2
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This post is a clarified version of the question in this one.

Consider an implicit function, $F(e,w,a,b,i,n)=0$, whose code is:

D[w - 1/((1 - e) w), e] ==  D[1 - e, e] ((-((-a (i + (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(1 - n e (1/(b (a e)^(b)))^(1/(b - 1)))) + (-1 + a^2) (-1 + e)^2 w - a (-1 + e) (i + (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(1 - n e (1/(b (a e)^(b)))^(1/(b - 1)))) w^2)/(a (-1 + e) i (1 - e + i + (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(1 - n e (1/(b (a e)^(b)))^(1/(b - 1)))) w))) - (-((-a (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(1 - n e (1/(b (a e)^(b)))^(1/(b - 1))) + (-1 + a^2) (-1 +   e) (-1 + e - i) w -  a (-1 + e) (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(
    1 - n e (1/(b (a e)^(b)))^(1/(b - 1))) w^2)/( a (-1 + e) i (1 - e + i + (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(1 - n e (1/(b (a e)^(b)))^(1/(b - 1)))) w))))

From the above implicit function, $e$ can be implicitly defined as a function of $w$ with the rest of the variables taken as constant.

My task takes three steps as follows.

Step 1: I numerically examined how this function $e=e(w)$ looks like for any given value of $a = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1$ along with the other parameters fixed at $n=1$, $b=0.7$, $i=0.1$. My code for this is:

Clear["Global`*"]
n = 1;
eqns[i_, b_] = {D[w - 1/((1 - e) w), e] == 
D[1 - e, e] ((-((-a (i + (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(
         1 - n e (1/(b (a e)^(b)))^(1/(b - 1)))) + (-1 + 
         a^2) (-1 + e)^2 w - 
      a (-1 + e) (i + (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(
         1 - n e (1/(b (a e)^(b)))^(1/(b - 1)))) w^2)/(
     a (-1 + e) i (1 - e + i + (
        n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(
        1 - n e (1/(b (a e)^(b)))^(1/(b - 1)))) w))) - (-((-a (
       n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(
       1 - n e (1/(b (a e)^(b)))^(1/(b - 1))) + (-1 + a^2) (-1 + 
         e) (-1 + e - i) w - 
      a (-1 + e) (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(
       1 - n e (1/(b (a e)^(b)))^(1/(b - 1))) w^2)/(
     a (-1 + e) i (1 - e + i + (
        n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(
        1 - n e (1/(b (a e)^(b)))^(1/(b - 1)))) w)))), 0 <= i <= 1, 0 <= a <= 1, 0 <= e <= 1, w >= 0, 0 <= b <= 1};
ContourPlot[Evaluate@ Table[Simplify@    eqns[1/10, 7/10][[1]], {a, {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1}}], {w, 0, 5}, {e, 0, 1}, PlotPoints -> 50, MaxRecursion -> 4, FrameLabel -> (Style[#, 14] & /@ {w, e}), RotateLabel -> False, PlotLegends -> Placed[LineLegend[{0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1}, 
LegendLabel -> "a"], {.7, .4}]]

And the result is: enter image description here

Step 2: I would like to use this $e=e(w)$ function numerically found as above to obtain the solution for $e$ and $w$ that solve $\frac{\partial e}{\partial w}=\frac{e}{w}$. That is, the pair of solutions $(e^* , w^* )$ for each $a$ will be obtained at the tangent point between the $e=e(w)$ function and the ray from the origin. Graphically, it will look like this (In the figure, I indicated the solution pairs in red dot):

enter image description here

As is clearly described in the above figure, I will be able to obtain a unique $(e^* , w^* )$ for each $a$.

Step 3: Finally, what I would like to produce two separate plots, one plotting $e^* $ against $a$ and the other plotting $w^* $ against $a$. These will look something like below. Step 1 is already done. Now, I need help on Steps 2 & 3. (But I don't need to produce the second figure, the one with the red dots added. All I need to produce are the last two plots.) Please help.

enter image description here enter image description here

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4
  • $\begingroup$ D[1 - e, e] seems a somewhat confusing way to write -1, and I wanted to check, is -1 correct? $\endgroup$
    – Michael E2
    Jul 5, 2023 at 15:40
  • $\begingroup$ @MichaelE2, yes, -1 would be a simpler way to put it. $\endgroup$
    – ppp
    Jul 5, 2023 at 21:46
  • $\begingroup$ @pppI get this, no errors: i.sstatic.net/ZP7RV.jpg -- What was the first error or two you got? (I'm using V13.2.1, if that matters.) $\endgroup$
    – Michael E2
    Jul 6, 2023 at 4:36
  • $\begingroup$ @MichaelE2, this is what I get: shorturl.at/fBHOX $\endgroup$
    – ppp
    Jul 6, 2023 at 5:31

