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On Riemann–Liouville Fractional Differentiability Of Continuous Functions And Its Physical Interpolation

Author

Listed:
  • WEI XIAO

    (School of Science, Nanjing University of Science and Technology, Xiaolingwei No. 200, Zhongshan Gate Street, Xuanwu District, Nanjing 210094, P. R. China)

  • YONG-SHUN LIANG

    (School of Science, Nanjing University of Science and Technology, Xiaolingwei No. 200, Zhongshan Gate Street, Xuanwu District, Nanjing 210094, P. R. China)

Abstract

In this paper, we mainly research on fractional differentiability of certain continuous functions with fractal dimension one. First, Riemann–Liouville fractional differential of differentiable functions must exist. Then, we prove the existence of Riemann–Liouville fractional differential of continuous functions satisfying the Lipschitz condition, which means that all of their Riemann–Liouville fractional integral of any positive orders in (0, 1) are differentiable. For continuous functions which do not satisfy the Lipschitz condition, we give counterexamples of certain continuous functions whose Riemann–Liouville fractional differential does not exist of certain positive order in (0, 1). Riemann–Liouville fractional differentiability of other one-dimensional continuous functions has also been investigated elementary. Fractional differentiability takes interpretation on physical problems like moving particle and transports through porous and percolation medium with residual memory.

Suggested Citation

  • Wei Xiao & Yong-Shun Liang, 2021. "On Riemann–Liouville Fractional Differentiability Of Continuous Functions And Its Physical Interpolation," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 29(08), pages 1-10, December.
    • Handle: RePEc:wsi:fracta:v:29:y:2021:i:08:n:s0218348x2150242x
    DOI: 10.1142/S0218348X2150242X
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