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Analysis Of Mixed Weyl–Marchaud Fractional Derivative And Box Dimensions

Author

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  • SUBHASH CHANDRA

    (School of Basic Sciences, Indian Institute of Technology Mandi, Kamand, Himachal Pradesh 175005, India)

  • SYED ABBAS

    (School of Basic Sciences, Indian Institute of Technology Mandi, Kamand, Himachal Pradesh 175005, India)

Abstract

The calculus of the mixed Weyl–Marchaud fractional derivative has been investigated in this paper. We prove that the mixed Weyl–Marchaud fractional derivative of bivariate fractal interpolation functions (FIFs) is still bivariate FIFs. It is proved that the upper box dimension of the mixed Weyl–Marchaud fractional derivative having fractional order γ = (p,q) of a continuous function which satisfies μ-Hölder condition is no more than 3 − μ + (p + q) when 0 < p, q < μ < 1, p + q < μ, which reveals an important phenomenon about linearly increasing effect of dimension of the mixed Weyl–Marchaud fractional derivative. Furthermore, we deduce box dimension of the graph of the mixed Weyl–Marchaud fractional derivative of a continuous function which is defined on a rectangular region in ℠2 and also, we analyze that the mixed Weyl–Marchaud fractional derivative of a function preserves some basic properties such as continuity, bounded variation and boundedness. The results are new and compliment the existing ones.

Suggested Citation

  • Subhash Chandra & Syed Abbas, 2021. "Analysis Of Mixed Weyl–Marchaud Fractional Derivative And Box Dimensions," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 29(06), pages 1-13, September.
    • Handle: RePEc:wsi:fracta:v:29:y:2021:i:06:n:s0218348x21501450
    DOI: 10.1142/S0218348X21501450
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