3 Answers 3

5
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This is clunky and can probably be optimized but it seems to work:

Define:

Clear["Global`*"]
f[e_, w_, a_, b_, i_, n_] = D[w - 1/((1 - e) w), e] - D[1 - e, e] ((-((-a (i + (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(1 - n e (1/(b (a e)^(b)))^(1/(b - 1)))) + (-1 + a^2) (-1 + e)^2 w - a (-1 + e) (i + (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(1 - n e (1/(b (a e)^(b)))^(1/(b - 1)))) w^2)/(a (-1 + e) i (1 - e + i + (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(1 - n e (1/(b (a e)^(b)))^(1/(b - 1)))) w))) - (-((-a (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(1 - n e (1/(b (a e)^(b)))^(1/(b - 1))) + (-1 + a^2) (-1 +   e) (-1 + e - i) w -  a (-1 + e) (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(
    1 - n e (1/(b (a e)^(b)))^(1/(b - 1))) w^2)/( a (-1 + e) i (1 - e + i + (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(1 - n e (1/(b (a e)^(b)))^(1/(b - 1)))) w))))

In other words, the contour curves you have plotted are those for which $$f(e(w),w) = 0. \tag{1}$$ Along such a curve, you will also have $$ \frac{d}{dw} \left[f(e(w),w)\right] = 0 \quad \Rightarrow \quad \frac{\partial f}{\partial e} \frac{de}{dw} + \frac{\partial f}{\partial w} = 0 $$ and the requirement that $de/dw = e/w$ then leads us to the condition $$ e \frac{\partial f}{\partial e} + w \frac{\partial f}{\partial w} = 0. \tag{2} $$ Equations (1) and (2) can be solved numerically for a fixed value of $a$, and the results can be plotted as a function of $a$ as follows:

With[{i = 1/10, b = 7/10, n = 1},
 sol[a_] := sol[a] =
   FindRoot[{
     f[e, w, a, b, i, n] == 0, 
     e D[f[e, w, a, b, i, n], e] + w D[f[e, w, a, b, i, n], w] == 0}, 
     {{e, 0.5}, {w, 0.5}}];
 {estar[a_], wstar[a_]} := {e /. sol[a], w /. sol[a]};
 estarplot = 
  Plot[estar[a], {a, 0.1, 1}, AxesLabel -> {a, SuperStar[e]}];
 wstarplot = 
  Plot[wstar[a], {a, 0.1, 1}, AxesLabel -> {a, SuperStar[w]}];
 ]
estarplot
wstarplot

enter image description here

enter image description here

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  • $\begingroup$ thanks for your help. It worked great. May I ask one more related question? The constant parameter values used are i=0.1, b=0.7, n=1. I'm also trying to use Manipulate to see how the results might change depending on the variations of these three parameters. Can you please help? $\endgroup$
    – ppp
    Jul 7, 2023 at 7:13
  • $\begingroup$ @ppp I think that adapting Michael E2's method to use ParametricNDSolve (with i, b, and n as parameters) would be your best bet in that case. My method is really not well-adapted to the sort of rapid updating that Manipulate requires. $\endgroup$ Jul 7, 2023 at 11:51
  • $\begingroup$ I was struggling to applying your method here to a similar problem but without success as you can see in this post: mathematica.stackexchange.com/questions/295283/… May I sincerely ask for your help once more? Thank you in advance! $\endgroup$
    – ppp
    Dec 20, 2023 at 1:15
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Aother way, using NDSolve to construct $w^*$ and $e^*$:

The basic ODE system is obtained by differentiating the OP's equation (myEq) with respect to a and constraining the line through the origin to be perpendicular to the gradient of the equation (at $(w(a),e(a))$:

D[{ (* (w,e) to satisfy the given equation *)
    myEQ /. Equal -> Subtract,
    (* vector (w,e) perpendicular to gradient *)
    {w, e} . D[myEQ /. Equal -> Subtract, {{w, e}}]
    } /. {w -> w[a], e -> e[a]}
  , a] == 0

Code:

myEQ = First@eqns[i, b];

(* solve for an initial point *)
icEQ = Block[{n = 1, i = 1/10, b = 7/10, a = 1/2},
      {#, Dt[#, w]} &@myEQ
      ] /. {e -> e[w]} /. {e'[w] -> e[w]/w} /. {e[w] -> e} // Simplify;
icEQ = icEQ /. Equal -> Subtract // Simplify;
icSOL = NSolve[icEQ == 0 && 0 < e < 1 && 0 < w < 2, {e, w}]
(*  {{e -> 0.577473, w -> 1.6782}}  *)

