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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 567401, 15 pages
doi:10.1155/2012/567401
Research Article
Local Fractional Fourier Series with Application to Wave Equation in Fractal Vibrating String
1Institute of Software Science, Zhengzhou Normal University, Zhengzhou 450044, China
2Department of Mathematics, Texas A and M University, Kingsville, TX 78363-8202, USA
3Department of Mathematics and Mechanics, China University of Mining and Technology, Jiangsu, Xuzhou 221008, China
Received 14 August 2012; Revised 25 October 2012; Accepted 8 November 2012
Academic Editor: Lan Xu
Copyright © 2012 Ming-Sheng Hu et al. 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.
Linked References
- R. Hilfe, Applications of Fractional Calculus in Physics, World Scientific, River Edge, NJ, USA, 2000. View at Publisher · View at Google Scholar
- J. Sabatier, O. P. Agrawal, and J. A. Tenreiro, Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, New York, NY, USA, 2007. View at Publisher · View at Google Scholar
- K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, London, UK, 1974.
- K. S. Miller and B. Ross, An Introduction To the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.
- I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press, San Diego, Calif, USA, 1999.
- S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, Switzerland, 1993.
- A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier, Amsterdam, The Netherlands, 2006.
- W. R. Schneider and W. Wyss, “Fractional diffusion and wave equations,” Journal of Mathematical Physics, vol. 30, no. 1, pp. 134–144, 1989. View at Publisher · View at Google Scholar
- O. P. Agrawal, “Solution for a fractional diffusion-wave equation defined in a bounded domain,” Nonlinear Dynamics, vol. 29, no. 1–4, pp. 145–155, 2002. View at Publisher · View at Google Scholar
- A. M. A. El-Sayed, “Fractional-order diffusion-wave equation,” International Journal of Theoretical Physics, vol. 35, no. 2, pp. 311–322, 1996. View at Publisher · View at Google Scholar
- H. Jafari and S. Seifi, “Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 5, pp. 2006–2012, 2009. View at Publisher · View at Google Scholar
- Y. Povstenko, “Non-axisymmetric solutions to time-fractional diffusion-wave equation in an infinite cylinder,” Fractional Calculus and Applied Analysis, vol. 14, no. 3, pp. 418–435, 2011. View at Publisher · View at Google Scholar
- F. Mainardi and G. Pagnini, “The Wright functions as solutions of the time-fractional diffusion equation,” Applied Mathematics and Computation, vol. 141, no. 1, pp. 51–62, 2003. View at Publisher · View at Google Scholar
- Y. Luchko, “Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1766–1772, 2010. View at Publisher · View at Google Scholar
- M. M. Meerschaert, D. A. Benson, H.-P. Scheffler, and B. Baeumer, “Stochastic solution of space-time fractional diffusion equations,” Physical Review E, vol. 65, no. 4, Article ID 041103, 4 pages, 2002. View at Publisher · View at Google Scholar
- S. D. Eidelman and A. N. Kochubei, “Cauchy problem for fractional diffusion equations,” Journal of Differential Equations, vol. 199, no. 2, pp. 211–255, 2004. View at Publisher · View at Google Scholar
- T. Sandev and Ž. Tomovski, “The general time fractional wave equation for a vibrating string,” Journal of Physics A, vol. 43, no. 5, Article ID 055204, 2010. View at Publisher · View at Google Scholar
- T. M. Atanackovic and B. Stankovic, “Generalized wave equation in nonlocal elasticity,” Acta Mechanica, vol. 208, no. 1-2, pp. 1–10, 2009.
