@Adam Ledger In windows you can right-click on the .dll and choose properties. There is some information there, which may or may not have the authors etc; that depends on the .dll author.

@Mohamed19 I don't understand. Can you give an example of what you want as an answer.

@Mohamed19 It was unclear what your notation meant, and so I made a guess. Now it is unclear what form you want the answer in, so you will need to give more details of what you are looking for.

I answered this in the other thread https://www.mapleprimes.com/questions/228082-Derivative-Of-BesselJalpha-Sqrtu2v22uvcosphi but you want a sum form. You will need to specify the problem more clearly because your notation above is not clear and it is also not clear what form you really want.

@Mohamed19 I'm assuming the arguments are to be derivative of increasing order. Need to have fixed values of _k1 and _k3. Maybe something like:
 

Download Bell.mw

@Zeineb I suggest you set up your conjecture as a Maple sum, and see if Maple can simplify to zero under the appropriate assumptions.

The code you copy-pasted earlier worked OK as I already showed. Now you have edited that out of your question and are asking something else that is not clear. Suggest you upload your worksheet - use the large green uparrow to load the worksheet with the problem and someone can take a look at it.

@mehdi jafari I agree that it is not in general, but it is if tau=0 and sqrt(sigma1^2) = sigma1, i.e., if sigma1 is positive, which is an important case. 

Suggest you upload your worksheet using the large green up-arrow. Then it will be easier for someone to diagnose.

@arshl If you just want to solve the pde with the given initial and boundary conditions, then laplace/fourier is not the way I would go in this case and I do not know how to do this. Maple's pdsolve uses many methods that could help you solve it without transforms.

For the initial condition, you need to pay attention to @Rouben Rostamian's use of Heaviside, and then you can get the right result:
fourier(Heaviside(x)*(1-exp(-x/sqrt(2))),x,omega);

@arshl You have incorrectly factored out u from diff(u(x,t)*diff(u(x,t),x),x). But in any case, finding laplace/fourier transforms of products such as u(x,t)*diff(u(x,t),x,x) or (diff(u(x,t),x,x)^2 is problematic, and Maple is not succeeding here.

@Carl Love Yes, that was careless. I redid with Laplace, but last term might still have a problem. Not sure what assumptions are required.

@Carl Love Well, the PDF has all the information, though you have to recognize it as the normal distirbution. There are other things you can get (as on the Distribution help page), but as far as I can see, you can't back out the name of the distribution. I had the same hope as you, but it was dashed...

@one pound A lot depends on what you want to do. The sin-cos form with only positive frequencies is equivalent to the exp form with positive and negative frequencies. I usually work with the series with regular Maple commands such as integration to find the coefficients for a given periodic function.

Download Fourier.mw

@Carl Love Isn't the whole process called "changing variables" :)