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Re: Manipulate a Plot of Evaluate DSolve

  • To: mathgroup at smc.vnet.net
  • Subject: [mg127523] Re: Manipulate a Plot of Evaluate DSolve
  • From: Bob Hanlon <hanlonr357 at gmail.com>
  • Date: Wed, 1 Aug 2012 04:56:31 -0400 (EDT)
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  • References: <20120729070521.C7CB3684D@smc.vnet.net>

Your diff eqn can be solved exactly with DSolve. Why use NDSolve?

ClearAll["Global`*"];

func[coef_, c_, t_] =
 (y[t] /. DSolve[{y'[t] == Cos[coef*t], y[0] == c},
     y[t], t][[1]])

(c*coef + Sin[coef*t])/coef

Manipulate[
 Plot[Evaluate[{n0*Sin[coef*x + time], -H2, H1,
    Table[func[coef, c, x], {c, -5, -1}]}],
  {x, -XMax, XMax},
  Frame -> True,
  Axes -> False,
  FrameLabel -> {"X", "Y"},
  Filling -> {1 -> {{2}, LightBlue}},
  PlotRange -> {-H2, H1},
  PlotLabel -> "Floating object in Sea Wave Model"],
 {{time, 0, "Time"}, 0, 10 Pi, 0.1,
  Appearance -> "Labeled", ImageSize -> Tiny},
 {{coef, 1, "Wave Frequency"}, 0.1, 5,
  Appearance -> "Labeled", ImageSize -> Tiny},
 {{n0, 1, "Wave Amplitude"}, 0.1, 5,
  Appearance -> "Labeled", ImageSize -> Tiny},
 {{H1, 3, "Air Height"}, 1, 10,
  Appearance -> "Labeled", ImageSize -> Tiny},
 {{H2, 10, "Water Depth"}, 1, 50,
  Appearance -> "Labeled", ImageSize -> Tiny},
 {{XMax, 10}, 1, 50,
  Appearance -> "Labeled", ImageSize -> Tiny},
 ControlPlacement -> Right]


