Exploring the Basics of Fractional Calculus: Prerequisites and Applications

[AdrianZ]

the title says everything. I encountered this term and I wanted to know what fractional calculus is and what it does.

 

[MathematicalPhysicist]

From the very little I read (and of which I can rememebr reading) from a textbook by Jerome Spanier and Oldham.

Fractional Calculus generalises the integration and differential operators.

For example have you ever wondered what $$\frac{d^{\frac{1}{2}}}{dx^{\frac{1}{2}}}$$ would stand for?

The book covers the theory and its application as I can remmber in stuff like diffusion equations and other stuff.

 

[AdrianZ]

I see, a complete answer would be appreciated.. what prerequisites I need to know to learn fractional calculus? do I need to have mastered partial differential equations to learn fractional calculus topics?

 

[JJacquelin]

An extension of derivatives and integrals to noninteger orders.
Fractional calculus is based on the Riemann-Liouville Integral Transform.
The literature is extensive. Just for introduction :
http://mathworld.wolfram.com/Riemann-LiouvilleOperator.html
http://mathworld.wolfram.com/FractionalCalculus.html
For general public (in French) : "La dérivation fractionnaire" in :
http://www.scribd.com/people/documents/10794575-jjacquelin

 

[MathematicalPhysicist]

I see, a complete answer would be appreciated.. what prerequisites I need to know to learn fractional calculus? do I need to have mastered partial differential equations to learn fractional calculus topics?

Well, the first few chapters you don't need more than Calculus 2-3 (chapter 1-5), chapters 6-7, it can be a plus if you have been exposed to ODE and special functions, the rest you really do need to know good PDE especially the last chapter which deals with diffusion.

 

[Matthollyw00d]

I first learned of it in a PDE course dealing with Sobolev spaces. Real Analysis by Folland and PDE by Evans both have a decent introduction in the latter part of the books. Folland gives the information to understand it in the previous part (though I found Folland to be quite difficult unless you already have a decent Reals background).

 

[JJacquelin]

what prerequisites I need to know to learn fractional calculus? do I need to have mastered partial differential equations to learn fractional calculus topics?

Not so much. Of course, you need to master Riemann integration. Riemann-Liouville Integral transform isn't more difficult than many other integral transforms, like Laplace-, Fourier-, etc.