rather than necessity.

I, personally, would just use

Pm := proc(...)
 ...
 a := Pn(...)
 ...
end proc:

and rely on Maple's scoping rules. If I ever did have the urge to reference the module itself, I'd probably use 'thismodule:-' rather than the module name

that you cannot force s>0,t>0 in your solving method, you can use the select() command, as in the attached.

Download sel.mw

 - not as "educational" as mmcdar, or as "elegant" as CarlLove - but it is short and hpefully simple

Download LS.mw

the output of the final execution group in the attached fulfil your requirements?

Can't display the ouptu of this worksheet on the site because plot arrays are used - and they have never rendered here.

TS.mw

Check the attached

Maybe a Maple version isue - which version are you running?

Download conj.mw

because it is much more efficient!

Like the help page for sum() says

To add a finite sequence of values, rather than compute a formula, use the add command.

Each sum() command will spend time looking for a symbolic solution, into which it can substitute the appropriate value values for the summation range. Only if it cannot find such a symbolic solution, will it revert to using a simple add().

It would not surprise me if in Maple 2021, the number of techniqes which the sum() command uses to find a symbolic solution was increased - so increasing the time taken to look for something which does not exist.

See the timing differences in the attached, running Maple2022 on a pretty ancient desktop

Download addsum.mw

you are missing a "multiplication sign" in the numerator. Using 2D input, this can either be a "space" or an explicit multiplication sign. Without either, the 2D-input parser is interpreting your numerator as a function application, where three functions M(), -n(), and 1() are applied.

See the differences in the attached

Download badMult.mw

although I think the sign of the term in u² in the textbook eq(8) may be wrong??

Download someInt2.mw

to produce numeric solutions and plots for your ODE system - see the attached.

In order to

" plot streamlines and isothermes and also 3D graphes of Nussult number and skin friction coefficient "

you are going to have to specify how each of these quantities can be calculated from from the information in this worksheet.

Download odeSys2.mw

be a lot simpler?

Download fillMat.mw

one of which (shown in the attached) produces a separate plot for each decade of frequency and then display these plots together.

This ought to ensure that Maple's plot() command will produce the same number of points in each decade.

Download logProb.mw

Your original worksheet defines the six functions, whihc I have organised in pairs

1.  sqrt(2.*sin(phi - 1.550000000) - 2.),
2. -sqrt(2.*sin(phi - 1.550000000) - 2.),
3.  1.414213562*sqrt(sin(phi - 1.550000000)),
4. -1.414213562*sqrt(sin(phi - 1.550000000)),
5.  sqrt(2.*sin(phi - 1.550000000) + 2.),
6. -sqrt(2.*sin(phi - 1.550000000) + 2.)

Now apply brain power

1. In functions (1) and (2) above the sin(phi - 1.550000000) factor can only produce values in the range -1..1, so 2.*sin(phi - 1.550000000) produces values in the range -2..2, so 2.*sin(phi - 1.550000000) - 2 produces values in the range -4..0. In other the argument of the square root function is always negative, and thus the values of these functions are purely imaginary, and Maple can't plot these on a "real" graph. So for these two functions, Maple will plot precisely nothing.
2. In functions (3) and (4) above, the sin(phi - 1.550000000) factor can only produce values in the range -1..1. In (1) above we showed that if the argument to the sqrt() function is negative, then Maple can't plot it on a real graph. The values of sin(phi - 1.550000000) will be positive in the range phi=1.55..Pi+1.55 (which as it happens, is very close to the range Pi/2..3*Pi/2). Values will also be positive in the range phi=-2*Pi+1.55..-Pi+1.55 (which as it happens, is very close to the range -3*Pi/2..-Pi/2). In these two ranges 1.55..Pi+1.55 and -2*Pi+1.55..-Pi+1.55, Maple will plot the function. In the three ranges -2*Pi.. -2*Pi+1.55, -Pi+1.55..1.55 and Pi+1.5..2*Pi, values of these functions will be purely imaginary and Maple will not plot them. So for these two functions, Maple will plot them when they are real, which means you get curves with "gaps", where the values are imaginary.
3. In functions (5) and (6) above the sin(phi - 1.550000000) factor can only produce values in the range -1..1, so 2.*sin(phi - 1.550000000) produces values in the range -2..2, so 2.*sin(phi - 1.550000000)  2 produces values in the range 0..4, so the aargument of the square root is always >=0, so Male can plot these two functions for any value of phi

In the attached, rather than using the values h=[-1, 0, 1] I have used the values h=[0, 1, 2], which generates the six functions

1.  1.414213562*sqrt(sin(phi - 1.550000000)),
2. -1.414213562*sqrt(sin(phi - 1.550000000)),
3.   sqrt(2.*sin(phi - 1.550000000) + 2.),
4. -sqrt(2.*sin(phi - 1.550000000) + 2.),
5.  sqrt(2.*sin(phi - 1.550000000) + 4.),
6. -sqrt(2.*sin(phi - 1.550000000) + 4.)

Now four of these (ie 3..6) in the above list always have arguments to the sqrt() which are >=0, and so can be plotted for all values of phi. Functions (1) and (2) iin the above list correspond to functions (3) and (4) in the earlier list. (Not surprising because both are generated by h=0.) Thus argument (2) in the above discussion applies, which mans that onec again you get curves with "gaps".

See the attached, for visualization.

Download implt.mw

it depends (a bit) on how accurately you want to know this maximum. The attached shows two possibilities

1. Extract the maximum point which is used on the graph
2. Curve fit around the maximum point determined above using splines, and differentiate the result wrt 't', which is (probably?) more accurate than (1) above

Download getMax.mw

you can use Maple's maximize() function to get the coordinates of the maximum of the difference between the two "gaussian" functions. This saves a fair bit of effort. See the attached

Download gMax2.mw

From a Microsoft help page, emphasis added

In the Windows API (with some exceptions discussed in the following paragraphs), the maximum length for a path is MAX_PATH, which is defined as 260 characters. A local path is structured in the following order: drive letter, colon, backslash, name components separated by backslashes, and a terminating null character. For example, the maximum path on drive D is "D:\some 256-character path string<NUL>" where "<NUL>" represents the invisible terminating null character for the current system codepage

I think there are ways to change this in Windows, but I suspect that Maple may not "honour" the change - it is more likely that Maple itself will check the path length, and issue an error if it is too long, rather than cause an error to be generated by the OS