15 minutes sounds about right.

@dharr The paper keeps 4 variables in A[i], with both diff(U(x,t),x) and diff(conjugate(U(x,t)),x) present. When I do that, I get the wrong coefficients for A[2], but the right number of terms, and the sums of subcripts = 2 as required. On the other hand in your post here you get the coefficients correct for A[2], but you do not have any diff(conjugate(U(x,t)),x).

There is definitely some difficulty about processing diff(conjugate(U(x,t)),x) that I do not understand - I have been careful to substitute for U(x,t) then conjugate so that the differentiation sees only expressions without conjugate. It's not clear to me how to proceed from here.

The whole process with the lambda's can be replaced by some recurrence formulas, but these have partial differentiation wrt the v[i] and again the conjugate variables make this hard to work out.

Download Laplace_Adomiian_Decomposition_method_3.mw

@dharr This is how I would approach it, but there is something I'm not understanding about Eq 8.12 (aside from it suggesting setting lambda=0 before differentiating wrt lambda. I'm out of time for the next few days.

Download Laplace_Adomiian_Decomposition_method_2.mw

@rlewis Click the green up-arrow; then choose the file, click upload, then click insert link. (Sometimes insert content doesn't work, though the link will usually work.

@janhardo You have 

uxx := diff(u[0](x), x $ 2);

but u[0] is already a function of x, so it should be

uxx := diff(u[0], x $ 2);

This is the basic way to do it, with everything in a big loop.

@salim-barzani As I said before, 

-diff(u[i], x $ 2)*I + A[i]

should have some t dependence or you can't expect to correctly take a Laplce transform wrt t. So you need to put the dependence of t in here (or just have it in from the beginning).

@salim-barzani I didn't see where you divided by 2. P[1] looks the same in both versions?

Download P1.mw

int(convert(1/A, Heaviside), x=-1..1);

@salim-barzani Your T[n] in DrD.mw has conjugate(u[k]) outside the summation, so it just stays as k.

Interesting. Nested radicals frequently give me problems. It is interesting that solve thinks they are the same.

expr:=sqrt(x+2*sqrt(x-1));
expr2:=sqrt(x-1)+1;
solve(expr2-expr,x);

gives x.

Probably you can use polygon from the plottools package for this, but if you upload your worksheet using the green up-arrow in the Mapleprimes editor, someone will help you.

@salim-barzani You didn't attempt to use Diff - all your Q's are zero because your B1 are zero.

@salim-barzani Does Diff work for you?

Download Diff.mw

@salim-barzani I didn't understand that. Can you give a very simple example of where conjugate doesn't allow you to do something which you want to do?

@salim-barzani You say "for diff(conjugate(u(x,t)),x) i use v[i,x]", but diff(conjugate(u(x,t)),x) is a function of t but v[i,x] is not. So for example the Laplace transform wrt t of v[i,x] will not be correct. If you fix these things up, I'll take another look.