Linked Questions

9
votes
4answers
662 views

What is the physical meaning of fractional calculus?

What is the physical meaning of the fractional integral and fractional derivative? And many researchers deal with the fractional boundary value problems, and what is the physical background? What ...
17
votes
7answers
884 views

Applications of Fractional Calculus

I've seen recently for the first time in Special Functions (by G. Andrews, R. Askey and R. Roy) the definitions of fractional integral $$(I_{\alpha }f)(x)=\frac{1}{\Gamma (\alpha )}\int_{a}^{x}(x-t)^...
0
votes
0answers
96 views

What is the relation between quantum calculus and fractional calculus?

Why did we need the fractional calculus? Does It have any relation with the quantum calculus or discrete quantum calculus?
9
votes
2answers
363 views

Do different methods of calculating fractional derivatives have to be equal?

Do different methods of calculating fractional derivatives have to be equal? Or do they sometimes end up differently? An example would be nice, and if possible, an explanation as too why such ...
0
votes
1answer
71 views

How to compute $(\int f(x) \, dx)^p$ with fractional number $p$?

It is well-known that one can say $(\int f(x) \, dx)^p = \int \prod_{i=1}^p f(x_i) \, dx_i$ if $p$ isa natural number. But what is if $p$ is a fractional ore even a real number? Is it possible to set $...
1
vote
2answers
1k views

How to define the Nabla-Operator

As I began to teach myself in differential geometry, I finally used to use the Nabla-Operator. I know and understand its usage as in $$ \nabla f := \left( \begin{matrix} \frac{∂f}{∂x_1} & \frac{∂...
7
votes
1answer
263 views

More unknown / underappreciated results of Euler

What are some of the more unknown and/or underappreciated things that Euler discovered? The man has done so much that there's bound to be notable results that most people aren't aware of. This could ...
7
votes
3answers
444 views

What is the half-derivative of zeta at $s=0$ (and how to compute it)?

[Update 3:] I gave a new partial answer following the ansatz in question Q3. I leave the other parts of the question untouched, they are also partially answered in specialized other questions in MSE. ...
2
votes
2answers
152 views

Differential operators confussion

I want to solve this PDE: $$u_t-6uu_x+u_{xxx} = 0\,(1)$$ with the Inverse Scattering Method. This method is based on showing that the above equation can be expressed as $$L_t=LB-BL,\,(2)$$ where $L$ ...
1
vote
2answers
178 views

How to make sense of this calculus notation, Advanced College Level

I have $f(x)$=$(2x,e^x)$ what does this notation mean? Notation: $Df(\frac{∂}{∂x})$ Certainly $Df(x)$=$(2,e^x)$ but how can I replace $x$ with $\frac{∂}{∂x}$? Particularly, how can I make sense of $...