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Higher-dimensional physical models with multimemory indices: analytic solution and convergence analysis.

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Advances in Difference Equations
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Summary
This summary is machine-generated.

This study introduces a novel analytical method using the γ̅-Maclaurin series to simulate Caputo fractional derivatives in higher-dimensional models. The findings suggest these derivatives may exhibit memory characteristics in physical systems.

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Area of Science:

  • Applied Mathematics
  • Fractional Calculus
  • Mathematical Physics

Background:

  • Fractional calculus offers advanced modeling capabilities for complex systems.
  • Higher-dimensional models require efficient analytical techniques for fractional derivatives.
  • Understanding the interplay of temporal and spatial fractional derivatives is crucial.

Purpose of the Study:

  • To analytically simulate the impact of temporal and spatial Caputo fractional derivatives in higher-dimensional physical models.
  • To develop and validate a novel analytical technique for solving fractional differential equations.
  • To investigate the memory characteristics potentially represented by Caputo fractional derivatives.

Main Methods:

  • Employing the γ̅-Maclaurin series, an extension of the power series technique.
  • Conducting convergence analysis for the proposed γ̅-Maclaurin series.
  • Applying the method to higher-dimensional heat and wave models with Caputo fractional derivatives.

Main Results:

  • Successfully solved higher-dimensional fractional heat and wave equations using the γ̅-Maclaurin series.
  • Demonstrated rapid convergence of the series solutions.
  • Validated the accuracy of the solutions by comparing projections with existing literature.
  • Graphical analysis indicated potential memory effects associated with Caputo fractional derivatives.

Conclusions:

  • The γ̅-Maclaurin series provides an effective analytical tool for higher-dimensional fractional models.
  • Caputo fractional derivatives in these models may represent physical memory phenomena.
  • The developed method shows promise for future applications in fractional dynamics.