@acer By the way, I realized that abs is not mandatory.

@WD0HHU 

I use Maple 2015 and it is very likely that some functions do not operate identically with Maple 2025.
Can you send me the your Maple 2025 worksheet in order I try to fix these issues?

More specificaly I want a worksheet which contains the ourputs of these commands:

core  := indets(solnum, function)[];

acore := expand(core) assuming y::real, z::real;

indets(acore, function);

test :=select(has, %, cos);

T1   := op(test);
T2   := test[];

U1   := op(T1);
U2   := op(T2);

coeff(U1, z);

It's clear that if z_period is not a numeric value all which follows will generate an error or another

simplify/trig worked with Maple 2015.

Download simplify.mw

By the way, the csgn(sin(x))*sin(x) result is obtained with Maple 2015 as the result of 

simplify(convert(A, sin))

@acer 

You're absolutely right about the remember option.

I had thought of using but finally evaded the question because I don't really understand what the procedure "remembers" does when used.
Whatever the results you get using it are impressive. I will treasure your program.

If you don't mind me using some of your time I gave two related questions to ask you

1. Why is the parameters option not allowed for bvp: is it a technical impossibility or simply something that has never been coded?
2. A lot of people use the output=listprocedure option instead of the default one.
   What is the advantage to using it beyond the fact that more quantities are directly available?

@KIRAN SAJJAN

Don't you think the least you could have done would have been to tell me if you were satisfied with what I did?
That said, you are accustomed to this type of attitude.

@Kitonum 

I vote up.

@Kitonum 

For_fun.mw

@Teep 

Thanks for your comment.

@Teep 

I used the output = solutionmodule option for the reason explained below, but you can of course omit it.

The idea behind this was to build a table gathering the computation, the optimum found, and the performances, in short something like this 
Some_tracks_5_5_history.mw


Some_tracks_5_5_history.mw

First point you plot a quantity which is define which is defined over a diabolo-shape domain, not over a rectangular domain: how could you get something like the picture you provide?

Second point: the function you contourplot is not symmetric wrt y=0: how could the filled regions have the same shading as in the image you provide?
(plot3d this same quantity if you don't believe me).

So, even if you "need [your] contour to look like the reference picture", they will never do.

No problems using Maple 2015 (and I guess newer versions) if things are written properly.
Do you use a legal Maple version? If yes which one ?
A good start would be to ubload your mw file using the big green up-arrow in the menubar.

Worksheet Mode: No_NULL_issues.mw

Document Mode: No_NULL_issues_2.mw

... did you try using the dethlimit option as adviced in the error message?

solution := Optimization:-LPSolve(
  C_max
  , constraints
  , assume = nonnegative
  , integervariables = binary_vars
  , variables = Vars
  , depthlimit = 300  # for instance
):

# or, to get some info about the time and memory used

solution := CodeTools:-Usage(
  Optimization:-LPSolve(
    C_max
    , constraints
    , assume = nonnegative
    , integervariables = binary_vars
    , variables = Vars
    , depthlimit = 300
  )
):

@Math-dashti 

What does this paper have to do with your original question?

What are the ranges for epsilon, u0 and v0 (those you want to investigate, not those you coded) ?

@KIRAN SAJJAN 

Non convergence appears when yu < yl, which depends on the values of x and t.

This can be shown in a very simple way.
Thus, for instance when t_fixed = 1 dsolve does not converge over the range x_fixed = 0..2 , but only over the subrange x_fixed = 0.366825212..1.633174787.

Read the attached file carefully and try to figure out what you can do with it XT_non_convergence_issue.mw