(************** Content-type: application/mathematica **************
                     CreatedBy='Mathematica 4.2'

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Notebook[{

Cell[CellGroupData[{
Cell["Euler's Method", "Section",
  FontSize->24],

Cell[TextData[{
  "We want to solve x' = t ",
  Cell[BoxData[
      \(TraditionalForm\`\@\(x\^2 + 1\)\)]],
  ", x(0) = 1, for  0 < t < 2."
}], "Text",
  FontSize->24],

Cell[BoxData[
    \(Clear[x, t]; f[t_, x_]\  = \ t*\@\(x\^2 + 1\)\)], "Input"],

Cell["Here's the code for Euler's method, with h=0.2", "Text",
  FontSize->24],

Cell[BoxData[{
    \(h = 0.2; ti = 0; tf = 2; 
    nsteps = \((tf - ti)\)/h;\), "\[IndentingNewLine]", 
    \(xlist\  = \ Table[0, {nsteps + 1}]; 
    xlist[\([1]\)] = 1;\), "\[IndentingNewLine]", 
    \(\(tlist\  = \ Table[0, {nsteps + 1}];\)\), "\[IndentingNewLine]", 
    \(Do[\[IndentingNewLine]xlist[\([j + 1]\)] = 
        xlist[\([j]\)] + 
          f[\ \ \ tlist[\([j]\)]\ , xlist[\([j]\)]\ \ \ ]*
            h; \[IndentingNewLine]tlist[\([j + 1]\)] = 
        tlist[\([j]\)] + h, \[IndentingNewLine]{j, 1, nsteps}]\)}], "Input"],

Cell["\<\
Here we compute the exact solution and plot it (in red) against the Euler \
approximation\
\>", "Text",
  FontSize->24],

Cell[BoxData[{
    \(\(sol[t_]\  = \ 
        x[t]\  /. \ First[
            First[DSolve[{\(x'\)[t] \[Equal] t*Sqrt[x[t]^2 + 1], 
                  x[0] \[Equal] 1}, x[t], t]]];\)\), "\[IndentingNewLine]", 
    \(\(pl1 = 
        ListPlot[Transpose[{tlist, xlist}], PlotJoined \[Rule] True, 
          DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", 
    \(\(pl2 = 
        Plot[sol[t], {t, 0, 2}, PlotStyle \[Rule] RGBColor[1, 0, 0], 
          DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", 
    \(Show[pl1, pl2, 
      DisplayFunction \[Rule] $DisplayFunction]\), "\[IndentingNewLine]", 
    \(\(Print["\<Error after one step = \>", 
        xlist[\([2]\)] - sol[0.2]];\)\), "\[IndentingNewLine]", 
    \(\(Print["\<Error at t=2 is \>", 
        xlist[\([nsteps + 1]\)] - sol[2]];\)\)}], "Input"]
}, Closed]],

Cell[CellGroupData[{

Cell["2nd-order Taylor series method", "Section",
  FontSize->24],

Cell["\<\
Now we solve the same problem with the 2nd order Taylor series method.  To do \
this, we first need to compute the partial derivatives of the right-hand side \
function\
\>", "Text",
  FontSize->24],

Cell[BoxData[{
    \(dfdt[t_, x_]\  = \ D[f[t, x], t]\), "\[IndentingNewLine]", 
    \(dfdx[t_, x_]\  = \ D[f[t, x], x]\)}], "Input"],

Cell["\<\
Here's the code for 2nd order Taylor iteration, again with h=0.2\
\>", "Text",
  FontSize->24],

Cell[BoxData[{
    \(h = 0.2; ti = 0; tf = 2; 
    nsteps = \((tf - ti)\)/h;\), "\[IndentingNewLine]", 
    \(xlist\  = \ Table[0, {nsteps + 1}]; 
    xlist[\([1]\)] = 1;\), "\[IndentingNewLine]", 
    \(\(tlist\  = \ Table[0, {nsteps + 1}];\)\), "\[IndentingNewLine]", 
    \(Do[\[IndentingNewLine]xlist[\([j + 1]\)] = 
        xlist[\([j]\)] + 
          f[\ \ \ tlist[\([j]\)]\ , \ xlist[\([j]\)]\ \ \ ]*h + 
          h^2/2*\((dfdt[\ \ \ tlist[\([j]\)]\ , xlist[\([j]\)]\ \ \ ]\  + \ 
                dfdx[\ \ \ tlist[\([j]\)]\ , \ xlist[\([j]\)]\ \ \ ]*
                  f[\ \ \ tlist[\([j]\)]\ , 
                    xlist[\([j]\)]\ \ \ ]\ \ )\); \
\[IndentingNewLine]tlist[\([j + 1]\)] = 
        tlist[\([j]\)] + h, \[IndentingNewLine]{j, 1, nsteps}]\)}], "Input"],

Cell["\<\
We plot the exact solution (in red) against the 2nd order Taylor \
approximation\
\>", "Text",
  FontSize->24],

