For example, suppose the first 100 lines of your file are the first row, the next 100 are the second row, etc, for 101 rows.

> phi:= ImportMatrix("E:\\POTENTIAL FOR MAPLE", transpose=false, datatype=float[8]); 
  B:= ArrayTools[Reshape](phi, 1..100,1..101,1..3);
  plots[surfdata](B);

For example, suppose the first 100 lines of your file are the first row, the next 100 are the second row, etc, for 101 rows.

> phi:= ImportMatrix("E:\\POTENTIAL FOR MAPLE", transpose=false, datatype=float[8]); 
  B:= ArrayTools[Reshape](phi, 1..100,1..101,1..3);
  plots[surfdata](B);

Interesting... I'm not sure exactly what's causing Maple to compute F1(x) or F2(x) as undefined for some values of x, but here's one piece of the puzzle:
with your command, Maple first evaluates F1(x) with x as a symbolic variable, obtaining

Int((-1.*sin(.5000000000*k)/(6.283185308+k)+sin(.5000000000*k)/(6.283185308-1.*k)+2.*sin(.5000000000*k)/k)*exp(-.5000000000e-2*k^2)*exp(-.1250000000e-2*k^2)*cos(t*k),k = Float(-infinity) .. Float(infinity))}

and that is what is sent to plot.  On the other hand, you can plot the function F1 rather than the expression F1(x) with

> plot(F1, -2 .. 2);

and this will produce a complete plot like that of F3(x+1/2), with no missing points.

... and indeed, if you plot it up to t = 1.23 or so, you'll see u(t) going off toward infinity, signalling the singularity.

Perhaps your initial conditions are not correct.

... and indeed, if you plot it up to t = 1.23 or so, you'll see u(t) going off toward infinity, signalling the singularity.

Perhaps your initial conditions are not correct.

In this form, possibly: I stopped Maple when memory usage got to 293 MB.  The standard thing to do in a central-force problem is to
take advantage of the fact that the motion will be in a plane, and choose coordinates so that will be the plane Z=0 (so phi = Pi/2).  Your differential equations become

{diff(R(t),`$`(t,2)) = -R(t)*(-diff(theta(t),t)^2+exp(-R(t)^2)), diff(theta(t),`$`(t,2)) = -2*diff(theta(t),t)*diff(R(t),t)/R(t)}

and then dsolve rapidly produces a solution (not explicit, unfortunately):

[{theta(t) = Int(RootOf(-Int(exp(_C1^2/_h)/(exp(_C1^2/_h)*_h*(-4*_C1^2*exp(_C1^2/_h)*_h+4+_C2*_C1^2*exp(_C1^2/_h)))^(1/2)/_h*_C1,_h = `` .. _Z)+t+_C3),t)+_C4, theta(t) = Int(RootOf(Int(exp(_C1^2/_h)/(exp(_C1^2/_h)*_h*(-4*_C1^2*exp(_C1^2/_h)*_h+4+_C2*_C1^2*exp(_C1^2/_h)))^(1/2)/_h*_C1,_h = `` .. _Z)+t+_C3),t)+_C4}, {R(t) = _C1/diff(theta(t),t)^(1/2)}]

In this form, possibly: I stopped Maple when memory usage got to 293 MB.  The standard thing to do in a central-force problem is to
take advantage of the fact that the motion will be in a plane, and choose coordinates so that will be the plane Z=0 (so phi = Pi/2).  Your differential equations become

{diff(R(t),`$`(t,2)) = -R(t)*(-diff(theta(t),t)^2+exp(-R(t)^2)), diff(theta(t),`$`(t,2)) = -2*diff(theta(t),t)*diff(R(t),t)/R(t)}

and then dsolve rapidly produces a solution (not explicit, unfortunately):

[{theta(t) = Int(RootOf(-Int(exp(_C1^2/_h)/(exp(_C1^2/_h)*_h*(-4*_C1^2*exp(_C1^2/_h)*_h+4+_C2*_C1^2*exp(_C1^2/_h)))^(1/2)/_h*_C1,_h = `` .. _Z)+t+_C3),t)+_C4, theta(t) = Int(RootOf(Int(exp(_C1^2/_h)/(exp(_C1^2/_h)*_h*(-4*_C1^2*exp(_C1^2/_h)*_h+4+_C2*_C1^2*exp(_C1^2/_h)))^(1/2)/_h*_C1,_h = `` .. _Z)+t+_C3),t)+_C4}, {R(t) = _C1/diff(theta(t),t)^(1/2)}]

Yes, I used Maple 13 as well.  What didn't work?  What results did you get?  Can you upload a worksheet?

Yes, I used Maple 13 as well.  What didn't work?  What results did you get?  Can you upload a worksheet?

If the ics are "correct", the answer is that there is no solution.  So your "approximate solution" is an approximation of something that doesn't exist.  I suggest you choose a problem that does have a solution.

If the ics are "correct", the answer is that there is no solution.  So your "approximate solution" is an approximation of something that doesn't exist.  I suggest you choose a problem that does have a solution.

> E0:=0.1; S0:=1; k[1]:=1; k[-1]:=1; k[2]:=10;
  v[0]:= (k[2]*E0*S0*k[1])/(S0*k[1]+k[-1]+k[2]);
  S:= S0*(1-v[0]/S0)^t;
  plot(S, t=0 .. 30); 
  

Note that I used S0 rather than S[0].  There can be problems when you use both a name and a subscripted form of the same name as variables. 

> E0:=0.1; S0:=1; k[1]:=1; k[-1]:=1; k[2]:=10;
  v[0]:= (k[2]*E0*S0*k[1])/(S0*k[1]+k[-1]+k[2]);
  S:= S0*(1-v[0]/S0)^t;
  plot(S, t=0 .. 30); 
  

Note that I used S0 rather than S[0].  There can be problems when you use both a name and a subscripted form of the same name as variables. 

I'm not sure what you mean by differentiation with respect to a function.  What is the answer to diff(sin(x)*x^2, sin(x)) supposed to be?    Do you mean something like this?

> implicitdiff({z = sin(x)*x^2, y = sin(x)},{z,x},z,y);

x*(cos(x)*x+2*sin(x))/cos(x)

Try it like this:

> with(plots); 
  m1 := implicitplot3d(9*x^2+4*y^2-36*z^2, x = -2 .. 2, 
     y = -3 .. 3, z = -1 .. 1, grid = [20, 20, 20]):
  n := animate(plot3d, [t, x = -2 .. 2, y = -3 .. 3], 
      t = -1 .. 1, background=m1):
  n;