@student_md You can save yourselt trouble of this sort by using a slightly different notation for (in this case) X[1] and w[1] in the actual computations. But that may be awkward. Delaying evaluation by using '  ' only does just that. Even in this case because the axis labels are kept in the plot structure and if that is evaluated again, then the assignment to X[1] (or w[1]) shows up instead.
Below I show a device that goes the other way: Changing the name of the label in an invisible way.
Replace w[1] by  ` w`[1] . That is backquote, space, w, backquote.

restart;
w[1]:=67: #Anything will do
plot1:=plot(x^(1/2),x=0..1,labels=[c,typeset('w[1]')]);
plot1; #67 becomes label now
plot1:=plot(x^2,x=0..1,labels=[c,typeset(` w`[1])]); #The little trick
plot2:=plot(x^(1/2),x=0..1,color=blue); #  If the labels are the same no need for them here.
plots:-display(plot1,plot2);

Finally I should mention that by using backquotes you can turn "anything" into a name, as in this silly example, where the name is `56`.
`56`:=123;
 

@sideshow To see what is going in that expression, try the same thing with an unassigned function F.
 

T:=add((D@@i)(F)(0)*m^i/i!,i=0..2);

D is the operator of differentiation of functions (procedures). So D(sin) returns cos.
D(sin)(Pi/2) therefore returns 0 since cos(Pi/2)=0.
(D@@2)(F) is the second derivative of F, so (D@@2)(sin) = -sin, and so (D@@2)(sin)(Pi/2) = -1.
For convenience (D@@0)(F) is simply F (or if you will, the 0th derivative of F).
That is used above in the add command where I have i running from 0 to 2.
## An ugly looking alternative (in my view) is this:
 

Td:=add(eval(diff(F(m),[m$i]),m=0)*m^i/i!,i=0..2);

For concrete functions like the function f defined in my answer above, T and Td return the same:
 

f:=m->tanh(sqrt(q)*z/y+x*m/y);
eval(T,F=f);
eval(Td,F=f);

You may have a look at the SignalProcessing package:
?SignalProcessing
I don't know anything about that, but maybe you will find what you need there.

Try removing expand (only those letters).  Assign the resulting expression to u, say.
Then do:
 

eval(u,{a=1,h=2,o=3,v=4});
eval(expand(u),{a=1,h=2,o=3,v=4});
w:=-2*a^23*h^40*o*v^2+2*a^23*h^40*v^3;
eval(w,{a=1,h=2,o=3,v=4});

You should find that the first 2 results are the same, but the result you report as the Maple 17 GUI and Mathematica result (w) is WRONG.
Also notice that I cannot confirm the difference you report between Maple 17 GUI and command line.
I use Maple 17.02.

@gkokovidis I tried the GUI and the command line version in Maple 17.02.
I got the same result as you report in both cases.
The same is true of Maple 2016.2.

@rath1 Yes, I expected that answer.
I my opinion it was a very bad mistake, when 2D-math input was introduced, to make a space mean multiplication. 
In particular in Maple, where parentheses () are used for two purposes:
(1) for function application as in f(x)
(2) for grouping of terms as in a*(b+c)
In Mathematica they can get away with it since function application uses square brackets:  f[x].

I'm not personally bothered by the problem since I use Maple notation (aka 1D-math) exclusively for input.
Let me add that because I use a Danish keyboard, I'm pleased with parentheses in function application, since the square brackets need Alt Gr + [ .
But then again square brackets are used extensively in Maple for other purposes.

@esmerindobernardes I'm just saying that your commands as I gave them return what seems to me to be a perfectly good answer. If I had thought I was really helping, I would have called this an answer!

For your information here are the details of my installation:

kernelopts(version);
   Maple 18.02, X86 64 WINDOWS, Oct 20 2014, Build ID 991181
interface(version);
 Standard Worksheet Interface, Maple 18.02, Windows 8, October 20 2014 Build ID 991181

My executed worksheet is here: MaplePrimes2017-02-24_Maple18.mw

I suppose that you won't be satisfied with using Maple Notation as output?
You can change to that in Tools/Options/Display/Output Display.
 