- J. Dong and M. Xu, “Space-time fractional Schrödinger equation with time-independent potentials,” Journal of Mathematical Analysis and Applications, vol. 344, no. 2, pp. 1005–1017, 2008. View at Publisher · View at Google Scholar
- S. Z. Rida, H. M. El-Sherbiny, and A. A. M. Arafa, “On the solution of the fractional nonlinear Schrödinger equation,” Physics Letters A, vol. 372, no. 5, pp. 553–558, 2008. View at Publisher · View at Google Scholar
- R. Scherer, S. L. Kalla, L. Boyadjiev, and B. Al-Saqabi, “Numerical treatment of fractional heat equations,” Applied Numerical Mathematics, vol. 58, no. 8, pp. 1212–1223, 2008. View at Publisher · View at Google Scholar
- S. Momani, “An explicit and numerical solutions of the fractional KdV equation,” Mathematics and Computers in Simulation, vol. 70, no. 2, pp. 110–118, 2005. View at Publisher · View at Google Scholar
- R. Metzler and T. F. Nonnenmacher, “Space- and time-fractional diffusion and wave equations, fractional Fokker-Planck equations, and physical motivation,” Chemical Physics, vol. 84, pp. 67–90, 2002.
- P. Paradisi, R. Cesari, F. Mainardi, and F. Tampieri, “The fractional Fick's law for non-local transport processes,” Physica A, vol. 293, no. 1-2, pp. 130–142, 2001.
- Y. Zhou and F. Jiao, “Nonlocal Cauchy problem for fractional evolution equations,” Nonlinear Analysis, vol. 11, no. 5, pp. 4465–4475, 2010. View at Publisher · View at Google Scholar
- V. E. Tarasov, “Fractional Heisenberg equation,” Physics Letters A, vol. 372, no. 17, pp. 2984–2988, 2008. View at Publisher · View at Google Scholar
- V. E. Tarasov and G. M. Zaslavsky, “Fractional Ginzburg-Landau equation for fractal media,” Physica A, vol. 354, no. 15, pp. 249–261, 2005.
- V. E. Tarasov, “Fractional hydrodynamic equations for fractal media,” Annals of Physics, vol. 318, no. 2, pp. 286–307, 2005. View at Publisher · View at Google Scholar
- J.-H. He, “Approximate analytical solution for seepage flow with fractional derivatives in porous media,” Computer Methods in Applied Mechanics and Engineering, vol. 167, no. 1-2, pp. 57–68, 1998. View at Publisher · View at Google Scholar
- Q. Wang, “Numerical solutions for fractional KdV-Burgers equation by Adomian decomposition method,” Applied Mathematics and Computation, vol. 182, no. 2, pp. 1048–1055, 2006. View at Publisher · View at Google Scholar
- G. C. Wu and E. W. M. Lee, “Fractional variational iteration method and its application,” Physics Letters A, vol. 374, no. 25, pp. 2506–2509, 2010. View at Publisher · View at Google Scholar
- J.-H. He, “A short remark on fractional variational iteration method,” Physics Letters A, vol. 375, no. 38, pp. 3362–3364, 2011. View at Publisher · View at Google Scholar
- Z.-B. Li and J.-H. He, “Fractional complex transform for fractional differential equations,” Mathematical & Computational Applications, vol. 15, no. 5, pp. 970–973, 2010.
- J.-H. He, S. K. Elagan, and Z. B. Li, “Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus,” Physics Letters A, vol. 376, no. 4, pp. 257–259, 2012. View at Publisher · View at Google Scholar
- Z. B. Li and J. H. He, “Application of the fractional complex transform to fractional differential equations,” Nonlinear Science Letters A, vol. 2, no. 3, pp. 121–126, 2011.
- Z. B. Li, W. H. Zhu, and J. H. He, “Exact solutions of time-fractional heat conduction equation by the fractional complex transform,” Thermal Science, vol. 16, no. 2, pp. 335–338, 2012.
- Q. L. Wang, J. H. He, and Z. B. Li, “Fractional model for heat conduction in polar bear hairs,” Thermal Science, vol. 16, no. 2, pp. 339–342, 2012.
