  We consider fractional relaxation and fractional oscillation equations
involving Erdelyi-Kober integrals. In terms of Riemann-Liouville integrals, the
equations we analyze can be understood as equations with time-varying
coefficients. Replacing Riemann-Liouville integrals with Erdelyi-Kober-type
integrals in certain 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.