I tried your code in my Maple 18.02.
 

restart;
Hso := Matrix(8, {(1, 4) = -x, (1, 6) = I*x, (2, 3) = x, (2, 5) = I*x, (3, 2) = x, (3, 5) = -I*z, (3, 8) = y, (4, 1) = -x, (4, 6) = I*z, (4, 7) = -y, (5, 2) = -I*x, (5, 3) = I*z, (5, 8) = -I*y, (6, 1) = -I*x, (6, 4) = -I*z, (6, 7) = -I*y, (7, 4) = -y, (7, 6) = I*y, (8, 3) = y, (8, 5) = I*y});
LinearAlgebra[Eigenvectors](Hso);

No problem. Large output.

@shesia A good book is the Programming Guide, which is available as a series of worksheets and as a pdf file.
In Maple itself you can access these worksheets by doing:
? Programming Guide

You can also download a pdf version from
http://www.maplesoft.com/documentation_center/

@shesia OK, I tried with the much larger set of data.
NOTE: Are these data really measured data? With that much accuracy! Or are they constructed by a teacher?
I revised Q slightly to make it:
(1) catch possible errors and in that case return the large list of residuals [10^6$(3*nops(T1))].
(2) print the parameter points for which Q is computed to show that something is going on.
 

Q:=proc(T1_p1, k1, keq, k4) option remember; local P1v,P2v,P2e_v,Sv,resid;
   res(parameters=[T1_p1, k1, keq, k4]);
   printf("Trying %a\n",[_passed]);
   try
     P1v:=P1fu~(T1);
     P2v:=P2fu~(T1);
     P2e_v:=P2e_fu~(T1);
     Sv:=Sfu~(T1);
     resid:=[P1v-P1d1,P2v+P2e_v-P2_P2ed1,Sv-Sd1];
     return [seq(seq(resid[i][j],i=1..3),j=1..nops(T1))]
   catch:
     return [10^6$(3*nops(T1))]
   end try
end proc: 

Then with q defined as

q:=[seq(subs(_nn=n,(proc(T1_p1, k1, keq, k4) Q(_passed)[_nn] end proc)),n=1..3*nops(T1))]: 

and the command

solLS:=Optimization:-LSSolve(q,initialpoint=[15,0.02,4,4]);

I got (after a while)

The plots don't look as good as before: That is not surprising if the data are real data (i.e. not constructed by a teacher as I remarked about above).
After all, isn't the differential equations model you have only a crude model?
A thing to keep in mind is that LSSolve doesn't necessarily find the global minimum!
For the same plot as I reported for the small data set I got:

@shesia The very simple reason is that you have 51 t-values, but only 24 values in the other lists:
nops~([T1,Sd1,P1d1,P2_P2ed1]);
                                                              [51, 24, 24, 24]

I noticed that when I copied your code s0 was set at 100000 which doesn't agree very well with the data.
I notice now that s0 is set at 10000, which is fine. I suppose you edited that in the meantime?
I shall edit my answer below to that effect.

In Maple 2016 the error message for your example (with D[1] as you have it) is more informative:

Error, (in PDEtools:-Solve) unable to handle functions of not names as in the case of the system passed, containing [_F2(-x)]

To see why there is talk about _F2(-x) try

pdsolve(PDE);
eval(%,t=0);

In Maple 12 (I don't have 13) the error message is

Error, (in pdsolve/sys) unable to handle PDE problem subject to boundary conditions {u(x, 0) = cos((1/2)*x), (D[1](u))(x, 0) = 0}

You don't need the package PDEtools, but if you do, the syntax is
with( PDEtools );

@Thomas Dean Yes, if D[2] is meant (i.e. the first derivative with respect to the second variable), then there is no problem:

pdsolve({PDE, u(x, 0) = cos((1/2)*x), D[2](u)(x, 0) = 0});