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

[@johnnyfonseca354]

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

[@barost2010]

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!

[@devarora3770]

Great nicely explained...!!

[@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!

[@DrunkenUFOPilot]

... and set the order of iteration of all those things to sqrt(i)

[@radhigam3091]

Sr, very nice explanation.

[@faizachishti5266]

good explanation

[@yuriabreu8791]

Thank you for this video. You teach well :)

[@siddharthshivarkar3903]

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

[@dr.shalabhkumarmishra3571]

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

[@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.

[@hamzaa.8082]

Thanks for uploading :) good luck!

[@AhmedIsam]

subscribe please 3:)

[@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?

[@DB-nl9xw]

More videos like this!

[@msdkabi4365]

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

[@DrunkenUFOPilot]

One of the great classics in fractional calculus is a book by Oldham & Spanier, "The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order"

[@DrunkenUFOPilot]

I list other literature in my Quora answer to the question "Differential Equations: What is d^(1/2)x / (dy)^(1/2) " (bad attempt to write LaTeX formula in plain ascii. SAD!) (note the questioner has y, x reversed from usual usage) If YT allows links, go to: qr.ae/TcthvP

[@djrobby20xx]

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) = exp(A1(t1-t0))(x(t0)+A1^-1*B)-A1^-1*B and x2(t2,t1)= exp(A2(t2-t1))(x(t1)+A2^-1*B)-A2^-1*B for one periodic orbit of the system. If d^(1/2)x/dt^1/2=Ax+B.... x1(t1,t0) =??? and x2(t2,t1)=???

[@sefatullapamiri3413]

hi brother if u have any monograoh or theiss about fractional differential equation please send me that by this email address , [e-mail]

[@djrobby20xx]

I wrote this two years ago... No answer obtained...

[@djrobby20xx]

@@sefatullapamiri3413 Alexsi in Tallinn did a masters thesis on Fractional order... my email [e-mail]

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

[@MuhannadGhazal]

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

[@AhmedIsam]

A Warm welcome ... Enjoy ^^

[@youcefyahiaoui1465]

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

[@Sinanmmd]

Sinh(x)

[@taimurzaman7322]

thumbs up :)

[@mohamedismail810]

What is the reference please

[@axe863]

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

[@macmos1]

axe863 what do you mean by tempered

[@nuclearrazorify]

Thank you!!!

[@sefatullapamiri3413]

hi brother if u have any monograoh or theiss about fractional differential equation please send me that by this email address , [e-mail]

[@leomico6394]

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

[@youmah25]

شكرا

[@alirezamirghasemi]

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

[@AhmedIsam]

Yes we can, that's a very interesting formula btw. Do you know how to derive it ?

[@alirezamirghasemi]

unfortunately I don't know how to derive it analytically. but creating a pattern is very easy. Sin(x) = Sin(x + 0) Cos(x) = Sin(x + (pi/2) -Sin(x) = Sin (x + pi) -Cos(x) = Sin(x+ (3*pi/2)) I think Cos(x) can be interpreted as a scaled version Sin(x) that causes function to shift by 90 degrees just like j in complex numbers, when is multiplied it shifts the angle of point by 90 degrees.

[@GAGANSACHDEVA04]

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

[@loganborghi5727]

yeah, when he did the 0.6th derivative of sin x he used sinh x formula

[@pettPette]

the vector v must be non-zero!

[@msdkabi4365]

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

[@AhmedIsam]

Check out this amazing "simple" Indian ebook www.springer.com/engineering/computational+intelligence+and+complexity/book/978-1-4020-6041-0 please subscribe to encourage me to do more.

[@msdkabi4365]

Tnx a lot doctor Ahmad