Dear expert,
We see that there are many defination of fractional integral operator such as Rimann-livoulle, Caputo, Saigo, AB fractional Caputo Frabirzo and ect. In which some due to Singular kernel and some due to non Singular kernel which is the best fractional integral among all.
Fractional operator depends on problems because, in many problems, I have seen different types of singular and nonsingular operator are good, but in real-world problems, the nonsingular operators are effective and reliable compared to the singular operator.
Generally, all of the definitions can be used every fields. Especially, Caputo Fabrizio fractional operator are applied to different types of singular and nonsingular operator. However, we must know that, function is need to be absolutely continuous every point. If you interested in fractional differential equations, I recommend two different sources. First source, "An introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications 1st Edition - October 21, 1998. Author: Igor Podlubny". Second source " Fractional Calculus: Models and Numerical Methods (Complexity, Nonlinearity and Chaos) by Kai Diethelm(Author), Dumitru Baleanu(Author), Enrico Scalas(Author)". These two different sources mention that Definition of Grünwal-Letnikov fractional operator, Rieman-Liouville fractional derivative, Caputo definition. You can look them. My regards, Vaijanath Laxmanrao Chinchane
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