@jalal 

Download Qop.mw

@mmcdara I agree with the OPs observations of errors in Maple 2023. I found them hard to track down, and got confused by the gammas. But please check, you have both gamma_i and gamma__i in the gap procedure. .I usually define the ode and ic set outside of the procedure to be passed to minimize, and do eval on them within the procedure.

@mmcdara Your 

solution := tanh((1/4)*sqrt(2)*x)

is I believe the solution the OP referred to. It is an exact solution for z(0)=0, z(infinity)=1 (as verified by taking the limit), but is only an approximate solution for z(0)=0, z(15)=1.

@mmcdara It is a confusing situation - while there isn't a help page for, say, `[]`, asking for help does take you to a page that descibes the user-level equivalent. And yet there is a help page for `?[]`.

@mmcdara I think the OP wanted documentation for their usage in the form `<,>`(1,2,3) and `<|>`(1,2,3).

@mmcdara Nice analysis, vote up. While having the procedure name integraal the same as the local integraal may be confusing, it is not an error, as the procedure name is global and does not conflict with the local variable within the procedure.

ps: the loop within optimalisatie looks to me like

for i while i < 10 and 1/10 < abs(int__sim - int__ex) do

(see above)

I recently looked for the documentation for `<|>` and couldn't find any. For `<,>`( sequence ) the user level <sequence> is equivalent and turns a sequence into a column vector (or perhaps matrix if the sequence has more complicated entries). Likewise `<|>`(sequence) turns the sequence into a row vector as though you entered <a|b|c...>. Programmatically, one may have generated a sequence and then the user level <sequence> is fine, which may be the reson that `<,>` is undocumented. On the other hand `<|>` is the only way to turn a sequence into a row vector, so there is a better case for documentation here.

Of course I may have just failed to find the documentation also...

If I paste the optimalisatie code into a code edit region I get the message that maplemint got about "These local variables were assigned a value, but otherwise unused:   int__sim_min", which is correct: You calculate this quantity (which looks like int with subscript sim_min) but then do not use it.

In the code edit region there is no message "invalid left hand side of assignment", so that seems to be spurious. [Edit; as @mmcdara points out this is assigning to Gamma__i_min and Gamma__i_max]

Note that the printed code for the procedure has many things such as int__sim_min looking like int[sim_min] as though you entered them as indexed variables. But pasting the 2-D code into a 1-D region or converting it to 1-D with the context menu shows that actually they are single variables displaying with subscripts, as you presumably intended. I've never seen that before.

In a code edit region it looks like

optimalisatie := proc(t__ex, c__ex, C__0, Q, V__r, m__b, rho, R, Gamma__i_min, Gamma__i_max)
	local Gamma__i, t__sim, C__i, int__sim, int__ex, C__i_min, C__i_max, C__i0_min, C__i0_max, C__i0, int__sim_min, int__sim_max, i;
	Gamma__i := 1/2*Gamma__i_min + 1/2*Gamma__i_max;
	t__sim, C__i := CC(C__0, Q, V__r, m__b, rho, R, Gamma__i);
	int__sim := integraal(t__sim, C__i);
	int__ex := integraal(t__ex, c__ex);
	for i while i < 10 and 1/10 < abs(int__sim - int__ex) do
		Gamma__i := 1/2*Gamma__i_min + 1/2*Gamma__i_max;
		t__sim, C__i_min := CC(C__0, Q, V__r, m__b, rho, R, Gamma__i_min);
		t__sim, C__i_max := CC(C__0, Q, V__r, m__b, rho, R, Gamma__i_max);
		t__sim, C__i := CC(C__0, Q, V__r, m__b, rho, R, Gamma__i);
		C__i0_min := normalisatie(C__i_min, C__0);
		C__i0_max := normalisatie(C__i_max, C__0);
		C__i0 := normalisatie(C__i, C__0);
		int__sim_min := integraal(t__sim, C__i0_min);
		int__sim_max := integraal(t__sim, C__i0_max);
		int__sim := integraal(t__sim, C__i0);
		if 0 < (int__sim - int__ex)*(int__sim_max - int__ex) then
			Gamma__i_max := Gamma__i;
		else
			Gamma__i_min := Gamma__i;
		end if;
	end do;
	return Gamma__i;
end proc;

Since none of the variables are now indexed, you need to check that was what you intended - was C or C__i intended to be an Array or Vector, for example?

@Scot Gould In @acer's detailed explanation its clear that it isn't just the plot form. The operator call form is robust and is my first go-to solution for plot difficulties, especially not needing to think about where and how many uneval quotes are needed. It has been around for as long as I can remember, and the plot help page refers to it as "The commonly used operator form of the calling sequence," even if it might not be "traditional" :)  

@acer Thanks for the detailed explanation. (I realised later that just entering the GAMMA equivalent didn't give the same result, and so the real situation was more complicated.)

@SHIVAS There is an attempt to do an inverse transformation at the end of my worksheet, which shows how to do it. It does not succeed but Eq 22 was different, Perhaps if it agreed with Eq 22 it would work. You will need to find the error in the eqns or conditions (assuming there is an error) to proceed.

@SHIVAS Yes, I did not get the same answer in eqn 22. I do not know what is wrong. If you think an assumption needs to be changed you will need to say which assumption it is and what it needs to be changed to, or change it yourself.

@SHIVAS I get a different transformed answer, which probably can't be inverse-transformed. Please check I didn't make any mistakes.

Download laplace.mw

@dharr If your objective is to follow the derivation in the paper, I can mostly derive the laplace transformed quantities, but I have to guess at the initial conditions - what are they? And why do x and y suddenly appear in the solution? Perhaps you can provide a link to the paper.

@Traruh Synred Somehow Export is being interpreted as ExportMatrix. I can't reproduce this in Maple 2023. @acer suggests you are using 2018, which had Export, but Maple 18 did not (use interface(version) to check your version). You can convert to a 4x1 Matrix and export it as you said, but saving to a .m file will save any Maple object and might be better for your application.