Study of Fractional Analytic Functions and Local Fractional Calculus

Authors

  • Chii-Huei Yu  School of Mathematics and Statistics, Zhaoqing University, Guangdong Province, China

DOI:

https://doi.org/10.32628/IJSRSET218482

Keywords:

Fractional Analytic Function, Local Fractional Calculus, Fundamental Theorem of Local Fractional Calculus, Product Rule, Quotient Rule, Chain Rule, Change of Variable, Integration by Parts.

Abstract

In this present paper, the role of fractional analytic function in local fractional calculus is studied. Some important properties and theorems in local fractional calculus are discussed, such as product rule, quotient rule, chain rule, fundamental theorem of local fractional calculus, change of variable, integration by parts and so on. In addition, we propose several examples and formulas of local fractional calculus.

References

  1. T. W. Korner, Fourier Analysis, Cambridge University Press, Cambridge, 1989.
  2. B. B. Mandelbrot, The Fractal Geometry of Nature, Freeman, New York, 1977.
  3. K. M. Kolwanka, A. D. Gangal, “Fractional differentiability of nowhere differentiable functions and dimensions,” Journal of Chaos, Vol. 6, pp. 505-513, 1996.
  4. K. M. Kolwanka, A. D. Gangal, “Holder exponents of irregular signals and local fractional deirvatives, ” PRAMANA-Journal of Physics, Vol. 48, No. 1, pp.49-68, 1997.
  5. X. -J. Yang, C. Cattani, G. Xie, “Local fractional calculus application to differential equations arising in fractal heat transfer,” Fractional Dynamics, pp.282-295, 2015.
  6. A. Carpinteri, P. Cornetti, “A fractional calculus approach to the description of stress and strain localization in fractal media,” Chaos, Solitons and Fractals, Vol. 13, pp. 85–94, 2002.
  7. A. -M. Yang, Y. -Z. Zhang, C. Cattani et al., “Application of local fractional series expansion method to solve Klein-Gordon equations on Cantor sets,” Abstract and Applied Analysis, Vol. 2014, Article ID 372741, 6 pages, 2014.
  8. X. -J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, NY, USA, 2012.
  9. J. H. He, “Local fractional variational iteration method for fractal heat transfer in silk cocoon hierarchy,” Nonlinear Science Letters A, Vol. 4, No. 1, pp. 15–20, 2013.
  10. X. -J. Yang and D. Baleanu, “Local fractional variational iteration method for Fokker-Planck equation on a Cantor set,” Acta Universitaria, Vol. 23, No. 2, pp. 3–8, 2013.
  11. C. -G. Zhao, A. -M. Yang, H. Jafari, and A. Haghbin, “The Yang-Laplace transform for solving the IVPs with local fractional derivative,” Abstract and Applied Analysis, Vol. 2014, Article ID 386459, 5 pages, 2014.
  12. E. Lutz, “Fractional transport equations for Levy stable processes,” Physical Review Letters, Vol. 86, No. 11, pp. 2208–2211, 2001.
  13. Y. Zhao, D. Baleanu, C. Cattani, et al., “Maxwell’s equations on Cantor sets: a local fractional approach,” Advances in High Energy Physics, Article ID686371, 6pages, 2013.
  14. A. M. Yang, C. Cattani, Zhang, et al., “Local fractional Fourier series solutions for nonhomogeneous heat equations arising in fractal heat flow with local fractional derivative,” Advances in Mechanical Engineering, 2014, Article ID514639, 5pages, 2014.
  15. X. –J. Yang, D. Baleanu, “Fractal heat conduction problem solved by local fractional variation iteration method,” Thermal Science, Vo. 17, No. 2, pp. 625–628, 2013.
  16. M. Li, X. –F. Hui, C. Cattani, X. –J. Yang, and Y. Zhao, “Approximate Solutions for Local Fractional Linear Transport Equations Arising in Fractal Porous Media,” Advances in Mathematical Physics, Vol. 2014, pp. 1–8, 2014.
  17. C. -H. Yu, “A new insight into fractional logistic equation,” International Journal of Engineering Research and Reviews, Vol. 9, Issue 2, pp.13-17, 2021.
  18. U. Ghosh, S. Sengupta, S. Sarkar, and S. Das, “Analytic solution of linear fractional differential equation with Jumarie derivative in term of Mittag-Leffler function,” American Journal of Mathematical Analysis, Vol. 3, No. 2, pp.32-38, 2015.
  19. C. -H. Yu, “Fractional mean value theorem and its applications,” International Journal of Electrical and Electronics Research, Vol. 9, Issue 2, pp. 19-24, 2021.
  20. C. -H. Yu, “Fractional Taylor series based on Jumarie type of modified Riemann-Liouville derivatives,” International Journal of Latest Research in Engineering and Technology, Vol. 7, Issue 6, pp.1-6, 2021.

International Journal of Scientific Research in Science, Engineering and Technology

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Published

2021-10-30

Issue

Volume 8, Issue 5, September-October-2021

Section

Research Articles

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How to Cite

[1]
Chii-Huei Yu "Study of Fractional Analytic Functions and Local Fractional Calculus" International Journal of Scientific Research in Science, Engineering and Technology (IJSRSET), Print ISSN : 2395-1990, Online ISSN : 2394-4099, Volume 8, Issue 5, pp.39-46, September-October-2021. Available at doi : https://doi.org/10.32628/IJSRSET218482