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Local Fractional Fourier Series with Application to Wave Equation in Fractal Vibrating String

Author

Listed:
  • Ming-Sheng Hu
  • Ravi P. Agarwal
  • Xiao-Jun Yang

Abstract

We introduce the wave equation in fractal vibrating string in the framework of the local fractional calculus. Our particular attention is devoted to the technique of the local fractional Fourier series for processing these local fractional differential operators in a way accessible to applied scientists. By applying this technique we derive the local fractional Fourier series solution of the local fractional wave equation in fractal vibrating string and show the fundamental role of the Mittag-Leffler function.

Suggested Citation

  • Ming-Sheng Hu & Ravi P. Agarwal & Xiao-Jun Yang, 2012. "Local Fractional Fourier Series with Application to Wave Equation in Fractal Vibrating String," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-15, December.
    • Handle: RePEc:hin:jnlaaa:567401
    DOI: 10.1155/2012/567401
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    Cited by:

    1. Guang-Sheng Chen & H. M. Srivastava & Pin Wang & Wei Wei, 2014. "Some Further Generalizations of Hölder′s Inequality and Related Results on Fractal Space," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    2. Yu Zhang, 2014. "Solving Initial‐Boundary Value Problems for Local Fractional Differential Equation by Local Fractional Fourier Series Method," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    3. Nabard Habibi & Zohre Nouri, 2020. "Application of Local Fractional Homotopy Perturbation Method in Physical Problems," Advances in Mathematical Physics, John Wiley & Sons, vol. 2020(1).
    4. Ai-Min Yang & Xiao-Jun Yang & Zheng-Biao Li, 2013. "Local Fractional Series Expansion Method for Solving Wave and Diffusion Equations on Cantor Sets," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    5. Yong-Ju Yang & Dumitru Baleanu & Xiao-Jun Yang, 2013. "Analysis of Fractal Wave Equations by Local Fractional Fourier Series Method," Advances in Mathematical Physics, John Wiley & Sons, vol. 2013(1).
    6. Shun-Qin Wang & Yong-Ju Yang & Hassan Kamil Jassim, 2014. "Local Fractional Function Decomposition Method for Solving Inhomogeneous Wave Equations with Local Fractional Derivative," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    7. Zhi-Yong Chen & Carlo Cattani & Wei-Ping Zhong, 2014. "Signal Processing for Nondifferentiable Data Defined on Cantor Sets: A Local Fractional Fourier Series Approach," Advances in Mathematical Physics, John Wiley & Sons, vol. 2014(1).
    8. Yong-Ju Yang & Dumitru Baleanu & Xiao-Jun Yang, 2013. "A Local Fractional Variational Iteration Method for Laplace Equation within Local Fractional Operators," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    9. Yong-Ju Yang & Liu-Qing Hua, 2014. "Variational Iteration Transform Method for Fractional Differential Equations with Local Fractional Derivative," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    10. Yang Zhao & Dumitru Baleanu & Mihaela Cristina Baleanu & De-Fu Cheng & Xiao-Jun Yang, 2013. "Mappings for Special Functions on Cantor Sets and Special Integral Transforms via Local Fractional Operators," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    11. Sheng-Ping Yan & Hossein Jafari & Hassan Kamil Jassim, 2014. "Local Fractional Adomian Decomposition and Function Decomposition Methods for Laplace Equation within Local Fractional Operators," Advances in Mathematical Physics, John Wiley & Sons, vol. 2014(1).
    12. Ya-Juan Hao & H. M. Srivastava & Hossein Jafari & Xiao-Jun Yang, 2013. "Helmholtz and Diffusion Equations Associated with Local Fractional Derivative Operators Involving the Cantorian and Cantor‐Type Cylindrical Coordinates," Advances in Mathematical Physics, John Wiley & Sons, vol. 2013(1).

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