0 years, 45 days

## Direct integration...

@Carl Love

Thanks.

(1) I actually want the spectrum, that is S(w)=|F(w)|^2 -- I tried doing this on the answer but it left it in too general a form  -- have a look at the last example on my attached sheet. I am sure that this could be simplified further, using trig and CylinderD identities ?

(2) In Maple, when a write or obtain get a function of a parameter such as in the last example g(w;a)=|A|^2 (w;a), where a is a parameter, how can I put in specific choice of a; e.g. g(a=0.3)? (Sorry, but I am comletely new to Maple!)

(3) How can I plot the fuction g(w;a=0.3) in the above example ?

Thanks

 (1)

 (2)

 (3)

 (4)

 (5)

 (6)

 (7)

 (8)

 (9)

 (10)

 (11)

## CylinderD / why fourier function does no...

Thanks.

What does CylinderD stand for?

Is there a reason why the "fourier" function does not do thi?

Strictly speaking, the function itself is singular at t=0, so really should be taking the FT of f(t)*Heavside(t). How would you evaluate the FT/integrals of the following functions using Maple:

(1) f(t)=(1/t^a)*Heavside(t)

(2) f(t)=(1/t^a)*exp(-t^2)*Heavside(t)

Thanks

 (1)

 (2)

 (3)

 (4)

 (5)

 (6)

 (7)

 (8)

 (9)

## InfinitesimalGenerator(3, u(x, t))...

Thanks. When I do "InfinitesimalGenerator(u(y, R))", I get the same error message.

A follow up question is, ifI want to impose the conditions on F1, F2, F3, F4 in She et al, how do I do that? Maybe that is what is required? In other words, explictly state the infinitesimals, like _zeta_y =y,  etc. Can that be done -- can you give me an example please.

Thanks

 (1)

 (2)

 (3)

 (4)

{InfinitesimalGenerator(u(y, R))}

## InfinitesimalGenerator(3, u(x, t))...

Dear Dharr,

I have followed this for a real problem in Z.-S. She et al, J. Fluid Mech. (2017),vol. 827, p.322. I get an error message below. (There is actually a line missing when I save, which I have put in by hand in curly brackets { - }.)

I note that when I do, infinies[-1], I do not get it printed out properly? Then InfinitesimalGenerator fails, which is what the error message refers to.

In She et al, they have F3-F4=0, and F2=1, F3=aR (a is a constant) -- but this is not eadily apparent from the output below? They then obtain the invariants, which is what I want to do using Phi=Invariants(infinies[-1],l,u)  - or equivalent - but I have not got there yet.

Thanks

 (1)

 (2)

 (3)

 (4)

 (5)

{InfinitesimalGenerator(infies[-1], u(y, R))}

## InfinitesimalGenerator(3, u(x, t))...

Thanks Carl. Things are getting better -- slowly.

## InfinitesimalGenerator(3, u(x, t))...

Actually, second thoughts, I will continue with the questions here - save me time having to repeat.

(Incidently, how do I attach a worksheet to this post? In fact how do I save a worksheet in Maple?)

After generating the infinitesimals as noted, I want to generate the group invariants. In the examples they refer to the last part of equation (??) -- which here in my worksheet is equation (3). I have tried, but I cannot get (3)_-1 (this is meant to be a subscript of -1 to (3). I tried this and got the error messaage:

InfinitesimalGenerator((3), u(x, t))
Error, invalid input: PDEtools:-InfinitesimalGenerator expects its 1st argument, S0, to be of type {procedure, list(`=`), list(algebraic)}, but received 3

The only way I can get the invariants is to explicitly put in the infinitesimals by hand first in to

InfinitesimalGenerator([...],u(x,t))

and then in to

Phi := Invariants([...],u(x,t))

where [...] is the explicit list of infinitesimals.

What am I doing wrong?

Thanks

## Thanks guys, restart: with(PDETools): ...

Thanks guys,

```restart:
with(PDETools):
U:= diff_table(u(x,t)):
declare(U[]);
PDE:= U[x,x] - U[t] = 0;  #Note the square brackets!
Infinitesimals(PDE);```

This works now, but it confuses me more because why didn't the standard example work?

What I gather from this is that declare is best done through declare(U[]) -- it's the only difference that I can see.

(Note: when a copy and paste on to this forum, some things are not correct, thus U_x,x actualy does appear correctly as Ux,x and U_t as Ut on my maple. There are other minor things like '" : " sometimes appears " ; " after copy.)

OK, that is dortec out, but other problems appear, but I'll raise them in another post.

Cheers