The generalized telegraph equation with moving harmonic source: Solvability using the integral decomposition technique and wave aspects.

The paper is devoted to study the frequency shift in the solution of the generalized telegraph equation with a moving point-wise harmonic source. This equation contains the nonlocality in time derivatives which is expressed by the memory functions η (t) and γ (t) , where η (t) smears the second time-derivative and γ (t) the first one. Moreover, in the Laplace domain we have η ˆ (s) = γ ˆ 2 (s). The generalized telegraph equation with an external source is solved by using the integral decomposition which allows us to write this solution as a product of the solution of the telegrapher equation with harmonic source and f γ ˆ (ξ , t) which is a function of the Laplace transform of memory function γ. Such obtained solution manifests the frequency shift which is illustrated in three examples of the memory functions γ (t) : the localized case, its mixture with power-law, and the power-law case only. We show that only the first two cases have the wave front and the Doppler-like shift. The third example, despite the lack of wave fronts, also manifests the frequency shift. Thus it turns out that the frequency shift occurs regardless of the existence of a wave front, but it is more visible when such a front exists. • The method of integral decomposition is applied to get solutions of the memory dependent evolution equations. • Jugging existence/non-existence of the Doppler shift is separated from the existence/non-existence the wave fronts. • It is shown that the Cattaneo-Vernotte equation may be used to describe the wave properties of the heat transport. [ABSTRACT FROM AUTHOR]