@janhardo I did some of it in first just I need to undrestand how they do it by maple in steps is clear what they did but how we do the same 

@dharr i will update some information from other paper, really  that one i provided from top is easiest one for explanation but i have other  paper and keep lookong for some thing better becuase i have to figure out and find this without this my papers gets rejected, so is important and in 2004 someone write the package for this i will put pictures of him too and another paper steps from 1993, i hope this is make to undrestand becuase if you get notion of topic you can solve anything , i can send you thus pdf in case your are intrested and the picture confuse you 

Moderator: papers removed for copyright reasons

https://doi.org/10.1016/j.amc.2004.10.055 (PDEPtest paper)
https://doi.org/10.1016/0895-7177(93)90224-M (Vlieg-Hulstman paper)
https://doi.org/10.1016/j.cpc.2004.04.005 (wkptest paper)

@acer  thanks a lot, in future i will use this if something come up i will ask again thank you so much....

@acer  but if all part be together  it will be better just linear and non-linear not seperate the one derivative two derivate and non-derivative so on..  and seperate by comma is becuase of that i needed to use in huge equation if seperated automatically like that i didn't use any other extra code 

Download L-N-L.mw

@janhardo the v is also a function so two  term in non linear v*u is non linear

@sand15  i try to explain in here , i hope find a way ,  just need a tactic of coding to arrange and trail what i explained i think it will be answer to this equation

same-problem.mw

@sand15  it is becuase of this term

(p[i]^2 - 1)*(diff(diff(f(x, y, z, w, t), x), x)/f(x, y, z, w, t) - diff(f(x, y, z, w, t), x)^2/f(x, y, z, w, t)^2)

in u if you watch is p[i] this p[i], i must be one or two
i think beside theta[1] is one and beside theta[2] is two but how arrange this

@sand15  your way is so good and really still i am watching your approach is not easy to do that, i already solve more than 30 pdes in this way and this is just begin when we find a[12],  thus problem are not solved before i have another equation  have same problem but we have to find this a[12] is not zero be sure about that is imposible to be zero,  in both of thus equation in top equation and in this one if you watch the R when we found it is not number  in other when i solve  is number and i didn't have problem , there must be a way for finding this a[12] 

same-problem.mw

@dharr  the result is same like before, i think this approach is not suatable for this equation i have to change it, Thanks Dr.

@dharr i did that, but answer is not what i  i am looking i am looking for a[0],a[1] and w,  i did a for a lot but this is problem come up i don't know how fix that

Thanks Dear Dr.David(@dharr) you are very helpfull for me on this topic, still i am work on this topic, and i did a great achivment and keep improving the work, this mehod  need a super computer for a long range of variable of (t) i saw your last graph which is so far from exact but if you make the range of t smaller it will be so small error or maybe zero,  your code is great like always, but i don't undrestand about this part when you write (infolevel[LAD] := 2) what mean of this 2 in the first example and we can change to 3 or more? i have mine code and i have to improve it, and i am intresting in your which is so compact, i don't know this method can hundle the hard one?  and if  need help  i will ask for help

@dharr  the pdetest i did it but have some condition need analytice method which in article i have to explain all of them 

num-Ex1_pdetest-satisfy.mw

please if possible make some pretty plot for me i needed i will work on it but nothing usefull get yet
also this table can be export?

table-e2.mw

@dharr i don't know how many term should i add to get that but must be there is a code do to infinite term ...

@janhardo  try to compare with result outcome is not usefull without  showing example?

@dharr same author paper, all paper of this author are the same, and i recognize one mistake  of paper which not multiply by (I=sqrt{-1}) but step for me is more important , please chekc your gmail too

e1.mw