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Fractional telegraph equation under moving time-harmonic impact.

Yuriy Povstenko1, Martin Ostoja-Starzewski2

  • 1Faculty of Science and Technology, Jan Dlugosz University in Czestochowa, Armii Krajowej 13/15, Czestochowa 42-200, Poland.

International Journal of Heat and Mass Transfer
|February 13, 2023
PubMed
Summary
This summary is machine-generated.

This study analyzes wave- and heat-type time-fractional telegraph equations with a moving harmonic source. The wave-type solution exhibits wave fronts and Doppler effects, unlike the heat-type.

Keywords:
Caputo derivativeFourier transformFractional calculusLaplace transformTelegraph equationTime-harmonic impact

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Area of Science:

  • Mathematical Physics
  • Fractional Calculus
  • Partial Differential Equations

Background:

  • The time-fractional telegraph equation models complex wave propagation phenomena.
  • Investigating fractional derivatives (order 1 < α < 2) is crucial for understanding anomalous diffusion and wave behavior.
  • The quasi-steady-state assumption is not applicable for these time-fractional models.

Purpose of the Study:

  • To analyze two distinct versions of the time-fractional telegraph equation: wave-type and heat-type.
  • To explore the mathematical behavior of solutions under a moving time-harmonic source.
  • To compare the characteristics of wave-type and heat-type fractional telegraph equations.

Main Methods:

  • The integral transform technique is employed to solve the governing fractional partial differential equations.
  • Two formulations are considered: one with second and Caputo fractional time-derivatives (wave-type), and another with first and Caputo fractional time-derivatives (heat-type).
  • Numerical simulations are performed to visualize the solutions for various dimensionless parameters.

Main Results:

  • The wave-type equation's solution demonstrates the presence of wave fronts and the Doppler effect.
  • The heat-type equation's solution does not exhibit these wave phenomena.
  • Solutions are distinct due to the different orders of fractional time-derivatives used.

Conclusions:

  • The fractional order and type of time-derivative significantly influence the wave propagation characteristics.
  • The integral transform method provides an effective analytical tool for these complex fractional models.
  • The study highlights the fundamental differences in behavior between wave-type and heat-type fractional telegraph equations.