@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

@dharr first don't worry about exact solution i solved a lot of this kind pde and harder than this, in thus pde have some condition about that parameter maybe becuase of that don't make it to zero, i can solve any pde and finding exact solution for you if needed, and there is a trick in physic which not all people know this,  which is if u(x,t)=f(x)*exp(#) then 

|u(x,t)|=f(x), in physic we do like that i never did that cahnge with conjugate when have power  is gonna |u(x,t)|^2=f(x)^2, i hope this is help full 
i have some 2025 article which i want to know  how you needed you want me to solve them then send to you or  you want i am uploaded the result here?

@dharr I can find a lot of exact solution is depend on analysis method in paper as you see the equation (2) and (9) are exact solution of pde, after we find the exact solution we apply laplace adomian decomposition method to get numerical solution, I saw a lot paper still fractional order remain I have to apply method for fractional order I will look for other paper which have more detail