@josephrajck The equation and the boundary conditions are correct, but Initial conditions are missing.  Additionally, you can't get a numerical solution if you leave the coefficient "a" and the length "l" inspecified.  Here I will take both of these to be 1.  You may change then as needed.

Download mw.mw

@vv Uh, OK, I see it now and I agree.  Also thanks to Preben for his explanations.

@Preben Alsholm Aren't all variables at the top level global by default?
restart;
type(a, `global`);          #true
type(whatever, `global`);   #true
type(_Z, `global`);         #true

I don't understand what it is so special about singling out _Z as being global.

@Preben Alsholm The error message that you have shown does not complain about "global".  It complains about "protected".  It may be a global variable, but that's beside the point.

In fact _Z is not unique in that respect.  Try any of:

_a := 12;
_b := 12;
_c := 12;

In general it's good to work under the assumption that all identifiers beginning with underscore are reserved for Maple's internal use, and avoid messing around with them.  Even if some of such identifiers are unprotected now, they may become unavailable in future releases.

This is exactly parallel to the C programming language where identifiers beginning with underscore are reserved for the compiler's internal use.  One may assign to them at one's own risk.

Download mw2.mw

@torabi In your differential equations we have F(t,x,y) but in the discretized equations we have F(t,x,y,z).  What is z? Something isn't quite right there.

Additionally, in order to do actual computations, we need to know the functions F and G. What are they?

@kainmuth Let w := s -> 1/abs(s), and then try:

int(w(x-y), x=0..1, y=0..1);

The result is infinity.  Consequently, your E is always infinity, regardless of the choice of f.  There is no hope of minimizing E.

See this:

https://www.mapleprimes.com/questions/224038-Adding-Arrows-To-A-Phase-Portrait

@torabi

Maple can solve the wave equation in one-dimensional space, but as far as I know it cannot handle the two-dimensional case, so you are out of luck here.

If you have the knowledge and ambition, you may set yourself the goal of solving the problem numerically with your own implementation of a finite element algorithm in Maple.  That will take some time and effort.

The commercial software Comsol Multiphysics provides ready-to-use finite element solvers for all sorts of PDEs.  If you have access to it, you may want to give it a try although it will take a few days just to learn the interface.  If you don't have Comsol, probably you cannot afford it since it is quite expensive.

I have no idea whether Matlab or Mathematica can solve your problem.  Ask someone who knows about them.  Good luck!

In the definition of E you have f(y).  What is y?  Perhaps you need a double integral that integrates on both x and y?

Additionally, what is w?

@deniscr OK, that's good.  Almost.  I see that you are entering your code in Maple's 2D input mode.  That's very much error-prone.  As is, you have two different "r" symbols in your worksheet.  They look the same but they are not the same.  Because of that if you try
simplify(U^+ . C . U - M) assuming positive;
you don't get zero.

To see where the problem lies, convert your worksheet to 1D input.  One way of doing this is through the Maple menus:
Edit -> Select All
Format -> Convert To -> 1-D Math Input

After you do that, look at your definition of the matrix M to see the problem.

To avoid such problems, consider making 1-D input the default for your worksheets.  Look at how acer and vv do it.

That said, I am puzzled why vv's calculations do not capture your solution U.  This requires a closer look at how Maple solves systems.

@deniscr Instead of posting an image of your code, please post the code itself.  One cannot execute an image.

@amcmaple I have already said that plotting a phase portrait for a function is not a meaningful concept.  One plots a phase portrait for a differential equation, not a function.

What you have shown appears to be the solution of a differential equation.  What you need to show is the differential equation itself.

Additionally, it will help if you would explain why you are interesterd in this question since, as you say, you don't know what a phase portrait is.

A  "phase portrait for a function" is not a meaningful concept.  Perhaps you mean a phase portrait for a differential equation?  If so, then tell us what differential equation you have in mind because the details of the answer will depend on that.

Your references to the plot of dX/dt versus X indicate that the differential equation must be of the second order.  But because of the missing information, the discussion has been diverted to plotting solution curves (not phase portraits) of first order equations. Be more specific if you want good answers.