Cell[BoxData[{
    \(\(pl1 = 
        ListPlot[Transpose[{tlist, xlist}], PlotJoined \[Rule] True, 
          DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", 
    \(\(pl2 = 
        Plot[sol[t], {t, 0, 2}, PlotStyle \[Rule] RGBColor[1, 0, 0], 
          DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", 
    \(Show[pl1, pl2, 
      DisplayFunction \[Rule] $DisplayFunction]\), "\[IndentingNewLine]", 
    \(\(Print["\<Error after one step = \>", 
        xlist[\([2]\)] - sol[0.2]];\)\), "\[IndentingNewLine]", 
    \(\(Print["\<Error at t=2 is \>", 
        xlist[\([nsteps + 1]\)] - sol[2]];\)\)}], "Input"]
}, Closed]],

Cell[CellGroupData[{

Cell["2nd order Runge Kutta", "Section",
  FontSize->21],

Cell["\<\
Now we solve the same problem with the 2nd order Runge Kutta method, still \
with h=0.2\
\>", "Text",
  FontSize->24],

Cell[BoxData[{
    \(ti = 0; tf = 2; h = 0.2; 
    nsteps = \((tf - ti)\)/h;\), "\[IndentingNewLine]", 
    \(xlist = Table[{}, {nsteps + 1}]; 
    tlist = Table[{}, {nsteps + 1}];\), "\[IndentingNewLine]", 
    \(xlist[\([1]\)] = 1; tlist[\([1]\)] = 0;\), "\[IndentingNewLine]", 
    \(Do[\[IndentingNewLine]x = xlist[\([j]\)]; 
      t = tlist[\([j]\)]; \[IndentingNewLine]xlist[\([j + 1]\)] = 
        x + h*f[t + h/2, x + h/2*f[t, x]]; \[IndentingNewLine]tlist[\([j + 
              1]\)] = t + h, \[IndentingNewLine]{j, 1, nsteps}]\)}], "Input"],

Cell["\<\
We plot the exact solution (in red) against the 2nd order Runge Kutta \
approximation\
\>", "Text",
  FontSize->24],

Cell[BoxData[{
    \(\(pl1 = 
        ListPlot[Transpose[{tlist, xlist}], PlotJoined \[Rule] True, 
          DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", 
    \(\(pl2 = 
        Plot[sol[t], {t, 0, 2}, PlotStyle \[Rule] RGBColor[1, 0, 0], 
          DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", 
    \(Show[pl1, pl2, 
      DisplayFunction \[Rule] $DisplayFunction]\), "\[IndentingNewLine]", 
    \(\(Print["\<Error after one step = \>", 
        xlist[\([2]\)] - sol[0.2]];\)\), "\[IndentingNewLine]", 
    \(\(Print["\<Error at t=2 is \>", 
        xlist[\([nsteps + 1]\)] - sol[2]];\)\)}], "Input"]
}, Closed]],

Cell[CellGroupData[{

Cell["4th order Runge Kutta", "Section",
  FontSize->22],

Cell["\<\
One last time, the same problem, now with the 4th order Runge Kutta method, \
still with h=0.2\
\>", "Text",
  FontSize->24],

Cell[BoxData[{
    \(ti = 0; tf = 2; h = 0.2; 
    nsteps = \((tf - ti)\)/h;\), "\[IndentingNewLine]", 
    \(xlist = Table[{}, {nsteps + 1}]; 
    tlist = Table[{}, {nsteps + 1}];\), "\[IndentingNewLine]", 
    \(xlist[\([1]\)] = 1; tlist[\([1]\)] = 0;\), "\[IndentingNewLine]", 
    \(Do[\[IndentingNewLine]x = xlist[\([j]\)]; 
      t = tlist[\([j]\)]; \[IndentingNewLine]k1 = 
        h*f[t, x]; \[IndentingNewLine]k2 = 
        h*f[t + h/2, x + k1/2]; \[IndentingNewLine]k3 = 
        h*f[t + h/2, x + k2/2]; \[IndentingNewLine]k4 = 
        h*f[t + h, x + k3]; \[IndentingNewLine]xlist[\([j + 1]\)] = 
        x + \((k1 + 2*k2 + 2*k3 + k4)\)/
            6; \[IndentingNewLine]tlist[\([j + 1]\)] = 
        t + h, \[IndentingNewLine]{j, 1, nsteps}]\)}], "Input"],

Cell["\<\
We plot the exact solution (in red) against the 4th order Runge Kutta \
approximation\
\>", "Text",
  FontSize->24],

Cell[BoxData[{
    \(\(pl1 = 
        ListPlot[Transpose[{tlist, xlist}], PlotJoined \[Rule] True, 
          DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", 
    \(\(pl2 = 
        Plot[sol[t], {t, 0, 2}, PlotStyle \[Rule] RGBColor[1, 0, 0], 
          DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", 
    \(Show[pl1, pl2, 
      DisplayFunction \[Rule] $DisplayFunction]\), "\[IndentingNewLine]", 
    \(\(Print["\<Error after one step = \>", 
        xlist[\([2]\)] - sol[0.2]];\)\), "\[IndentingNewLine]", 
    \(\(Print["\<Error at t=2 is \>", 
        xlist[\([nsteps + 1]\)] - sol[2]];\)\)}], "Input"]
}, Closed]],

Cell[CellGroupData[{

Cell["Estimating Error In Euler's Method by halving the step size", "Section",
  FontSize->21],

Cell["\<\
Here's our familiar differential equation, with its exact solution \
computed:\
\>", "Text",
  FontSize->18],

