Distributed-order equations; composite derivative; wave equation; Mittag-Leffler functions; Fox H-function; numerical Laplace inversion; Tauberian theorem

In this paper we investigate the solution of generalized distributed-order wave equations with composite time fractional derivative and external force, by using the Fourier-Laplace transform method. We represent the corresponding solutions in terms of infinite series in three parameter (Prabhakar), Mittag-Leffler and Fox H-functions, as well as in terms of the so-called Prabhakar integral operator. Generalized uniformly distributed-order wave equation is analysed by using the Tauberian theorem, and the mean square displacement is graphically represented by applying a numerical Laplace inversion algorithm. The numerical results and asymptotic behaviors are in good agreement. Some interesting examples of distributed-order wave equations with special external forces by using the Dirac delta function are also considered.