@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.

@spalinowy

@spalinowy Where do you see a nonlinearity?  The expression
              
is linear in u(t). That is what is meant by linearization.

@MargoMaples You say x_n+1 =x_n- (f(x_n)*d) / (f(x_n+d)-f(x_n)) and "d is a really small number".

Your problem of two intersecting circles leads to a nonlinear system of two equations in two unknowns. Consequently, the variable x_n in that iteration scheme would be a point in R².

This raises the question: What do you mean by x_n+d where x_n is in R² and d is a number?

Do you see the problem?  You cannot add a point and a number!

That's why I am asking for either a precise description of iteration scheme that you wish to apply, or an easily accessible reference where that description can be found.

Or... ask the person who gave you this problem for clarification.

@MargoMaples What is d?

It will help if you either describe precisely the algorithm that you have in mind, or refer to a web page that describes it.  Offering various permutations of the words Newton, Raphson, Modified, Secant is not a good substitute.