Cell[BoxData[
    \(Clear[x, t]; f[t_, x_]\  = \ \((t + 1)\)*\@\(x\^2 + 1\)\)], "Input"],

Cell[BoxData[
    \(sol[t_]\  = \ 
      x[t]\  /. \ First[
          First[DSolve[{\(x'\)[t] \[Equal] f[t, x[t]], x[0] \[Equal] 1}, 
              x[t], t]]]\)], "Input"],

Cell["Here's one step of Euler with stepsize 0.2", "Text",
  FontSize->15],

Cell[BoxData[{
    \(h = 0.2; t = 0; nsteps = 1; x = 1;\), "\[IndentingNewLine]", 
    \(x = x + f[t, x]*h; t = t + h;\), "\[IndentingNewLine]", 
    \(\(\(Print["\<One-step Euler estimate is  \>", 
      x, "\<;  Exact answer is \>", sol[0.2]]\)\(\[IndentingNewLine]\)
    \)\)}], "Input"],

Cell["\<\
Now here are 2 steps of Euler with stepsize 0.1.  Note that difference \
between the new Euler estimate and the old is roughly equal to the difference \
between the 2-step estimate and the exact result.  
In addition, it is roughly equal to twice the difference between the 1-step \
estimate and the exact result.  If we are not concerned with such factors of \
two, but only an order-of-magnitude sense of the size of the error, the \
difference between the estimates gives us a good (conservative) sense of the \
difference between the 1-step estimate and the exact result.
\
\>", "Text",
  FontSize->15],

Cell[BoxData[{
    \(h = 0.1; t = 0; nsteps = 1; x = 1;\), "\[IndentingNewLine]", 
    \(x = x + f[t, x]*h; t = t + h;\), "\[IndentingNewLine]", 
    \(x = x + f[t, x]*h; t = t + h;\), "\[IndentingNewLine]", 
    \(\(\(Print["\<Two-step Euler estimate is  \>", 
      x]\)\(\[IndentingNewLine]\)
    \)\)}], "Input"],

Cell["", "Text"],

Cell["\<\
(If we really wanted to be clever, we could use the error analysis to design \
an extrapolation scheme; note that the linear extrapolation between the \
one-step and two-step Euler estimates gives a quite good estimate of the \
exact answer.) \
\>", "Text",
  FontSize->16]
}, Closed]],

Cell[CellGroupData[{

Cell["\<\
Estimating Error In Euler's Method by using higher-order method\
\>", "Section",
  FontSize->21],

Cell[BoxData[
    \(Clear[x, t]; f[t_, x_]\  = \ \((t + 1)\)*\@\(x\^2 + 1\)\)], "Input"],

Cell[BoxData[
    \(sol[t_]\  = \ 
      x[t]\  /. \ First[
          First[DSolve[{\(x'\)[t] \[Equal] f[t, x[t]], x[0] \[Equal] 1}, 
              x[t], t]]]\)], "Input"],

Cell["Here's one step of Euler with stepsize 0.2", "Text",
  FontSize->15],

Cell[BoxData[{
    \(h = 0.2; t = 0; nsteps = 1; x = 1;\), "\[IndentingNewLine]", 
    \(x = x + f[t, x]*h; t = t + h;\), "\[IndentingNewLine]", 
    \(Print["\<Euler estimate is  \>", x, "\<;  Exact answer is \>", 
      sol[0.2]]\)}], "Input"],

Cell["\<\
Now here is one step of 2nd order Taylor with stepsize 0.2.  The difference \
between estimates is roughly equal to the difference between the Euler \
estimate and the exact result\
\>", "Text",
  FontSize->15],

Cell[BoxData[{
    \(Clear[t, x]; dfdt[t_, x_]\  = \ D[f[t, x], t]; 
    dfdx[t_, x_]\  = \ D[f[t, x], x];\), "\[IndentingNewLine]", 
    \(h = 0.2; t = 0; nsteps = \((tf - ti)\)/h; 
    x = 1;\), "\[IndentingNewLine]", 
    \(x = 
      x + f[t, x]*h + 
        h^2/2*\((dfdt[\ t, x\ ]\  + \ dfdx[\ t, x\ ]*f[t, x]\ \ )\); 
    t = t + h;\), "\[IndentingNewLine]", 
    \(Print["\<2nd order Taylor estimate is  \>", x]\)}], "Input"]
}, Closed]],

Cell[CellGroupData[{

Cell["Fehlberg's ugly 4th and 5th order Runge Kutta algorithms", "Section",
  FontSize->22],

Cell[BoxData[
    \(Clear[x, t]; f[t_, x_]\  = \ \((t + 4)\)*\@\(x\^2 + 1\)\)], "Input"],

Cell[BoxData[
    \(sol[t_]\  = \ 
      x[t]\  /. \ First[
          First[DSolve[{\(x'\)[t] \[Equal] f[t, x[t]], x[0] \[Equal] 1}, 
              x[t], t]]]\)], "Input"],

