@Rana47 

I've just seen that  @SHIVAS question has been removed and the the link
https://www.mapleprimes.com/questions/236755-How-To-Run-Or-Implement-HPM-Method-In-Maple

I sent you is obsolete.
Here is the file I attached to my answer to @SHIVAS

Download hpm_error_sand15.mw

@Rana47 

If you "do not know the other guys", why does your code contain the same names of variables, the same useless declare statement, the same way to write the ansatz, the same errors ?

"Please help to produce results mentioned in the pictures":

I do not inderstand what you are saying.
I answered @SHIVAS a few hours ago and uploaded a code.
Go and see the answer here
https://www.mapleprimes.com/questions/236755-How-To-Run-Or-Implement-HPM-Method-In-Maple

Another advice: search for the name homotopy on this cite, you will find several answers.
To cite few
https://www.mapleprimes.com/questions/223044-Homotopy-Perturbation-Method
https://www.mapleprimes.com/questions/227515-Homotopy-Perturbation-Method
https://www.mapleprimes.com/questions/232586-How-To-Solve-System-Of-Nonlinear-ODE

@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.