractional partial differential equations,” Applied Mathematics Letters, vol. 24, no. 7, pp. 1046–1050, 2011.
- H. Jafari, C. M. Khalique, and M. Nazari, “An algorithm for the numerical solution of nonlinear fractional-order Van der Pol oscillator equation,” Mathematical and Computer Modelling, vol. 55, no. 5-6, pp. 1782–1786, 2012.
- H. Jafari, M. Nazari, D. Baleanu, and C. M. Khalique, “A new approach for solving a system of fractional partial differential equations,” Computers & Mathematics with Applications, vol. 66, no. 5, pp. 838–843, 2013.
- J.-S. Duan, T. Chaolu, and R. Rach, “Solutions of the initial value problem for nonlinear fractional ordinary differential equations by the Rach-Adomian-Meyers modified decomposition method,” Applied Mathematics and Computation, vol. 218, no. 17, pp. 8370–8392, 2012.
- J. S. Duan, R. Rach, D. Baleanu, and A. M. Wazwaz, “A review of the Adomian decomposition method and its applications to fractional differential equations,” Communications in Fractional Calculus, vol. 3, pp. 73–99, 2012.
- M. Caputo, “Linear models of dissipation whose Q is almost frequency independent—part II,” Geophysical Journal of the Royal Astronomical Society, vol. 13, no. 5, pp. 529–539, 1967.
- R. L. Bagley and P. J. Torvik, “A generalized derivative model for an elastomer damper,” The Shock and Vibration Bulletin, vol. 49, no. 2, pp. 135–143, 1979.
- H. Beyer and S. Kempfle, “Definition of physically consistent damping laws with fractional derivatives,” Zeitschrift für Angewandte Mathematik und Mechanik, vol. 75, no. 8, pp. 623–635, 1995.
- F. Mainardi, “Fractional relaxation-oscillation and fractional diffusion-wave phenomena,” Chaos, Solitons & Fractals, vol. 7, no. 9, pp. 1461–1477, 1996.
- R. Gorenflo and F. Mainardi, “Fractional calculus: integral and differential equations of fractional order,” in Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi, Eds., vol. 378 of CISM Courses and Lectures, pp. 223–276, Springer, New York, NY, USA, 1997.
- B. N. N. Achar, J. W. Hanneken, and T. Clarke, “Response characteristics of a fractional oscillator,” Physica A, vol. 309, no. 3-4, pp. 275–288, 2002.
- A. Al-rabtah, V. S. Ertürk, and S. Momani, “Solutions of a fractional oscillator by using differential transform method,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1356–1362, 2010.
- M. Li, S. C. Lim, and S. Chen, “Exact solution of impulse response to a class of fractional oscillators and its stability,” Mathematical Problems in Engineering, vol. 2011, Article ID 657839, 9 pages, 2011.
- S. C. Lim, L. Ming, and L. P. Teo, “Locally self-similar fractional oscillator processes,” Fluctuation and Noise Letters, vol. 7, no. 2, pp. L169–L179, 2007.
- S. C. Lim and L. P. Teo, “The fractional oscillator process with two indices,” Journal of Physics A, vol. 42, no. 6, Article ID 065208, 34 pages, 2009.
- M. Li, “Approximating ideal filters by systems of fractional order,” Computational and Mathematical Methods in Medicine, vol. 2012, Article ID 365054, 6 pages, 2012.
- E. Kaslik and S. Sivasundaram, “Non-existence of periodic solutions in fractional-order dynamical systems and a remarkable difference between integer and fractional-order derivatives of periodic functions,” Nonlinear Analysis: Real World Applications, vol. 13, no. 3, pp. 1489–1497, 2012.
- J.-S. Duan, Z. Wang, Y.-L. Liu, and X. Qiu, “Eigenvalue problems for fractional ordinary differential equations,” Chaos, Solitons & Fractals, vol. 46, pp. 46–53, 2013.
- R. P. Agarwal, B. de Andrade, and C. Cuevas, “Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations,” Nonlinear Analysis: Real World Applications, vol. 11, no. 5, pp. 3532–3554, 2010.
- C. Li and Y. Ma, “Fractional dynamical system and its linearization theorem,” Nonlinear Dynamics, vol. 71, no. 4, pp. 621–633, 2013.
- B. Davies, Integral Transforms and Their Applications, Springer, New York, NY, USA, 3rd edition, 2001.