Cell["Here's one step of Fehlberg's RK4", "Text",
  FontSize->16],

Cell[BoxData[{
    \(t = 0; h = 0.2`20; x = 1.0`20;\), "\[IndentingNewLine]", 
    \(\(k1 = h*f[t, x];\)\), "\[IndentingNewLine]", 
    \(\(k2 = h*f[t + h/4, x + k1/4];\)\), "\[IndentingNewLine]", 
    \(\(k3 = 
        h*f[t + 3*h/8, x + 3*k1/32 + 9*k2/32];\)\), "\[IndentingNewLine]", 
    \(\(k4 = 
        h*f[t + 12*h/13, 
            x + 1932*k1/2197 - 7200*k2/2197 + 
              7296*k3/2197];\)\), "\[IndentingNewLine]", 
    \(\(k5 = 
        h*f[t + h, 
            x + 439*k1/216 - 8*k2 + 3680*k3/513 - 
              845*k4/4104];\)\), "\[IndentingNewLine]", 
    \(\(x4 = 
        x + 25*k1/216 + 1408*k3/2565 + 2197*k4/4104 - 
          k5/5;\)\), "\[IndentingNewLine]", 
    \(\(t = t + h;\)\), "\[IndentingNewLine]", 
    \(\(Print["\<4th order estimate is  \>", 
        N[x4, 18]];\)\), "\[IndentingNewLine]", 
    \(Print["\<Exact answer is        \>", 
      N[sol[0.2`20], 18]]\), "\[IndentingNewLine]", 
    \(Print["\<4th order error is     \>", x4 - sol[0.2`20]]\)}], "Input"],

Cell["\<\
Here's one step of Fehlberg's RK5; notice little additional computation is \
required\
\>", "Text",
  FontSize->16],

Cell[BoxData[{
    \(t = 0; h = 0.20`20; x = 1.0`20;\), "\[IndentingNewLine]", 
    \(\(k6 = 
        h*f[t + h/2, 
            x - 8*k1/27 + 2*k2 - 3544*k3/2565 + 1859*k4/4104 - 
              11*k5/40];\)\), "\[IndentingNewLine]", 
    \(\(x = 
        x + 16*k1/135 + 6656*k3/12825 + 28561*k4/56430 - 9*k5/50 + 
          2*k6/55;\)\), "\[IndentingNewLine]", 
    \(\(t = t + h;\)\), "\[IndentingNewLine]", 
    \(\(Print["\<5th order estimate is  \>", 
        N[x, 18]];\)\), "\[IndentingNewLine]", 
    \(Print["\<Exact answer is        \>", 
      N[sol[0.20`20], 18]]\), "\[IndentingNewLine]", 
    \(Print["\<4th and 5th order estimates differ by \>", 
      Abs[x4 - x]]\)}], "Input"]
}, Closed]],

Cell[CellGroupData[{

Cell["Adaptive time-steps (in Fehlberg algorithm)", "Section",
  FontSize->22],

Cell["\<\
Here's a new differential equation, for which we can find an exact solution, \
and observe that it has a shape that might be easier to track in some \
regions, harder in others.\
\>", "Text",
  FontSize->18],

Cell[BoxData[
    \(Clear[x, t]; 
    f[t_, x_]\  = \(x*\((t + 4)\)\)\/\(\((x + 1)\)*\((t + 1)\)*\((1 + \((t - \
4)\)\^2)\)\)\)], "Input"],

Cell[BoxData[
    \(sol[t_]\  = \ 
      x[t]\  /. \ First[
          First[DSolve[{\(x'\)[t] \[Equal] f[t, x[t]], x[0] \[Equal] 1}, 
              x[t], t]]]\)], "Input"],

Cell[BoxData[
    \(Plot[sol[t], {t, 0, 10}]\)], "Input"],

Cell[TextData[{
  "We'll strive for ",
  Cell[BoxData[
      \(TraditionalForm\`\(\(10\^\(-2\)\)\(\ \)\)\)]],
  "error overall.  By a crude estimate, Fehlberg will give one-step error of \
",
  Cell[BoxData[
      \(TraditionalForm\`h\^5\)]],
  ", so that from t=0 to t=20, which will take 20/h steps, we expect error of \
around 20 ",
  Cell[BoxData[
      \(TraditionalForm\`h\^4\)]],
  ".  So, to get overall error of ",
  Cell[BoxData[
      \(TraditionalForm\`10\^\(-2\)\)]],
  ", we imagine that 20 ",
  Cell[BoxData[
      \(TraditionalForm\`h\^4\)]],
  "= ",
  Cell[BoxData[
      \(TraditionalForm\`10\^\(-2\)\)]],
  ", or ",
  Cell[BoxData[
      \(TraditionalForm\`h\^4\)]],
  "= ",
  Cell[BoxData[
      \(TraditionalForm\`10\^\(-3\)\)]],
  "/2, or h = 0.15.  We'll start with that h.  If the estimate of the \
one-step error is greater than ",
  Cell[BoxData[
      \(TraditionalForm\`\((0.15)\)\^5\)]],
  "=",
  Cell[BoxData[
      \(TraditionalForm\`10\^\(-4\)\)]],
  ", we'll insist on cutting in half the step size, while if it's less than \
",
  Cell[BoxData[
      \(TraditionalForm\`10\^\(-6\)\)]],
  ", we'll double the step size "
}], "Text",
  FontSize->16],

