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A Geometric Based Connection between Fractional Calculus and Fractal Functions

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Abstract

Establishing the accurate relationship between fractional calculus and fractals is an important research content of fractional calculus theory. In the present paper, we investigate the relationship between fractional calculus and fractal functions, based only on fractal dimension considerations. Fractal dimension of the Riemann–Liouville fractional integral of continuous functions seems no more than fractal dimension of functions themselves. Meanwhile fractal dimension of the Riemann–Liouville fractional differential of continuous functions seems no less than fractal dimension of functions themselves when they exist. After further discussion, fractal dimension of the Riemann–Liouville fractional integral is at least linearly decreasing and fractal dimension of the Riemann–Liouville fractional differential is at most linearly increasing for the Hölder continuous functions. Investigation about other fractional calculus, such as the Weyl-Marchaud fractional derivative and the Weyl fractional integral has also been given elementary. This work is helpful to reveal the mechanism of fractional calculus on continuous functions. At the same time, it provides some theoretical basis for the rationality of the definition of fractional calculus. This is also helpful to reveal and explain the internal relationship between fractional calculus and fractals from the perspective of geometry.

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  1. School of Mathematics and Statistic, Nanjing University of Science and Technology, Nanjing, 210094, P. R. China

    Yong Shun Liang

  2. Department of Mathematics, Nanjing University, Nanjing, 210023, P. R. China

    Wei Yi Su

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  1. Yong Shun Liang
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Supported by National Natural Science Foundation of China (Grant No. 12071218) and the Fundamental Research Funds for the Central Universities (Grant No. 30917011340)

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Liang, Y.S., Su, W.Y. A Geometric Based Connection between Fractional Calculus and Fractal Functions. Acta. Math. Sin.-English Ser. 40, 537–567 (2024). https://doi.org/10.1007/s10114-023-1663-3

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