Inverse Laplace transforms often require assumptions to work. Here assuming alpha is positive works, at least for Ex 1.

Ex1.mw

file menu, bottom right corner - labeled options.

@salim-barzani I didn't use your ugly code because you did not have the correct x,t dependence in it at the time, though you since fixed that. But I also wanted general code - there is no reason to have special P,Q,R for every case when you can go directly to A[n].

Thanks for pointing out the missing factorial. You may be suggesting something else is missing but if so I didn't understand what you meant. So I think this code is correct now. The conclusion is now the same as @acer's in the thread here, i.e., that the paper has an error.

Download Laplace_Adomiian_Decomposition_method_4.mw

The error message indicates a problem with floats. Ideally, you should do the whole thing symbolically and then substitute in values, but that takes too long, probably because the expressions get too complicated. So you can convert the float parameters to rational; run through the loop and then evalf the results. Then you will need to change the parameters and run it again. It's possible that with numerical parameters you don't need the assumptions and/or the simplify (one or both), you you could try that for faster operation.

error-problem2.mw

It means there are some assumptions involving these variables. You can avoid displaying them with Options->Display->Assumed variables and choose No Annotation (or you might prefer Phrase).

There is no BlockDiagonalMatrix command. When the output of a command has the same command in it, that is a hint that the command does not exist. However, DiagonalMatrix does what you want.

Y8 := DiagonalMatrix([Y4, Y4_R]);

I don't understand the logic of what you are trying to do well enough to answer your other questions.

pdetest_explained.mw

pdetest_fixed.mw

Download parts.mw

Edit: version that accepts the set of parts as a third argument

f := proc(p::posint, n::{posint, range(posint), list(posint)}, parts::set(nonnegint))
  local sys, letters, s, w, addition, ser, z, u, letts, _s, i, cu;
  letts := seq(cat(_s, i), i in parts);
  sys := {letters = 'Union'(letts), w = 'Set'(letters), letts =~ 'Atom'};
  addition := (map2(apply, s, {letts}) =~ parts) union {s(w) = 'Set'(s(letters))};
  Order := p + 1;
  ser := combstruct:-agfseries(sys, addition, 'unlabeled', z, [[u, s]])[w(z, u)];
  ser := collect(convert(ser, 'polynom'), [z,u], 'recursive');
  cu := coeff(ser, z, p);
  if n::posint then
    coeff(cu, u, n)
  else
    [seq(coeff(cu, u, i), i = n)]
  end if
end proc:

e.g., f(15, 100, {0, 1, 4, 6, 8, 9}); gives 129.

f(15, 90 .. 100, {0, 1, 4, 6, 8, 9}); gives [201, 187, 186, 175, 172, 159, 158, 147, 143, 133, 129]

f(15, [90, 92, 100], {0, 1, 4, 6, 8, 9}); gives [201, 186, 129]

The following is just a programatic version of how you made U. When you say "generalize to higher dimensions" I'm assuming you mean larger matrices, not things in more than two dimensions that wouldn't be matrices.

MMflatten := proc(Q::Matrix(Matrix))
  local m, n, i, j;
  m, n := op(1, Q);
  <seq(`<|>`(seq(Q[i, j], j = 1 .. n)), i = 1 .. m)>
end proc:

The first error message says "Warning, expecting only range variable x1 in expression 29239.76608*Ex*x1+794956140.3-29239.76608*Ma-131578.9474*Ey to be plotted but found names [Ex, Ey, Ma]". So you have not given numerical values to Ma, Ex, and Ey. The thing you are trying to plot is

29239.76608*Ex*x1 + 7.949561403*10^8 - 29239.76608*Ma - 131578.9474*Ey

but it can have only the name x1 in. You can try, say 

and you do get a plot, but I don't know what values of these parameters would be sensible.

Download solve.mw

I don't think you can expect sum to give an analytical result, but you can use add to get a numerical result for summing up a large but finite number of terms. (Of course you can use ":" before the evalf to suppress the fraction.) There are probably ways of speeding this up if you want more terms and are only interested in a floating answer.

Download igcd.mw

I usually use a colon at the end of an assignment statement storing a plot in a variable, then display it:

plt0 := implicitplot3d(pln, x = -3 .. 3, y = -3 .. 3, z = -3 .. 3, style = patchnogrid, transparency = 0.6):
display(plt0);

What you see with a semicolon in the first line has changed over various versions (plot structure, miniplot, full plot); I didn't check here. But also I would store the plot and not the display of the plot.

That format is hardcoded somewhere hard to find. Here's a hack that takes the existing output and substitutes different strings. Here I use the 5.1f format to get one decimal place.

VennHack.mw

You can collect your Usol values in a table by indexing them. See in red.

ode.mw