Cell[BoxData[{
    \(\(Clear[t, x, h];\)\), "\[IndentingNewLine]", 
    \(\(FehlbergEstimate[t_, x_, h_]\  := \ 
        Block[{}, \[IndentingNewLine]k1 = h*f[t, x]; \[IndentingNewLine]k2 = 
            h*f[t + h/4, x + k1/4]; \[IndentingNewLine]k3 = 
            h*f[t + 3*h/8, x + 3*k1/32 + 9*k2/32]; \[IndentingNewLine]k4 = 
            h*f[t + 12*h/13, 
                x + 1932*k1/2197 - 7200*k2/2197 + 
                  7296*k3/2197]; \[IndentingNewLine]k5 = 
            h*f[t + h, 
                x + 439*k1/216 - 8*k2 + 3680*k3/513 - 
                  845*k4/4104]; \[IndentingNewLine]x4 = 
            x + 25*k1/216 + 1408*k3/2565 + 2197*k4/4104 - 
              k5/5; \[IndentingNewLine]k6 = 
            h*f[t + h/2, 
                x - 8*k1/27 + 2*k2 - 3544*k3/2565 + 1859*k4/4104 - 
                  11*k5/40]; \[IndentingNewLine]x5 = 
            x + 16*k1/135 + 6656*k3/12825 + 28561*k4/56430 - 9*k5/50 + 
              2*k6/55; \[IndentingNewLine]{x4, 
            x5}\[IndentingNewLine]];\)\[IndentingNewLine]\), "\
\[IndentingNewLine]", 
    \(tlist = {0}; h = 0.15`20; xlist = {1.0`20}; nsteps = 1; 
    emin\  = \ 10\^\(-6\); emax = 10\^\(-4\);\), "\[IndentingNewLine]", 
    \(While[
      tlist[\([nsteps]\)] < 20, \[IndentingNewLine]t = tlist[\([nsteps]\)]; 
      x = xlist[\([nsteps]\)]; \[IndentingNewLine]newxvals\  = \ 
        FehlbergEstimate[t, x, h]; \[IndentingNewLine]x4 = newxvals[\([1]\)]; 
      x5 = newxvals[\([2]\)]; \[IndentingNewLine]While[
        Abs[x4 - x5] > emax, \[IndentingNewLine]h = 
          h/2; \[IndentingNewLine]newxvals\  = \ 
          FehlbergEstimate[t, x, h]; \[IndentingNewLine]x4 = 
          newxvals[\([1]\)]; 
        x5 = newxvals[\([2]\)];\[IndentingNewLine]]; \[IndentingNewLine]xlist \
= Append[xlist, x5]; tlist = Append[tlist, t + h]; 
      nsteps = nsteps + 1; \[IndentingNewLine]If[Abs[x4 - x5] < emin, 
        h = h*2]\[IndentingNewLine]]\), "\[IndentingNewLine]", 
    \(tlist\), "\[IndentingNewLine]", 
    \(xlist\)}], "Input"],

Cell[BoxData[{
    \(\(pl1 = 
        ListPlot[Transpose[{tlist, xlist}], PlotJoined \[Rule] True, 
          DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", 
    \(\(pl2 = 
        ListPlot[Transpose[{tlist, xlist}], PlotJoined \[Rule] False, 
          PlotStyle \[Rule] PointSize[0.02], 
          DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", 
    \(pl3 = 
      Show[pl1, pl2, DisplayFunction \[Rule] $DisplayFunction, 
        PlotLabel \[Rule] "\<Numerical Solution (both dots and curve)\>"]\), \
"\[IndentingNewLine]", 
    \(\(pl4 = 
        Plot[sol[t], {t, 0, 20}, 
          DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", 
    \(Show[pl2, pl4, DisplayFunction \[Rule] $DisplayFunction, 
      PlotLabel \[Rule] "\<Dots=Numerical Solution, Curve=Exact \
Solution\>"]\)}], "Input"]
}, Closed]],

Cell[CellGroupData[{

Cell[TextData[StyleBox["Instability", "Text"]], "Section",
  FontSize->24],

Cell["\<\
Here's a new IVP, which will illustrate the idea of instability:\
\>", "Text",
  FontSize->18],

Cell[BoxData[
    \(Clear[t, x]; 
    f[t_, x_]\  = \ 5*x\  - \ 6\ \[ExponentialE]\^\(-t\);\)], "Input",
  FontSize->18],

Cell[BoxData[
    \(Clear[t, x]; 
    sol[t_]\  = \ 
      x[t]\  /. \ DSolve[{\(x'\)[t] == f[t, x[t]], x[0] \[Equal] 1}, x[t], 
          t]\)], "Input"],

Cell[TextData[{
  "On general principles, we might expect RK4 to get the answer accurate to \
within around ",
  Cell[BoxData[
      \(TraditionalForm\`h\^4\)]],
  ", but..."
}], "Text",
  FontSize->18],

