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Fractal Derivatives, Fractional Derivatives and q-Deformed Calculus
Airton Deppman1, Eugenio Megías2, Roman Pasechnik3
1Instituto de Física, Universidade de São Paulo, São Paulo 05508-090, Brazil.
Entropy (Basel, Switzerland)
|July 29, 2023
Summary
This study clarifies fractal derivatives and fractional derivatives, highlighting their connection to fractal geometry and Hausdorff dimensions. It shows how q-calculus and Caputo derivatives approximate fractal derivatives, with implications for diffusion and epidemic modeling.
Area of Science:
- Mathematics
- Fractal Geometry
- Non-integer Calculus
Background:
- Fractional derivatives and fractal derivatives are distinct mathematical concepts.
- Fractal derivatives relate to Hausdorff's fractional dimension geometry.
- Understanding their differences is crucial for applications in complex systems.
Purpose of the Study:
- To analyze and differentiate between fractional and fractal derivatives.
- To explore continuous approximations of fractal derivatives.
- To establish connections between fractal derivatives and existing calculus frameworks like q-calculus and Caputo derivatives.
Main Methods:
- Comparative analysis of fractional and fractal derivative definitions.
- Investigation of continuous approximations for fractal derivatives.
- Demonstration of relationships using q-calculus and Caputo's derivative.
Main Results:
- The fractal derivative is linked to fractal dimension geometry.
- The q-calculus derivative approximates the fractal derivative of a fractal function.
- Caputo's derivative is proportional to a continuous approximation of the fractal derivative.
Conclusions:
- This work clarifies the relationship between fractional and fractal derivatives.
- Identified approximations have implications for fractional differential equations and anomalous diffusion.
- Findings contribute to understanding phenomena in fractal systems and fractal geometry.
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