Bob Hanlon


On Tue, Jul 31, 2012 at 5:34 PM, Juan Barandiaran
<barandiaran.juan at gmail.com> wrote:
> Thanks again Bob,
>
> It looks a promising approach as it works and has the differential equation
> outside the Manipulate and Plot blocks.
> The only problem that I find now is that this solution is incredibly slow as
> soon as you include it in a list of Plots.
> I'll give a simple example, of the model I'm building of some objects moving
> in the sea waves.
>
> First a simple user interface taking advantage of the great options we have
> in Manipulate and Plot:
>
> ClearAll["Global`*"];
>
> Manipulate[
>  Plot[
>   {n0*Sin[coef*x + time], -H2, H1,
>    n0*Sin[coef*x + time] + c /. c -> Range[-5, -1]},
>   {x, -XMax, XMax}, AxesLabel -> {"X", "Y"},
>   Filling -> {1 -> {{2}, LightBlue}}, PlotRange -> {-H2, H1},
>   AspectRatio -> Automatic,
>   PlotLabel -> "Floating object in Sea Wave Model"],
>  {{time, 0, "Time"}, 0, 5*2*Pi,
>   Appearance -> "Labeled"}, {{coef , 1, "Wave Frequency"}, 0.1, 5,
>   Appearance -> "Labeled"}, {{n0, 0.5, "Wave Amplitude"}, 0.1, 5,
>   Appearance -> "Labeled"}, {{H1, 3, "Air Height"}, 1, 10,
>   Appearance -> "Labeled"}, {{H2, 10, "Water Depth"}, 1, 50,
>   Appearance -> "Labeled"}, {{XMax, 10}, 1, 50,
>   Appearance -> "Labeled"}, ControlPlacement -> Right]
>
> If you move the sliders of Manipulate it works fast and is responsive.
>
> Then we add a simple differential equation with the proposed solution:
>
> func[coef_?NumericQ, c_?NumericQ,
>   xLim_?NumericQ] := (y[t] /.
>    NDSolve[{y'[t] == Cos[coef*t], y[0] == c}, y, {t, -xLim, xLim}][[
>     1]])
>
> Manipulate[
>  Plot[Evaluate[
>    Table[func[coef, c, 10] /. t -> x, {c, -5, -1}]], {x, -10, 10},
>   PlotRange -> {-10, 5}], {{coef, 1}, 0.1, 5, 0.01,
>   Appearance -> "Labeled"}]
>
> It continues to work fast. So far so good.
>
> Now we try to integrate both things in a Manipulate block (or a Plot, I
> don't think the problem is with manipulate):
>
> Manipulate[
>  Plot[
>   {n0*Sin[coef*x + time], -H2, H1,
>    Evaluate[Table[func[coef, c, XMax] /. t -> x, {c, -5, -1}]]},
>   {x, -XMax, XMax}, AxesLabel -> {"X", "Y"},
>   Filling -> {1 -> {{2}, LightBlue}}, PlotRange -> {-H2, H1},
>   AspectRatio -> Automatic,
>   PlotLabel -> "Floating object in Sea Wave Model"],
>  {{time, 0, "Time"}, 0, 5*2*Pi,
>   Appearance -> "Labeled"}, {{coef , 1, "Wave Frequency"}, 0.1, 5,
>   Appearance -> "Labeled"}, {{n0, 1, "Wave Amplitude"}, 0.1, 5,
>   Appearance -> "Labeled"}, {{H1, 3, "Air Height"}, 1, 10,
>   Appearance -> "Labeled"}, {{H2, 10, "Water Depth"}, 1, 50,
>   Appearance -> "Labeled"}, {{XMax, 10}, 1, 50,
>   Appearance -> "Labeled"}, ControlPlacement -> Right]
>
> And at least in my system, the interface becomes incredibly slow... WHY ????
> Any workaround? Is it necessary to use a Table?
>
> Thanks once more and best regards,
>
> Juan
>
>
>
> 2012/7/31 Bob Hanlon <hanlonr357 at gmail.com>
>>
>> Not sure what happened. Try this
>>
>> Clear[func]
>>
>> func[coef_?NumericQ, c_?NumericQ] :=
>>  (y[t] /. NDSolve[
>>      {y'[t] == Cos[coef*t], y[0] == c},
>>      y, {t, -10, 10}][[1]])
>>
>> Manipulate[
>>  Plot[
>>   Evaluate[
>>    Table[
>>     func[coef, c] /. t -> x,
>>     {c, 5}]],
>>   {x, -10, 10},
>>   PlotRange -> {-5, 10}],
>>  {{coef, 1}, 0.1, 5, 0.01,
>>   Appearance -> "Labeled"}]
>>
>>
>> Bob Hanlon
>>
>>
>> On Mon, Jul 30, 2012 at 9:17 PM, Juan Barandiaran
>> <barandiaran.juan at gmail.com> wrote:
>> > Thanks Bob,
>> >
>> > Yes, I could use NDSolve instead of DSolve, in my case I'm only getting
>> > numerical outputs, so it should do the job.
>> >
>> > But in the example that you send me, why is nothing plotted?
