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Fractal Mellin transform and non-local derivatives

  • Alireza Khalili Golmankhaneh EMAIL logo , Kerri Welch , Cristina Serpa und Palle E. T. Jørgensen
Veröffentlicht/Copyright: 20. November 2023
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Abstract

This paper provides a comparison between the fractal calculus of fractal sets and fractal curves. There are introduced the analogues of the Riemann–Liouville and Caputo integrals and derivatives for fractal curves, which are non-local derivatives. Moreover, the concepts analogous to the fractional Laplace operator to address fractal non-local differential equations on fractal curves are defined. Additionally, in the paper it is introduced the fractal local Mellin transform and fractal non-local transform as tools for solving fractal differential equations. The results are supported with tables and examples to demonstrate the findings.

MSC 2020: 28A80; 34B05; 34M45

Funding statement: Cristina Serpa acknowledges partial funding by national funds through FCT-Foundation for Science and Technology, project reference: UIDB/04561/2020.

A Appendix

In this section, several formulas are provided, which are referenced from [49, 44]. These formulas play a crucial role in the analysis and calculations presented in the paper.

The Mittag–Leffler one-parameter function is defined by

E α ( x ) = k = 0 x k Γ ( α k + 1 ) ( α > 0 , β > 0 ) .

The Mittag–Leffler two-parameter function is defined by

E α , β ( x ) = k = 0 x k Γ ( α k + β ) ( α > 0 , β > 0 ) ,

where E α , 1 ( x ) = E α ( x ) .

The gamma function is defined by

Γ ( x ) = 0 e - τ τ x - 1 𝑑 τ .

The beta function is defined by

𝐁 ( x , ν ) = 0 1 τ x - 1 ( 1 - τ ) ν - 1 d τ ( Re ( x ) > 0 , Re ( ν ) > 0 ) .

The beta function relates to the gamma function by the following equation:

𝐁 ( x , ν ) = Γ ( x ) Γ ( ν ) Γ ( x + ν ) .

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Received: 2023-03-16
Revised: 2023-06-22
Accepted: 2023-06-28
Published Online: 2023-11-20
Published in Print: 2024-06-01

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