Output of plots from within Do[ ] Command

Question

I am wondering why the following does not plot the way it's written and exactly how does the Do[] function operate? I think I could be using it incorrectly.

In[1]:= u[x_, t_] := (1/2) (Exp[-(x + t)^2] + Exp[-(x - t)^2])
        Do[
           Plot[u[x, k], {x, -5, 5}, 
           AxesLabel -> {x, u}, 
           PlotRange -> {0, 1}],
           {k, 0, 3}
          ]

---

EDIT

In[2]:= Clear["Global`*"]
        a=1;
        b=1;
        \[Phi][\[Xi]_,\[Nu]_,t_]:=\!\(
        \*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(10\)]\(
        \*UnderoverscriptBox[\(\[Sum]\), \(j = 1\), \(10\)]\((
        \*FractionBox[\(5\ \((\(-1\) + 
        \*SuperscriptBox[\((\(-1\))\), \(j\)])\)\ \((\(-1\) + 
        \*SuperscriptBox[\((\(-1\))\), \(i\)])\)\), \(j\ i\ 
        \*SuperscriptBox[\(\[Pi]\), \(2\)]\)] Cos[
        \*FractionBox[\(1\), 
        SuperscriptBox[\(\[Pi]\), \(4\)]] 
        \*SqrtBox[\(
        \*SuperscriptBox[\(i\), \(4\)] 
        \*SuperscriptBox[\(\[Pi]\), \(4\)] + 
        \*SuperscriptBox[\(j\), \(4\)] 
        \*SuperscriptBox[\(\[Pi]\), \(4\)]\)] t])\) Sin[i\ \[Pi]\ \[Xi]] Sin[j\ \[Pi]\                   
        \[Nu]]\)\);
        Do[Print@
           Plot3D[\[Phi][\[Xi],\[Nu],m],{\[Xi],0,a},{\[Nu],0,b},
           Mesh:>Automatic,
           PlotLabel:>"Surface Plot of Solution at t = "<>ToString[m],
           ColorFunction:>{"BlueGreenYellow"}],
           {m,0,10,2}
          ]

Answer 1

The reason is that since version 6 of Mathematica, the Plot returns its output just like any other function. It does not print is as a side effect. Do is meant to be used with operations that have side effects and simply discards the output, so you never see the plot.

You can either use Table, which collects the values in a list:

Note how you get a list of graphics objects. Graphics are expressions like any other, they're simply shown using a special formatting. You can always see their internal structure by using InputForm on them or pressing Ctrl-Shift-I on the containing cell.

Or you can use Print explicitly:

Do[Print@Plot[u[x, k], {x, -5, 5}, AxesLabel -> {x, u}, 
   PlotRange -> {0, 1}], {k, 0, 3}]

---

Note: If as @JM noted, you are trying to animate the plot, use Animate:

Animate[Plot[u[x, k], {x, -5, 5}, AxesLabel -> {x, u}, 
  PlotRange -> {0, 1}], {k, 0, 3, 1}]

Answer 2

Another way is to use DisplayFunction, something before the time of graphic input/output of version 6. The following line prints the same way as Szabolcs's 2nd example.

Do[Plot[u[x, k], {x, -5, 5}, AxesLabel -> {x, u}, PlotRange -> {0, 1},
   DisplayFunction -> Print], {k, 0, 3}]

DisplayFunction -> f wraps the actual output of any Graphics function (Plot here) into f. This way one can directly send the plot to any pipe, for example to a stream or file. If you are version 6+ though, the best solution is to use Table, just as Szabolcs has said.

Answer 3

Since this is getting a bit lengthy for comments:

What is the most efficient way of using the iterator for the Do command. Because at most it take three arguments, e.g. Do[expr,{k,min,max,step-size}]. But what if you have a function such as ϕ[ξ,ν,t], and you want to plot for value of time say: t=0,0.3,0.7,0.10,0.14,0.16. As you can notice, the time instances (steps) are not by the same amount, so how could you account for this

The syntax you are looking for is:

Do[f[x], {x, {0, 0.3, 0.7, 0.10, 0.14, 0.16}}]

This also works for Table. In simple cases such as this example you could use Scan or Map:

Scan[f, {0, 0.3, 0.7, 0.10, 0.14, 0.16}]

Do each graphic have a internal name associated with them or could one be assigned to them. Because I know that GraphicsGrid[{{g_11,g_12,...},...}] needs to have variables (functions) hence -> (g11,g12) assigned for input to create the 2-D grid.

No, the elements of the table given to GraphicsGrid do not need to be assigned to symbols first. Here is an example using the function given in your question:

u[x_, t_] := (1/2) (Exp[-(x + t)^2] + Exp[-(x - t)^2])

plots =
 Table[
   Plot[u[x, k], {x, -5, 5}, AxesLabel -> {x, u}, PlotRange -> {0, 1}],
   {k, 0, 3}
 ];

Partition[plots, 2] // GraphicsGrid