Fractional telegraph equation under moving time-harmonic impact

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.