Fractals and fractional integrals: Are there some accurate relationships between them?
https://doi.org/10.26907/mrsej-24105
Abstract
This study establishes precise links between the fractional integral of the RL-type and the averaging technique of a smooth function over 1D-fractal sets. These findings were previously reported in the works [1], [2]. To draw in the interest of other experts operating in the NMR/EPR zones, it is helpful to repeat them again. The physical meaning of these acquired formulas is explained and numerical verifications are performed with the purpose of confirming the analytical results. Furthermore, results were achieved for a combination of fractal circuits with a discrete set of fractal dimensions that were generalized. We suppose that these new results help understand deeper the intimate links between fractals and fractional integrals of different types, especially in applications of the fractional operators in complex systems.
Keywords
1D-Fractals,
Riemann-Liouville fractional integrals,
averaging procedure over 1D Cantor sets,
the generalized self-similar electric circuits
Supporting agencies: Dr. Prof. Len Dissado and Dr. Prof. Robert Hill, two English researchers, provided significant inspiration for the substance of this study. Profound conversations and seminars with them during the author’s (in 1982-1983 years) visit to Chelsea College (London University) enabled the author of this study to comprehend this issue and formulate some problems in this interesting area of research.
About the Author
R. R. Nigmatullin
Kazan National Research Technical University (KAI)
Russian Federation
Russian Federation
Kazan 420111
References
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Review
For citations:
Nigmatullin R.R. Fractals and fractional integrals: Are there some accurate relationships between them? Magnetic Resonance in Solids. 2024;26(1):24105 (14 pp.). https://doi.org/10.26907/mrsej-24105
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