- G. Jumarie, “On the representation of fractional Brownian motion as an integral with respect to ,” Applied Mathematics Letters, vol. 18, no. 7, pp. 739–748, 2005. View at Publisher · View at Google Scholar
- G. Jumarie, “Probability calculus of fractional order and fractional Taylor's series application to Fokker-Planck equation and information of non-random functions,” Chaos, Solitons and Fractals, vol. 40, no. 3, pp. 1428–1448, 2009. View at Publisher · View at Google Scholar
- G. Jumarie, “Laplace's transform of fractional order via the Mittag-Leffler function and modified Riemann-Liouville derivative,” Applied Mathematics Letters, vol. 22, no. 11, pp. 1659–1664, 2009. View at Publisher · View at Google Scholar
- G.-C. Wu, “Adomian decomposition method for non-smooth initial value problems,” Mathematical and Computer Modelling, vol. 54, no. 9-10, pp. 2104–2108, 2011. View at Publisher · View at Google Scholar
- K. M. Kolwankar and A. D. Gangal, “Local fractional Fokker-Planck equation,” Physical Review Letters, vol. 80, no. 2, pp. 214–217, 1998. View at Publisher · View at Google Scholar
- A. Carpinter and A. Sapora, “Diffusion problems in fractal media defined on Cantor sets,” ZAMM Journal of Applied Mathematics and Mechanics, vol. 90, no. 3, pp. 203–210, 2010.
- K. M. Kolwankar and A. D. Gangal, “Fractional differentiability of nowhere differentiable functions and dimensions,” Chaos, vol. 6, no. 4, pp. 505–513, 1996. View at Publisher · View at Google Scholar
- X. R. Li, Fractional calculus, fractal geometry, and stochastic processes [Ph.D. thesis], University of Western Ontario, Ontario, Canada, 2003.
- A. Babakhani and V. D. Gejji, “On calculus of local fractional derivatives,” Journal of Mathematical Analysis and Applications, vol. 270, no. 1, pp. 66–79, 2002. View at Publisher · View at Google Scholar
- A. Parvate and A. D. Gangal, “Calculus on fractal subsets of real line. I. Formulation,” Fractals, vol. 17, no. 1, pp. 53–81, 2009. View at Publisher · View at Google Scholar
- F. Ben Adda and J. Cresson, “About non-differentiable functions,” Journal of Mathematical Analysis and Applications, vol. 263, no. 2, pp. 721–737, 2001. View at Publisher · View at Google Scholar
- A. Carpinteri, B. Chiaia, and P. Cornetti, “The elastic problem for fractal media: basic theory and finite element formulation,” Computers & Structures, vol. 82, no. 6, pp. 499–508, 2004.
- A. Carpinteri and P. Cornetti, “A fractional calculus approach to the description of stress and strain localization in fractal media,” Chaos, Solitons, Fractals, vol. 13, no. 1, pp. 85–94, 2002.
- Y. Chen, Y. Yan, and K. Zhang, “On the local fractional derivative,” Journal of Mathematical Analysis and Applications, vol. 362, no. 1, pp. 17–33, 2010. View at Publisher · View at Google Scholar
- A. Carpinteri, B. Chiaia, and P. Cornetti, “On the mechanics of quasi-brittle materials with a fractal microstructure,” Engineering Fracture Mechanics, vol. 70, no. 6, pp. 2321–2349, 2003.
- X. J. Yang, “Local fractional integral transforms,” Progress in Nonlinear Science, vol. 4, pp. 1–225, 2011.
- X. J. Yang, Local Fractional Functional Analysis and Its Applications, Asian Academic Publisher, Hong Kong, 2011.
- X. J. Yang, “Local fractional calculus and its applications,” in Proceedings of the 5th IFAC Workshop Fractional Differentiation and Its Applications, (FDA '12), pp. 1–8, Nanjing, China, 2012.
- X. J. Yang, M. K. Liao, and J. W. Chen, “A novel approach to processing fractal signals using the Yang-Fourier transforms,” Procedia Engineering, vol. 29, pp. 2950–2954, 2012.
- X. J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, NY, USA, 2012.
- S. Zhang and H.-Q. Zhang, “Fractional sub-equation method and its applications to nonlinear fractional PDEs,” Physics Letters A, vol. 375, no. 7, pp. 1069–1073, 2011. View at Publisher · View at Google Scholar
- J. H. He, “Analytical methods for thermal science—an elementary introduction,” Thermal Science, vol. 15, pp. S1–S3, 2011.
- J. H. He, “A new fractal derivation,” Thermal Science, vol. 15, pp. S145–S147, 2011.
- J. H. He, “Asymptotic methods for solitary solutions and compactons,” Abstract and Applied Analysis, vol. 2012, Article ID 916793, 130 pages, 2012.