A Question about fractional calculus

[itex]E_\mu^t[/itex] is a function similar to [itex] e^t [/itex] in that [itex]D_t^\mu E_\mu^t= E_\mu^t [/itex]

Its often useful to define the inverse function. [itex]E_\mu^t[/itex] is similar to [itex] e^t [/itex] so we suppose that the inverse of [itex]E_\mu^t[/itex] should be similar to the inverse of [itex] e^t [/itex].

We call this new inverse function the generalized natural log, [itex]\ln_\mu[/itex], to highlight this similarity. In the case where [itex]\mu=1[/itex], [itex]E_1^t=e^t[/itex] and [itex]\ln_1 x= \ln x[/itex].

 

No. The function [itex] e^t[/itex] is always [itex] e^t [/itex], its definition does not change. The function [itex] E_\mu^t[/itex] is a different function, it sounds like it can be calculated using Equation 28 in the reference you posted (you didn't post the page that contains this equaion).

[itex] E_\mu^t[/itex] has the property [itex] E_1^t=e^t[/itex], but this equaility does not hold for all [itex]\mu [/itex]. For instance [itex] E_{1.3333}^t \ne e^t[/itex].

 

When [itex] \mu \ne 1[/itex] the function [itex] E_\mu^t [/itex] is defined by equation 28 in the reference you are using. The function [itex]\ln_\mu x [/itex] is its inverse. What part of this is unclear?

 

There are many examples of use in physics which are reported in the book of K.B.OLdham & J.Spanier "The fractional Calculus" AcademicPress, N.-Y.

The littérature on the subject is extensive. A short bibliography is given p.12 in the paper for general public : https://fr.scribd.com/doc/14686539/The-Fractional-Derivation-La-derivation-fractionnaire . Also, this paper shows an use of fractional differ-integration to generalize the basic electrical components.

 

Ralf Metzler has also worked a lot on these things. Here is a fairly straight forward introduction:
The random walk's guide to anomalous diusion: a fractional dynamics approach
http://www.tau.ac.il/~klafter1/258.pdf