Dear @Praveen, I am a bit familiar with the application of fractional calculus operators in engineering, especially in the are of system dynamics and control. I will attach some papers that might be helpful for this thread!In last twenty years, this approach becomes more significant with wide variety of engineering applications!
I think that you can find the best response for your question in books "I. Podlubny, Fractional differential equations" and "K. Diethelm, Analysis of fractional differential equations".
I found a fractional behaviour for the first time in an ECG, but I thought it was due to the electronic devices used in the aquisition system. Later I discovered the same behaviour in other signals. Recently I collaborate in modelling a capacitor taht is really fractional and revised papers with fractional sensors. In the last 20 years I've followed the publications on the subject and written a book and some papers. The most interesting engineering applications I found were done by Prof. Oustaloup. He was at the FSS2013 in Ghent and explained how to construct automobile dampeers.
I profit the opportunity to say that I am firmly convinced that the most used Riemann-Liouville and Caputo derivatives are objectively almost useless in applications and factors that impede the enlargement of the application fields since they avoid using classic results.
My research field is anomalous diffusion. I think the advantages of fractional calculus are good data fitting, non-locality description and easy-to-use, the disadvantages include physical interpretation, parameters determination and three dimensional analysis.