Fractional order derivative of a function & fractional numbers' factorial.

So in this video, some of the fractional calculus, fractional order derivative (you don't need any background to get it) and how to find the factorial of a fractional number (you need to cover the FT and LT to fully understand it).

[johnnyfonseca]

The exponential form of sine has 2j in the denominator, not 2.

[barost]

excellent  explanation! please do more videos about fractional calculus and it's application in engineering. another topic i am interested in are PDE's. maybe you are familiar with this topic and you can teach about it

[DrunkenUFOPilot]

Cool, fractional derivatives and fractional Fourier in the same video!

[DrunkenUFOPilot]

Okay, now lets see you do fractionally iterated exponential functions. And of course, fractional derivatives of them, and using them in something resembling a Fourier transform, and of course fractionally iterating that!

[MrJohnsurf]

Let's take the case of the half-derivative and consider the following desirable properties for half-derivatives:

Let D stand for the derivative operator.

Desirable properties (1), (2) & (3) for half-derivatives:

(1) D^m (f) is single-valued and exists for polynomials and m can be 1/2 or any positive integer.
(2) D^1/2(kf) = kD^1/2(f) (where k is a constant -- Constant Multiple Rule)
(3) D^m(D^n(f)) = D^(m+n)(f) Law of Exponents)

Then we will have the following sequence of consequences from (1), (2) & (3) above:

Lemma 1: D^1/2(0) = 0
proof: D^1/2(0) = D^1/2(0*0) = 0*D^1/2(0) = 0 by (2) and assuming that D^1/2(0) is finite.

Lemma 2: D^1/2(1) = k for some constant k
proof: D(D^1/2(1)) = D^(1+1/2)(1) = D^1/2(D(1)) = D^1/2(0) = 0,
so D^1/2(1) is a constant -- call it k.

Lemma 3: D^1/2(1) = 0 (the above constant, k, is 0)
proof: D^1/2(1) = k, so
D^1/2(k) = D^1/2(D^1/2(1)) = D(1) = 0, but
D^1/2(k) = D^1/2(k*1) = k*D^1/2(1) = k*k by (2)
therefore k*k = 0, so k = 0

Lemma 4: D^1/2(x) = constant
D(D^1/2(x)) = D^(1+1/2)(x) by (3) = D^1/2(D(x)) = D^1/2(1) = 0

Lemma 5: 1 = 0
1 = D(x) = D^1/2(D^1/2(x)) = D^1/2(constant) = D^1/2(constant*1) =
constant*D^1/2(1) = constant*0 = 0

Therefore we can't have both the Constant Multiple Rule and the Law of Exponents holding for half-derivatives.

So my questions to you are:

1) What restrictions must you apply on the set of functions you are trying to define half-derivatives on so that you don't run into the above contradiction? Polynomials have problems as shown above.

2) If you want to keep the set of functions that half-derivatives can be applied to include polynomials, then which of the laws are you willing to give up -- the Constant Multiple Rule or the Law of Exponents? Because of the above contradiction, you can't keep both rules... yet they seem like very natural rules you would want to keep and they hold for ordinary derivatives. Maybe D(D^1/2(f)) does not equal D^1/2(D(f)) and the half-derivative operation does not commute with ordinary derivatives -- in which case the Law of Exponents doesn't hold.

It appears that you must give up the Law of Exponents or the Constant Multiple Rule or find some clever way to restrict the set of functions that fractional calculus is applied to. Which way from here for the fractional calculus? Do we give up commutation of fractional derivatives with ordinary derivatives and work without the Law of Exponents?

[dr.shalabhkumarmishra]

Excellent Work!!
Can you share some video on "how to solve polynomials with fractional power" and "fractional derivative of Log (x)"

[siddharthshivarkar]

Fourier transform is with respect to dw not with respect to dt

[msdkabi]

Thank you very much,
Can u suggest a book about this subject?

hello, where can i get i similar explanation wrriten? any biobliography sugested?

[yuriabreu]

Thank you for this video.
You teach well :)

[kindofmagicmike]

My one problem with this video is the fact that it ignores the inconsistency of the definition of Riemann and Euler definitions of the fractional derivative.

[axe]

Even though fractional calculus is awesome, tempered fractional calculus is more applicable to real world applications. 

[MuhannadGhazal]

hi my friend, here i am, subscribing your own channel, sorry for being late to, i'll watch all your series later,  

[leomico]

At least, it shows the relationship between the gamma function and Laplace transformation

[youcefyahiaoui]

oops! you forgot "j" in the denominator. It's actually divided by 2j for sin(x)...

[ministerdixon]

Hi Ahmed! Thanks a lot and keep up the good work... Ahmed! How can I find or use the fractional order in state space equation for its general solution? Example: dx/dt = Ax+B, whose solution is x1(t1, t0) = and x2(t2, t1)= for one periodic orbit of the system.
If d^(1/2)x/dt^1/2=Ax+B.... x1(t1, t0) =??? and x2(t2, t1)=???

[alirezamirghasemi]

Can`t we present the n`th derivative of Sin(x) as Sin(x + n(pi/2))?

[GAGANSACHDEVA]

Hi. There are some mistakes in your video. like Sine value in terms of exponential and also in fourier transforms....

[msdkabi]

Thank you very much,
Can u suggest a book about this subject?