Fractional relaxation and fractional oscillation models involving Erdelyi-Kober integrals

We consider fractional relaxation and fractional oscillation equations involving Erdelyi--Kober integrals. In terms of the Riemann--Liouville integrals, the equations we considered can be understood as equations with time-varying coefficients. Replacing Riemann-Liouville integrals with Erdelyi--Kober-type integrals in fractional oscillation models, we obtain some more general integro-differential equations. The corresponding Cauchy-type problems can be solved numerically, and, in some cases analytically, in terms of Saigo--Kilbas Mittag--Leffler functions. The numerical results are obtained by a treatment similar to that developed by K.~Diethelm and N.J.~Ford to solve the Bagley--Torvik equation. Novel results about the numerical approach to the fractional damped oscillator equation with time-varying coefficients are also presented.