Cell[BoxData[{
    \(ti = 0; tf = 2; h = 0.2; 
    nsteps = \((tf - ti)\)/h;\), "\[IndentingNewLine]", 
    \(xlist = Table[{}, {nsteps + 1}]; 
    tlist = Table[{}, {nsteps + 1}];\), "\[IndentingNewLine]", 
    \(xlist[\([1]\)] = 1; tlist[\([1]\)] = 0;\), "\[IndentingNewLine]", 
    \(Do[\[IndentingNewLine]x = xlist[\([j]\)]; 
      t = tlist[\([j]\)]; \[IndentingNewLine]k1 = 
        h*f[t, x]; \[IndentingNewLine]k2 = 
        h*f[t + h/2, x + k1/2]; \[IndentingNewLine]k3 = 
        h*f[t + h/2, x + k2/2]; \[IndentingNewLine]k4 = 
        h*f[t + h, x + k3]; \[IndentingNewLine]xlist[\([j + 1]\)] = 
        x + \((k1 + 2*k2 + 2*k3 + k4)\)/
            6; \[IndentingNewLine]tlist[\([j + 1]\)] = 
        t + h, \[IndentingNewLine]{j, 1, nsteps}]\)}], "Input"],

Cell[BoxData[{
    \(\(pl1 = 
        ListPlot[Transpose[{tlist, xlist}], PlotJoined \[Rule] True, 
          DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", 
    \(\(pl2 = 
        Plot[sol[t], {t, 0, 2}, PlotStyle \[Rule] RGBColor[1, 0, 0], 
          DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", 
    \(Show[pl1, pl2, 
      DisplayFunction \[Rule] $DisplayFunction]\), "\[IndentingNewLine]", 
    \(Print["\<Error at t=2 is \>", 
      xlist[\([nsteps + 1]\)] - sol[2]]\)}], "Input"],

Cell["\<\
Here are a bunch of solutions to this differential equation at various \
initial conditions near x=1\
\>", "Text",
  FontSize->16],

Cell[BoxData[{
    \(Clear[t, x]; 
    gensol[t_, cval_]\  = \ 
      First[\((x[t]\  /. \ DSolve[\(x'\)[t] == f[t, x[t]], x[t], t])\)\  /. \ 
          C[1] \[Rule] cval]\), "\[IndentingNewLine]", 
    \(Clear[t]; 
    nearby[t_]\  = \ 
      Table[gensol[t, cval], {cval, \(-3\)/1000, 3/1000, 
          1/1000}];\), "\[IndentingNewLine]", 
    \(Plot[Evaluate[nearby[t]], {t, 0, 2}, 
      PlotRange \[Rule] {\(-2\), 2}]\)}], "Input"],

Cell["\<\
For this example, we can see at a glance that df/dx=5, so we can expect \
instability (and in fact we see it).
Let's also check the example in the previous section.  The formula for f(t,x) \
is pretty complicated, so we can't see at a glance what sign of df/dx is, but \
we can have algorithm compute it:\
\>", "Text",
  FontSize->16],

Cell[BoxData[{
    \(Clear[x, t]; 
    f[t_, x_]\  = \(x*\((t + 4)\)\)\/\(\((x + 1)\)*\((t + 1)\)*\((1 + \((t - \
4)\)\^2)\)\);\), "\[IndentingNewLine]", 
    \(dfdx[t_, x_]\  = \ D[f[t, x], x]\)}], "Input"],

Cell[BoxData[{
    \(\(Clear[t, x, h];\)\), "\[IndentingNewLine]", 
    \(\(FehlbergEstimate[t_, x_, h_]\  := \ 
        Block[{}, \[IndentingNewLine]k1 = h*f[t, x]; \[IndentingNewLine]k2 = 
            h*f[t + h/4, x + k1/4]; \[IndentingNewLine]k3 = 
            h*f[t + 3*h/8, x + 3*k1/32 + 9*k2/32]; \[IndentingNewLine]k4 = 
            h*f[t + 12*h/13, 
                x + 1932*k1/2197 - 7200*k2/2197 + 
                  7296*k3/2197]; \[IndentingNewLine]k5 = 
            h*f[t + h, 
                x + 439*k1/216 - 8*k2 + 3680*k3/513 - 
                  845*k4/4104]; \[IndentingNewLine]x4 = 
            x + 25*k1/216 + 1408*k3/2565 + 2197*k4/4104 - 
              k5/5; \[IndentingNewLine]k6 = 
            h*f[t + h/2, 
                x - 8*k1/27 + 2*k2 - 3544*k3/2565 + 1859*k4/4104 - 
                  11*k5/40]; \[IndentingNewLine]x5 = 
            x + 16*k1/135 + 6656*k3/12825 + 28561*k4/56430 - 9*k5/50 + 
              2*k6/55; \[IndentingNewLine]{x4, 
            x5}\[IndentingNewLine]];\)\[IndentingNewLine]\), "\
\[IndentingNewLine]", 
    \(tlist = {0}; h = 0.15`20; xlist = {1.0`20}; nsteps = 1; 
    emin\  = \ 10\^\(-6\); emax = 10\^\(-4\);\), "\[IndentingNewLine]", 
    \(While[
      tlist[\([nsteps]\)] < 20, \[IndentingNewLine]t = tlist[\([nsteps]\)]; 
      x = xlist[\([nsteps]\)]; \[IndentingNewLine]newxvals\  = \ 
        FehlbergEstimate[t, x, h]; \[IndentingNewLine]x4 = newxvals[\([1]\)]; 
      x5 = newxvals[\([2]\)]; \[IndentingNewLine]While[
        Abs[x4 - x5] > emax, \[IndentingNewLine]h = 
          h/2; \[IndentingNewLine]newxvals\  = \ 
          FehlbergEstimate[t, x, h]; \[IndentingNewLine]x4 = 
          newxvals[\([1]\)]; 
        x5 = newxvals[\([2]\)];\[IndentingNewLine]]; \[IndentingNewLine]xlist \
= Append[xlist, x5]; tlist = Append[tlist, t + h]; 
      nsteps = nsteps + 1; \[IndentingNewLine]stablist = 
        Append[stablist, dfdx[t + h, x5]]; \[IndentingNewLine]If[
        Abs[x4 - x5] < emin, 
        h = h*2]\[IndentingNewLine]]\), "\[IndentingNewLine]", 
    \(tlist\), "\[IndentingNewLine]", 
    \(xlist\), "\[IndentingNewLine]", 
    \(\(Print["\<Here are the values of the stability indicator df/dx along \
the trajectory:\>"];\)\), "\[IndentingNewLine]", 
    \(stablist\)}], "Input"]
}, Closed]],

