Fractional telegraph equation under moving time-harmonic impact

The time-fractional telegraph equation with moving time-harmonic source is considered on a real line. We investigate two characteristic versions of this equation: the wave-type with the second and Caputo fractional time-derivatives as well as the heat-type with the first and Caputo fractional time-derivatives. In both cases the order of fractional derivative 1 < alpha < 2 . For the time-fractional telegraph equation it is impossible to consider the quasi-steady-state corresponding to the solution being a product of a function of the spatial coordinate and the time-harmonic term. The considered problem is solved using the integral transforms technique. The solution to the wave-type equation contains wave fronts and describes the Doppler effect contrary to the solution for the heat-type equation. Numerical results are illustrated graphically for different values of nondimensional parameters. (c) 2021 Elsevier Ltd. All rights reserved.

Automated summary

This study investigates two characteristic versions of the time-fractional telegraph equation with a moving time-harmonic source on a real line, and successfully solves the problem using the integral transforms technique. The solution to the wave-type equation contains wave fronts and describes the Doppler effect, while the solution to the heat-type equation does not contain wave fronts.