Fractional telegraph equation under moving time-harmonic impact. (January 2022) - Povstenko, Yuriy; Ostoja-Starzewski, Martin
Highlights: The time-fractional telegraph equation with moving time-harmonic source is considered on a real line. 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 are investigated. The solution to the "wave-type" equation contains wave fronts and describes the Doppler effect contrary to the solution for the "heat-type" equation. 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 derived solutions can be successfully used when the source term can be expanded into a Fourier series. Abstract: 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 < α < 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.