@janhardo the mistake is here, when we define adomian my adomian is wrong, how   becuase for equation one A[0]=invlaplace[1/s*laplace(v[0]w[0])] i can't define be two function w and v becuase we have initial condition for u[0] v[0] and w[0] in finding u[1] we use v[0] and w[0] in finding v[1] we use u[0] and w[0] in finding w[1] we use u[0] and v[0] !

the equation is true but i did another mistake in adomian which we take derivative adomian for finding u(x,y,t) the nonlinear term diff(v,x) *diff(w,x) is like that diff(v,x)*diff(w,y) also for v(x,y,t) and w(x,y,t) it is clear in picture and also in mw.

@janhardo you are work hard on it, i have shame if i ask you another question about  this topic, is so intrested and so fun if  undrestand each part in this adomian it is  a minut for you but for me is so hard, for second example i did each part but this adomian make a problem for me i can't make it true with two diferent function if have time just look at  it if intrested give me a hand, hhh ofcurse you did all of this tahnks again..

adomian have a lot rule to written i have some other but this is easiest one but can't apply it for all function

ex2.mw

@janhardo watch example is so simple watch this part  A[0] and B[0] are the same 

before taking laplace give us zero and exp(x)*t is our initial after taking laplace inverse which is u[0], you did adition is minus between them 

and in using adomian if we have both it will be better the function before substitute and after A[0]=u[0]^2 then A[0]=exp(x)*t like that 

@janhardo you did a great job , specialy when you calculated this u[0] with substitute, but u[1] must be zero i know you put them in equation and all satisfy that is so good is emazing really,but this is importan for future example we need reach that u[1] must  be zero  becuase A[0] and B[0] are the same with different sign so add them will be zero, and we use this u[1] for future u[2] so when u[1] is zero u[2] automatically will be zero, are you so near for get results but my openion for you  if you undrestand the definition of adomian polynomial and LDM(laplace decomposition method) you can easily make it so better

@janhardo  i will check more for your code i need some time but in this example u[1](x,t) and other u[2] and so one all of them are zero as you see in picture i will upload paper  you can watch more for more undrestanding like that will be better but please if you done the work upload  here i need that is so intrested

3_2.pdf

this is have same example

@janhardo  is so good but need a little bit more compact, in final we need that u[1] and u[0] in function term not symbolic which we find one by one of them in each step seperatly u[0] is clear is initial condition with RHS of function when we take laplace inverse, but for u[1] for this equation we just have nonlinear function at RHS so, we calculate like that u[1]=invlaplace(-1/s^2*A[0]+1/s^2B[0]) this is will be u[1] for finding u[2] we have same procces u[2]=invlaplace(-1/s^2*A[1]+1/s^2B[1]) then if we calculate the final result we get a function if we substitute in our PDE must satisfy i calculate 3 of them but in fact it is go to infinity, for any other question and more undrestanding of this  topic please tell me i can help with full detail 

@janhardo  you have talent to make a good procedure i don't have that yet, i know each step to make procedure but i can't because of coding your procedure is good but have some issue like in final results not substitute u[0] u[1] and so on maybe @acer , @nm , @dharr , @Kitonum , @mmcdara , can give us a hand ofcurse they will if they have a time .

@janhardo I am looking for solved whole example by procedure and I think no one can do that 

@acer in fact i don't know anything in front you, your code are so cool, if you have any lecture note of yourself please provide for me, i wanna learn slowly slowly i want teach maple in university

@janhardo ................................

@Alfred_F  this ode equation is coming out from PDE equation have three step this is second step which i must find the solution of ode by the exist method but in some paper they find solution of this auxiliary equation and then by one case they can find 20 or even more solution so i want know how to do that then i can do that too it will be so easier if for each auxiliary equation can do that and the number solution will be increase by this we can modified method  and do sub equation methods....

@acer you did a great job but

in your case why calculate seperatly not all our parameter toghether map must be [4]? and a[0] if zero is ok but each parameter of ODE equation and in series solution assumption a[1] must not be zero which multiply by G(xi) ! in paper assumption he get 15 and more  how he achived them he must use limit i think he used limit for auxiliary equation

@janhardo ..................................................

@nm  this is your last answer? How if we don't have data like that i want find more we can finded by changing the limit or not? Like you did if we say m->1 left say m->2 right what will happen?

@janhardo I don't know why they did that