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Bridging the gap between models based on ordinary, delayed, and fractional differentials equations through integral

Noemi Zeraick Monteiro1, Rodrigo Weber Dos Santos1, Sandro Rodrigues Mazorche2

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This study unifies classical, delayed, and fractional models using a gamma Mittag-Leffler kernel for evolution equations. The framework incorporates historical data, enhancing mathematical modeling for systems like COVID-19 dynamics.

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Area of Science:

  • Mathematical Biology
  • Dynamical Systems
  • Fractional Calculus

Background:

  • Evolution equations with convolution operators are established, but a gap exists in linking specific kernels to advanced models.
  • Classical, delayed, and fractional differential equations often exist in isolation, lacking a unified theoretical framework.

Purpose of the Study:

  • To develop a unified framework for evolution equations using a general convolution kernel.
  • To demonstrate that classical, delayed, and fractional models are special cases of this framework.
  • To analyze the properties and applications of the generalized model, including its ability to incorporate historical data.

Main Methods:

  • Utilized a gamma Mittag-Leffler memory kernel to generalize evolution equations.
  • Classified different kernel types and analyzed the asymptotic behavior of the general model.
  • Performed numerical simulations and parameter analysis for kernel classification.

Main Results:

  • Established a unified framework encompassing classical, delayed, and fractional models.
  • Fractional models constructed maintain dimensional balance and explicitly link fractional orders to past data.
  • Demonstrated the model's ability to reproduce COVID-19 infection dynamics in Australia, Brazil, and Peru.

Conclusions:

  • The proposed unified framework effectively incorporates historical data via integro-differential equations.
  • This approach expands mathematical modeling capabilities for systems described by ordinary, delayed, or fractional differential equations.
  • The gamma Mittag-Leffler kernel provides a versatile tool for modeling complex dynamics with memory effects.