>> >
>> > Thanks again and best regards,
>> >
>> > Juan
>> >
>> >
>> > 2012/7/30 Bob Hanlon <hanlonr357 at gmail.com>
>> >>
>> >> If DSolve cannot solve the equations then use NDSolve.
>> >>
>> >> func[coef_?NumericQ, c_?NumericQ, x_?NumericQ] :=
>> >>  y[t] /. NDSolve[{y'[t] == Cos[coef*t], y[0] == c},
>> >>      y[t], {t, -10, 10}][[1]] /. t -> x
>> >>
>> >> Manipulate[Plot[
>> >>   Evaluate[func[coef, c, x] /. c -> Range[5]],
>> >>   {x, -10, 10},
>> >>   PlotRange -> {-5, 10}],
>> >>  {{coef, 1}, 0.1, 5, 0.01,
>> >>   Appearance -> "Labeled"}]
>> >>
>> >>
>> >> Bob Hanlon
>> >>
>> >>
>> >> On Sun, Jul 29, 2012 at 6:12 PM, Juan Barandiaran
>> >> <barandiaran.juan at gmail.com> wrote:
>> >> > Thanks for your answer Bob,
>> >> > Of course your solution works, but I still don't understand why mine
>> >> > doesn't
>> >> > and I cannot use your proposed approach because the way you write the
>> >> > problem it is easy for Mathematica to solve the DSolve.
>> >> > And this is just a simple example, in my real problem the DSolve
>> >> > cannot
>> >> > be
>> >> > solved analytically.
>> >> > This is why I tried to express the function as:
>> >> >
>> >> > {{y -> Function[{x}, DSolve[y'[x] == Cos[coef *x], y, x]]}}
>> >> >
>> >> > , which is something like the output I get from my DSolve.
>> >> >
>> >> > Thanks for your help.
>> >> >
>> >> > Juan
>> >> >
>> >> >
>> >> > 2012/7/29 Bob Hanlon <hanlonr357 at gmail.com>
>> >> >>
>> >> >> Clear[func];
>> >> >>
>> >> >> func[coef_, c_, x_] =
>> >> >>  y[x] /. DSolve[{y'[x] == Cos[coef*x], y[0] == c}, y[x], x][[1]] //
>> >> >>   Simplify
>> >> >>
>> >> >> c + Sin[coef*x]/coef
>> >> >>
>> >> >>
>> >> >> Manipulate[Plot[Evaluate[
>> >> >>    func[coef, c, x] /.
>> >> >>     c -> Range[5]],
>> >> >>   {x, -10, 10},
>> >> >>   PlotRange -> {-5, 10}],
>> >> >>  {{coef, 1}, 0.1, 5, 0.01,
>> >> >>   Appearance -> "Labeled"}]
>> >> >>
>> >> >>
>> >> >> Bob Hanlon
>> >> >>
>> >> >>
>> >> >> On Sun, Jul 29, 2012 at 3:05 AM,  <barandiaran.juan at gmail.com>
>> >> >> wrote:
>> >> >> > Hi,
>> >> >> >
>> >> >> > I'm trying to Manipulate a Plot of a quite difficult function
>> >> >> > which
>> >> >> > involves solving a differential equation, but cannot be solved
>> >> >> > analytically.
>> >> >> >
>> >> >> > To try to simplify the example and simulate it, let's assume that
>> >> >> > we
>> >> >> > have the following function:
>> >> >> >
>> >> >> > func[coef_] = {{y -> Function[{x}, DSolve[y'[x] == Cos[coef *x],
>> >> >> > y,
>> >> >> > x]]}}
>> >> >> >
>> >> >> > Manipulate[Plot[{{Evaluate[y[x] /. func[coef] /. C[1] ->
>> >> >> > {Range[-5,
>> >> >> > 0]}]}}, {x, -10, 10}], {{coef , 1}, 0.1, 5}]
>> >> >> >
>> >> >> > I get an error: DSolve::dsvar: "-9.99959 cannot be used as a
>> >> >> > variable"
>> >> >> >
>> >> >> > I think that this is because Manipulate assigns a value to x (=
>> >> >> > -9.99959) BEFORE solving the DSolve, even though to avoid it I'm
>> >> >> > using the
>> >> >> > Evaluate function, which should process the function before
>> >> >> > assigning
>> >> >> > a
>> >> >> > value to x.
>> >> >> >
>> >> >> > But the thing is that the "coef" to be Manipulated is at the same
>> >> >> > "level" as the x in the Manipulate block, so probably if I need
>> >> >> > the
>> >> >> > coef to
>> >> >> > solve the DSolve, I also have the x that gives me an error.
>> >> >> >
>> >> >> > Is there any workaround? I guess I'm not understanding properly
>> >> >> > how
>> >> >> > Mathematica processes these simple expressions.
>> >> >> >
>> >> >> > Thanks, Juan
>> >> >> >
>> >> >
>> >> >
>> >
>> >
>
>



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