@nm It works after a proper simplification:

Q((-(x*y)/2+sqrt(x^2*y^2-4*y)/2)*y, x,y):
expand(rationalize(%));
#                               -2

Edit. Q fails only for f(x,y) of the form f(x,y) = g(x)*y, 
but then g  must be g(x) = A/x, so, only the functions f(x,y) = A*y/x are not identified.

@nm The values given by P are correct too. 

restart;
f:=(x,y)->sqrt(x*y):
'p'=P(f(x,y), x,y):
p:=-1;
t^r=simplify(f(t*x, t^p*y)/f(x,y)) assuming t>0;
p:=2;
t^r=simplify(f(t*x, t^p*y)/f(x,y)) assuming t>0;
#                            p := -1
#                            t^r  = 1
#                            p := 2
#                            t^r  = t^(3/2)     

Actually, in this example p can be arbitrary!

@AHSAN I think you don't understand what I mean. When you try to compute e.g. int(x+y, x), you may obtain any of the expressions  x^2/2 + x*y,  x^2/2 + x*y + 7, x^2/2 + x*y + y^2 - ln(y),  x^2/2 + x*y + y^12 + sinh(y^2-2)  etc.
How are you going to interpret the plot, without knowing which expression is plotted?

 

@AHSAN I have told you that int(v,x) cannot be expressed by elementary functions. Maple probably tries to find a (huge and useless for you) expression containing sums over RootOfs (based on the residue theorem).
But even if B existed, it would be only modulo a function of y, so, your plot does not make much sense.

@AHSAN It's not the same thing. The new v is a simple rational function in x which has an elementary antiderivative.

@AHSAN  Sorry, I do not understand. Be more specific.

@DanishMapleFan  For example, if g is even and h is not even then f does not exist.

To adapt for earlier versions it is (probably) enough to replace a += b with a := a + b,
To map into [0,1]^2  just divide the result of d2xy  by n.
For an arbitry dimension, more work is needed, and I do not have the time for this. Note that it is easy to implement a Lebesgue filling curve in any dimension (just a few lines in Maple).

@acer In fracdiff, alpha should be not an integer. For alpha a float integer (such as 0. and 1.), fracdiff returns 0. I wonder whether this is intentional.

@JAMET The answer contains a numerical example. 

@kiraMou Ok, I see now that you are using Maple 18. Please replace that line with

print(Sol[crt] = eval(X, SOL[i][j])); crt:=crt+1;

I hope it will work for you too.

@mmcdara 

For me,  answer -> a (possible) solution,  reply -> comment (not containing in general a solution).

restart;
# a simplistic Functional Form "à la dismantle"
FF := proc(e, n:=0)
  if e::atomic then printf("%s\n", cat(" "$n, e)); return fi;
  printf("%s\n", cat(" "$n, op(0,e)));
  map(FF, [op(e)], n+4);
  NULL
end:
FF((x+1)/(x-1)^2+sqrt((exp(-x^2)-1)/(x+1)));
+
    *
        +
            x
            1
        ^
            +
                x
                -1
            -2
    ^
        *
            +
                exp
                    *
                        -1
                        ^
                            x
                            2
                -1
            ^
                +
                    x
                    1
                -1
        1/2

@nm It is not just that. In Mathematica the two "forms" are equivalent, the first one being graphical. In Maple they can be obtained easily with op().
The result of dismantle cannot be obtained this way. It reflects the internal representation, not very useful for a regular user.

dismantle(5*x+7*y);

SUM(5)
   NAME(4): x
   INTPOS(2): 5
   NAME(4): y
   INTPOS(2): 7

dismantle(a*x+b*y);

POLY(6)
   EXPSEQ(5)
      NAME(4): a
      NAME(4): b
      NAME(4): x
      NAME(4): y
   DEGREES(HW): ^2 ^1 ^0 ^1 ^0 
   INTPOS(2): 1
   DEGREES(HW): ^2 ^0 ^1 ^0 ^1 
   INTPOS(2): 1

From op()'s point of view, the two expressions have the same "structure": 

`+`(`*`(5,x), `*`(7,y));
`+`(`*`(a,x), `*`(b,y));

@snowman Do not increase Digits. Increase numpoints or set gridrefine=3 in implicitplot.

p:=plots:-implicitplot(G, -5..5, -5..5, numpoints=10000):