Cell[CellGroupData[{

Cell["RK4 on a System of ODEs", "Section",
  FontSize->18],

Cell["\<\
Consider the system of two ODEs modeling the population growths of predator \
jackals (j, in thousands) and prey rabbits (r, in thousands):
\t\tr ' = 0.25 r - 0.3 r j
\t\tj ' = 0.05 r j - 0.08 j\
\>", "Text",
  FontSize->19],

Cell[TextData[{
  "Below I'll set up the right-hand side as a vector function ",
  Cell[BoxData[
      \(TraditionalForm\`\(\(\(f\)\(\ \)\)\& \[Rule] \)\)]],
  "(equal to (r',j')), with inputs t (scalar) and ",
  Cell[BoxData[
      \(TraditionalForm\`\(x\& \[Rule] \)\)]],
  " (vector equal to (r,j)):"
}], "Text",
  FontSize->19],

Cell[BoxData[
    \(Clear[r, j]; 
    f[t_, x_]\  := \ {0.25  x[\([1]\)]\  - \ 0.3*x[\([1]\)]*x[\([2]\)], 
        0.05*x[\([1]\)]*x[\([2]\)] - 0.08*x[\([2]\)]};\)], "Input"],

Cell[TextData[{
  "Now I use the RK4 code ",
  StyleBox["exactly as it was before",
    FontSlant->"Italic"],
  "; x and f now stand for vectors, but the formulas are exactly the same"
}], "Text",
  FontSize->19],

Cell[BoxData[{
    \(ti = 0; tf = 100; h = 0.2; 
    nsteps = \((tf - ti)\)/h;\), "\[IndentingNewLine]", 
    \(xlist = Table[{0, 0}, {nsteps + 1}]; 
    tlist = Table[{}, {nsteps + 1}];\), "\[IndentingNewLine]", 
    \(xlist[\([1]\)] = {2, 0.5}; 
    tlist[\([1]\)] = 0;\), "\[IndentingNewLine]", 
    \(Do[\[IndentingNewLine]x = xlist[\([j]\)]; 
      t = tlist[\([j]\)]; \[IndentingNewLine]k1 = 
        h*f[t, x]; \[IndentingNewLine]k2 = 
        h*f[t + h/2, x + k1/2]; \[IndentingNewLine]k3 = 
        h*f[t + h/2, x + k2/2]; \[IndentingNewLine]k4 = 
        h*f[t + h, x + k3]; \[IndentingNewLine]xlist[\([j + 1]\)] = 
        x + \((k1 + 2*k2 + 2*k3 + k4)\)/
            6; \[IndentingNewLine]tlist[\([j + 1]\)] = 
        t + h, \[IndentingNewLine]{j, 1, nsteps}]\)}], "Input"],

Cell["\<\
Now plot the path of the number of jackals versus the number of foxes as time \
runs forward\
\>", "Text",
  FontSize->17],

Cell[BoxData[
    \(ListPlot[xlist, PlotJoined \[Rule] True, 
      PlotRange \[Rule] {{0, 5}, {0, 2}}, 
      AxesLabel \[Rule] {"\<rabbits (in thous)\>", "\<jackals (in thous)\>"}]\
\)], "Input"]
}, Closed]],

