ON RIEMANN–LIOUVILLE FRACTIONAL DIFFERENTIABILITY OF CONTINUOUS FUNCTIONS AND ITS PHYSICAL INTERPOLATION

Wei Xiao () and Yong-Shun Liang
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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

FRACTALS (fractals), 2021, vol. 29, issue 08, 1-10

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.

Keywords: Box Dimension; Lipschitz Condition; Hölder Condition; Variation; Riemann–Liouville Fractional Calculus (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1142/S0218348X2150242X

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