The advantage of fractional calculus over integer order calculus is that it can explain the hereditary behavior of the model, which enhance the study of the particular model problem. My question is "How to explain this property if we take two different values of the fractional derivative (say, alpha_1=0.5 and alpha_2=0.75)? In other words what is the basic difference between these two values in physical point of view?
Very hard to answer this. My experience indicates that non-locality is a question often discussed in
physics, quantum theory, but I have never seen fractional calculus applied to this question.
So I think two completly different issues are involved here.
Regarding relevant applications, fractional calculus is still not very often used. It is a proceedure without any very clear
geometric meaning. Perhaps start with the simplest fractions first.
A fractional derivative is a function. As function, it has a domain. The fact that a function has a local property signifies that respective function has this property only on a restreint part of his domain( on a neighborhood of a point of his domain, for example).
If the function has a property relative to its whole domain, then we say that the property is global.
Thank you @Dinu Teodorescu for your response. But how it changes with change in the fractional order?
Generally, the properties of fractional derivatives are global(non-local) and can be deduced from properties of Euler's Gamma function, from properties of integrals and from considered value of fractional order.
For example, in case of Caputo fractional derivatives, properties change when the exponent a+1-n goes from positive to negative!
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