Cell[CellGroupData[{

Cell["The 3-body problem", "Section",
  FontSize->18],

Cell["\<\
fx[x1_,y1_,z1_,x2_,y2_,z2_,m1_,m2_] = \
-(x1-x2)*m1*m2/((x1-x2)^2+(y1-y2)^2+(z1-z2)^2)^(3/2);
fy[x1_,y1_,z1_,x2_,y2_,z2_,m1_,m2_] = \
-(y1-y2)*m1*m2/((x1-x2)^2+(y1-y2)^2+(z1-z2)^2)^(3/2);
fz[x1_,y1_,z1_,x2_,y2_,z2_,m1_,m2_] = \
-(z1-z2)*m1*m2/((x1-x2)^2+(y1-y2)^2+(z1-z2)^2)^(3/2);
rhs[x_,m1_,m2_,m3_] := 
Block[{},
x1=x[[1]];y1=x[[2]];z1=x[[3]];
x2=x[[4]];y2=x[[5]];z2=x[[6]];
x3=x[[7]];y3=x[[8]];z3=x[[9]];
vx1=x[[10]];vy1=x[[11]];vz1=x[[12]];
vx2=x[[13]];vy2=x[[14]];vz2=x[[15]];
vx3=x[[16]];vy3=x[[15]];vz3=x[[18]];
{vx1,vy1,vz1,vx2,vy2,vz2,vx3,vy3,vz3,
fx[x1,y1,z1,x2,y2,z2,m1,m2]+fx[x1,y1,z1,x3,y3,z3,m1,m3],
fy[x1,y1,z1,x2,y2,z2,m1,m2]+fy[x1,y1,z1,x3,y3,z3,m1,m3],
fz[x1,y1,z1,x2,y2,z2,m1,m2]+fz[x1,y1,z1,x3,y3,z3,m1,m3],
fx[x2,y2,z2,x1,y1,z1,m2,m1]+fx[x2,y2,z2,x3,y3,z3,m2,m3],
fy[x2,y2,z2,x1,y1,z1,m2,m1]+fy[x2,y2,z2,x3,y3,z3,m2,m3],
fz[x2,y2,z2,x1,y1,z1,m2,m1]+fz[x2,y2,z2,x3,y3,z3,m2,m3],
fx[x3,y3,z3,x2,y2,z2,m3,m2]+fx[x3,y3,z3,x1,y1,z1,m3,m1],
fy[x3,y3,z3,x2,y2,z2,m3,m2]+fy[x3,y3,z3,x1,y1,z1,m3,m1],
fz[x3,y3,z3,x2,y2,z2,m3,m2]+fz[x3,y3,z3,x1,y1,z1,m3,m1]}
]\
\>", "Input"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(ti = 0; tf = 200; h = 0.02; 
    nsteps = \((tf - ti)\)/h;\), "\[IndentingNewLine]", 
    \(x = {0, 0, 0, \ \ \ \ 15, 0, 0, \ \ \ \ \(-25\), 10, 2, \ \ \ \ 0, 0, 
        0, \ \ \ 0, Sqrt[10/15], 0, \ \ \ \ 0\ , \ 
        Sqrt[10/Sqrt[729]]*Cos[0.1], Sqrt[10/Sqrt[729]]*Sin[0.1]}; 
    t = 0;\), "\[IndentingNewLine]", 
    \(m1 = 5; m2 = 1; m3 = 0.5;\), "\[IndentingNewLine]", 
    \(\(Show[
        Graphics[{PointSize[0.03], \[IndentingNewLine]{RGBColor[1, 0, 0], 
              Point[{x[\([1]\)], x[\([2]\)]}]}, \[IndentingNewLine]{RGBColor[
                0, 1, 0], 
              Point[{x[\([4]\)], x[\([5]\)]}]}, \[IndentingNewLine]{RGBColor[
                0, 0, 1], Point[{x[\([7]\)], x[\([8]\)]}]}}, 
          PlotRange \[Rule] {{\(-3\), 3}, {\(-3\), 
                3}}]];\)\), "\[IndentingNewLine]", 
    \(Do[\[IndentingNewLine]k1 = 
        h*rhs[x, m1, m2, m3]; \[IndentingNewLine]k2 = 
        h*rhs[x + k1/2, m1, m2, m3]; \[IndentingNewLine]k3 = 
        h*rhs[x + k2/2, m1, m2, m3]; \[IndentingNewLine]k4 = 
        h*rhs[x + k3, m1, m2, m3]; \[IndentingNewLine]x = 
        x + \((k1 + 2*k2 + 2*k3 + k4)\)/6; \[IndentingNewLine]If[
        Mod[j, 100] \[Equal] 
          0, \[IndentingNewLine]xcm\  = \ \((m1*x[\([1]\)] + m2*x[\([4]\)] + 
                m3*x[\([7]\)])\)/\((m1 + m2 + 
                m3)\); \[IndentingNewLine]ycm\  = \ \((m1*x[\([2]\)] + 
                m2*x[\([5]\)] + m3*x[\([8]\)])\)/\((m1 + m2 + 
                m3)\); \[IndentingNewLine]Show[
          Graphics[{PointSize[0.03], \[IndentingNewLine]{RGBColor[1, 0, 0], 
                Point[{x[\([1]\)] - xcm, 
                    x[\([2]\)] - ycm}]}, \[IndentingNewLine]{RGBColor[0, 1, 
                  0], Point[{x[\([4]\)] - xcm, 
                    x[\([5]\)] - ycm}]}, \[IndentingNewLine]{RGBColor[0, 0, 
                  1], Point[{x[\([7]\)] - xcm, x[\([8]\)] - ycm}]}}, 
            PlotRange \[Rule] {{\(-30\), 30}, {\(-30\), 
                  30}}]]]; \[IndentingNewLine]t = 
        t + h, \[IndentingNewLine]{j, 1, nsteps}]\)}], "Input"],

Cell[BoxData[
    \({0, 0, 0, 0, \@\(2\/3\), 0, 0, 0, 0.06075668254475505`, 1562\/98415, 
      50\/19683, 10\/19683, \(-\(1\/45\)\) - 25\/\(426\ \@426\), 
      25\/\(1704\ \@426\), 5\/\(1704\ \@426\), 
      125\/19683 + 25\/\(426\ \@426\), \(-\(50\/19683\)\) - 
        25\/\(1704\ \@426\), \(-\(10\/19683\)\) - 
        5\/\(1704\ \@426\)}\)], "Print"],

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