@Rana47 

Did the three of you guys @Rana47 , @SHIVAS and @Madhukesh J K work together to ask the same question, or are you all the same person?

In any case I think that a correct attitude would be to reply the answers instead of opening a new thread on the same subject , with exactly the same bugged code.

Look here
https://www.mapleprimes.com/questions/236758-HPM-Error-Invalid-Input
and here
https://www.mapleprimes.com/questions/236747-Error-In-The-Program-HPM

As your code is strangely close to a recent code from another OP (are you and  @Madhukesh J Kthe same person?), I advice you to look here
https://www.mapleprimes.com/questions/236747-Error-In-The-Program-HPM
to see how HPM can be done.

@arashghgood 


You didn't answer all my remarks, for instance remark 2: where does a term like

(3/4)*k*omega*eta(k,t)*eta(k,t)

comes from?
It can come only from the Fourier transform of zeta(x, t) by its derivative wrt x:

Download Remark_2b.mw

Does the"main equation in real space" contain such a convolution product?

More suspect is the last term in your transformed ode: what is it term in real space where it come from?

Download Remark_3.mw

I think you should provide us the "main equation in real space" with its initial and/or boundary conditions.

in the system you dsolve.
Did you mean equa?

@acer 

@Hullzie16 code worked perfect well with Maple 2015.2.
Maybe this information will help the development team to find where the bug comes from.

@dharr 

I voted up, but I believe there are a few points you could consider.

Firstly I don't thik writting K[3] = K[1]*k[3] is appropriate because neither K[1]nor K[3] intervene by themselves in the equation:  eqn only depends on the difference K[3]-K[1].
I think it would have been better to set K[3] = K[1] + k instead, which reduces the number ot parameters by 1.

Next, dimension analysis shows that gamma[1] and gamma[2] have the same dimension: this could lead to a simpler final equation with still one less parameter.

(last point: it is easy to show that theta is dimensionless).

Completing @acer's reply:

• You meant probably k*alpha instead of ?
  Copy-pasting into a 1D mode worksheet mode

  `kα`

• There is no need do declare GAMMA as local as soon as you define GAMMA... like the original (when its argument is an integer).
  By the way

  GAMMA(k*alpha+1) / GAMMA(k*alpha+alpha+1) = GAMMA(k+1) / GAMMA(k+2) = 1/(k+1)
  

• Within your solve command, the first argument is

  1/(k+1)*(diff(U[k](x), x, x)+2*sum(U[r](x)*(diff(U[k-r](x), x)), r = 0 .. k)-(diff(sum(U[r](x)*U[k-r](x), r = 0 .. k), x)))
  

  (
  I used sum instead of add to "show" what this term [TERM] looks like
  
  )
  What result do you expect by doing

  solve(TERM, U[k+1](x))

  given TERM doesn't contain U[k+1](x) ?

@raj2018 

Integrating formally f1^4 is a dead end for the same reason that integrating f2 is.
f1^4 contains terms of the form:

(1-x/a)^(-kc+1/2)*(1-x/b)^(-kh+1/2)

So the only possibility is to use a numerical integration.

Several errors:

1. No termination of the loop (end do is missing).
2. What do you expectto do writting f[i] = f[i+1] ?
3. n being undefined there is no chance that an hypothetical result matches the expected one (which doesn't contain n).
   Are those n typos?
    

Point 1 put apart, your problem has nothing to do with Maple: this is more a question about how you are capable or not to write correctly your recurrence relations.
Fix all these points and come again with something more consistent if you still get errors.

@Anthrazit 
Good luck

@Anthrazit 

I see...
I can't do more for you and I hope someone else will fix your problem.
As a last resort, even if I know this is a lot of work, could it be an option for you to use only Maplets?

@dharr 

For information:
Concerning the square root of symmetric positive definite matrices I submitted a question here
https://www.mapleprimes.com/questions/235834-MatrixPower-Doesnt--Give-The-Right-Answer

Extremely useful!
I agree with @C_R, a post would seem to me more appropriate and would give it more exposure.

@Preben Alsholm 

Given the definition of the integration over an infinite domain (simply definition in the sequel):

int(C, x=0..+infinity) = limit(int(C, x=0..a), a=+infinity);

one gets

limit(int(C, x=0..a), a=+infinity);
eval(%, C=0)
                       signum(C) infinity
                           undefined

In the other way

limit(int(0, x=0..a), a=+infinity);
                               0

Does this difference comes from the way eval acts?
---------------------------------------------------------------------------------------------------
Now consider this

int(0, y=0..1, x=0..+infinity);
                           undefined

But, applying the definition:

int(limit(int(0, x=0..a), a=+infinity), y=0..1);
limit(int(int(0, y=0..1), x=0..a), a=+infinity);
limit(int(0, y=0..1, x=0..a), a=+infinity);
                               0
                               0
                               0

So, do we have to consider that Maple doesn't apply the definition when it directly computes the integral?

@shreen 

Approximate_Solution.mw