Abstract
The current manuscript examines the effect of the fractional temporal variation on the vibration of waves on non-homogeneous elastic substrates by applying the Laplace integral transform and the asymptotic approach. Four different non-homogeneities, including linear and exponential forms, are considered and scrutinized. In the end, it is reported that the fractional temporal variation significantly affects the respective vibrational fields greatly as the vibrations increase with a decrease in the fractional-order mu. Besides, the two approaches employed for the cylindrical substrates are also shown to be in good agreement for very small non-homogeneity parameter a. More so, the present study is set to play a vital role in the fields of material science, and non-homogenization processes to state a few.