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Analysis of fractal dimension of mixed Riemann-Liouville integral

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Abstract

In this article, we provide a rigorous study on the fractal dimension of the graph of the mixed Riemann-Liouville fractional integral for various choices of continuous functions on a rectangular region. We estimate bounds for the box dimension and the Hausdorff dimension of the graph of the mixed Riemann-Liouville fractional integral of the functions which belong to the class of continuous functions and the class of Hölder continuous functions. We also show that the box dimension of the graph of the mixed Riemann-Liouville fractional integral of two-dimensional continuous functions is also two. Furthermore, we give the construction of unbounded variational continuous functions. Later, we prove that the box dimension and the Hausdorff dimension of the graph of the mixed Riemann-Liouville fractional integral of unbounded variational continuous functions are two. Moreover, we illustrate our results by using some examples.

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Funding

The first author has received the financial support from the CSIR, India (file no.: 09/1058(0012)/2018-EMR-I).

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  1. School of Basic Sciences, Indian Institute of Technology Mandi, Kamand, H.P., 175005, India

    Subhash Chandra & Syed Abbas

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  1. Subhash Chandra
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  2. Syed Abbas
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Correspondence to Syed Abbas.

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Chandra, S., Abbas, S. Analysis of fractal dimension of mixed Riemann-Liouville integral. Numer Algor 91, 1021–1046 (2022). https://doi.org/10.1007/s11075-022-01290-2

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  • DOI: https://doi.org/10.1007/s11075-022-01290-2

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