Fractional Klein-Gordon equation for linear dispersive phenomena: analytical methods and applications

In this paper we discuss some explicit results related to the fractional Klein--Gordon equation involving fractional powers of the D'Alembert operator. By means of a space-time transformation, we reduce the fractional Klein--Gordon equation to a case of fractional hyper-Bessel equation. We find an explicit analytical solution by using the McBride theory of fractional powers of hyper-Bessel operators. These solutions are expressed in terms of multi-index Mittag-Leffler functions studied by Kiryakova and Luchko. A discussion of these results within the framework of linear dispersive wave equations is provided. We also present exact solutions of the fractional Klein--Gordon equation in the higher dimensional cases. Finally, we suggest a method of finding travelling wave solutions of the nonlinear fractional Klein--Gordon equation with power law nonlinearities.