A Question about fractional calculus

[bubblewrap]

Okay, maybe not really fractional calculus but I don't know what this stands for. Its in the black circle (more like an ellipse though), what does the mu under the natural logarithm mean?

 

[the_wolfman]

E_\mu^t is a function similar to e^t in that D_t^\mu E_\mu^t= E_\mu^t

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

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

 

[bubblewrap]

What happens when$\mu$ is not 1? I don't understand how this is used. For example when $\mu$ is 2, does it mean that you use $e$ defined in $E^t_\mu$? What does that mean though?

 

[the_wolfman]

What happens when$\mu$ is not 1? I don't understand how this is used. For example when $\mu$ is 2, does it mean that you use $e$ defined in $E^t_\mu$? What does that mean though?

No. The function e^t is always e^t, its definition does not change. The function E_\mu^t 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).

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

 

[bubblewrap]

Yeah but as I said what happens when mu is not 1, other than it not being $e^t$?

 

[the_wolfman]

Yeah but as I said what happens when mu is not 1, other than it not being $e^t$?

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

 

[bubblewrap]

I think I understand now. One more thing, which area of physics uses fractional calculus? Is fractional calculus important in physics?

 

[JJacquelin]

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.

 

[Strum]

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