{e\[FivePointedStar], w\[FivePointedStar]} = NDSolveValue[{
   ode = Solve[
      Block[{n = 1, i = 1/10, b = 7/10},
       D[{
           myEQ /. Equal -> Subtract,
           {e, w} . D[myEQ /. Equal -> Subtract, {{e, w}}]
           } /. {w -> w[a], e -> e[a]}
         , a] == 0
       ],
      {e'[a], w'[a]}
      ] /. Rule -> Equal,
   e[1/2] == (e /. First@icSOL), w[1/2] == (w /. First@icSOL)},
  {e, w}, {a, $MachineEpsilon, 1}]

(* cp = OP's ContourPlot[..] *)
Show[
  cp,
  Graphics[{
    PointSize@Medium,
    Point[
     Table[{w\[FivePointedStar][a], e\[FivePointedStar][a]}, {a, 0.1, 
       0.9, 0.1}]],
    Opacity[0.3],
    Line[
     Table[{{0, 0}, {4, 
        4 e\[FivePointedStar][a]/w\[FivePointedStar][a]}}, {a, 0.1, 1,
        0.1}]]
    }],
  ParametricPlot[{w\[FivePointedStar][a], e\[FivePointedStar][a]}, 
   Evaluate@Flatten@{a, e\[FivePointedStar]["Domain"]}, 
   PlotStyle -> Magenta, PlotRange -> All], Axes -> True,
  AspectRatio -> Automatic, ImageSize -> Large, 
  PlotRange -> {{0, 4}, {-0.2, 1.1}}
  ] /. {0.7`, 0.4`} -> {0.88`, 0.4`}

enter image description here

ListLinePlot[{e\[FivePointedStar], w\[FivePointedStar]}, 
 PlotLegends -> 
  Block[{e\[FivePointedStar], w\[FivePointedStar]}, 
   HoldForm /@ {e\[FivePointedStar], w\[FivePointedStar]}], 
 PlotRange -> All]

enter image description here


In response to comments: Large image of a session in V13.3.0 for Mac OS X ARM (64-bit) (June 3, 2023); code cut and pasted from MSE (the code for cp from the OP as indicated and the code from "Code" in this answer):

https://i.sstatic.net/fCKPo.jpg

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  • $\begingroup$ Michael, thanks! I tried to reproduce your result, but your code yields errors. Can you please confirm? $\endgroup$
    – ppp
    Jul 6, 2023 at 4:15
  • $\begingroup$ I still don't reproduce your result. Can you please confirm your code? Thank you very much! $\endgroup$
    – ppp
    Jul 18, 2023 at 10:04
  • $\begingroup$ "I still don't reproduce your result." NONE of it worked? $\endgroup$
    – Michael E2
    Jul 18, 2023 at 23:22
  • $\begingroup$ correct, this is what I get: shorturl.at/fBHOX $\endgroup$
    – ppp
    Jul 19, 2023 at 5:01
  • $\begingroup$ I would really appreciate if you could help me on this whenever you have time. Thank you so much in advance! $\endgroup$
    – ppp
    Jul 22, 2023 at 7:45
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This needs some work, that needs be done for every tangent point. As an example I show it for a==1. To start, we define a function of e,w and a:

f[e_, w_, a_] = Simplify[eqns[1/10, 7/10]][[1, 1]];

The next step is to create some points on the curve around the tangent point. For a==1 this will be points in the region 2<w<.2.8 and 0.4<e<0.7:

pts = Table[{w, e /. Evaluate@
     NSolve[{f[e, w, 1] == 0, 0.4 < e < 0.7}, e, Reals][[1]]}, {w, 2, 2.8, 0.01}]

Next we fit some suitable function through the points. I choose a 4th order poly:

poly = LinearModelFit[pts, {1, x, x^2, x^3, x^4}, x]

enter image description here

Finally we can solve the tangent equation:

xt = x /. Solve[{fun'[x] == poly[x]/x, 2 < x < 3}, x, Reals][[1]]

2.53014

Lets check if this worked by plotting the curves:

ContourPlot[Evaluate[f[e, w, 1] == 0], {w, 0, 5}, {e, 0, 1}, 
 Epilog -> {Point[{xt, fun[xt]}], Line[{{0, 0}, {xt, fun[xt]}}]}]

enter image description here

Now this has to be repeated for the other tangent points.

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