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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.
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.
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