@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

@KIRAN SAJJAN 

You asked for a plot in the range x=0..600.
In your last wotksheet you took x=5 and claim "it's converging"... yes indeed, for this value of x.

Now take x=6 and tell me if "it's converging".

@KIRAN SAJJAN 

you understood what I said when I told you to try and fix yourself the "Newton non converging" issue

Upload your worksheet by using the big green up-arrow in the menubar

@Math-dashti 

Have you given up on this question, or is it still of interest to you?

Is it you who deleted your last question, which I spent the time to answer?

If that's the case, let me tell you that you are being particularly impolite and scornful. 

For the record your question was

To which I answered