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Note on fractional Mellin transform and applications.

Adem Kılıçman1, Maryam Omran2

  • 1Department of Mathematics, Institute for Mathematical Research, University Putra Malaysia, 43400 Serdang, Selangor Malaysia.

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Summary
This summary is machine-generated.

This study introduces the fractional Mellin transform using Riemann-Liouville fractional integral and Caputo fractional derivative. Researchers explored its properties and extended existing Mellin transform properties into the fractional domain.

Keywords:
Caputo fractional derivativeFractional calculusMellin transformRiemann–Liouville fractional integral and derivative

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Area of Science:

  • Mathematics
  • Fractional Calculus
  • Integral Transforms

Background:

  • Integral transforms are fundamental in solving differential equations.
  • Fractional calculus extends the concept of differentiation and integration to non-integer orders.
  • The Mellin transform is a generalization of the Laplace and Fourier transforms.

Purpose of the Study:

  • To define and investigate the properties of the fractional Mellin transform.
  • To extend existing properties of the Mellin transform to the fractional calculus domain.
  • To establish a foundation for applying fractional calculus concepts to integral transforms.

Main Methods:

  • Definition of the fractional Mellin transform utilizing the Riemann-Liouville fractional integral operator.
  • Definition of the fractional Mellin transform employing the Caputo fractional derivative.
  • Analysis of the fundamental properties of the newly defined fractional transform.

Main Results:

  • The fractional Mellin transform is successfully defined using both Riemann-Liouville and Caputo definitions.
  • Several key properties of the fractional Mellin transform were derived and studied.
  • Existing properties of the classical Mellin transform were successfully extended to the fractional domain.

Conclusions:

  • The fractional Mellin transform provides a new tool within fractional calculus.
  • The study establishes the theoretical underpinnings for further research in fractional integral transforms.
  • This work opens avenues for applying fractional Mellin transforms in various scientific and engineering fields.