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Cell[TextData[{
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 ": Using the above initial conditions, solve the given PDE system, with \
diffusion coefficient = 0.05 and with periodic boundary conditions. Do \
density plots of the solutions ",
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 " and ",
 StyleBox["w", "Input"],
 ". Hint: to compute and plot the solution, you can refer to the notebook as \
given in lecture."
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 ": repeat the above with ",
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Cell["Kinetic Wave: Wave-Speed Dependence on Initial Condition", "Section",
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 "In the  case that the diffusion coupling is small, one can in fact get \
waves with speeds that depends on the particular form of initial conditions. \
These are the characteristics of ",
 StyleBox["kinetic waves",
  FontSlant->"Italic"],
 ", as opposed to the bistable, ",
 StyleBox["travelling wave",
  FontSlant->"Italic"],
 ".\n\nIn particular, we will now look at the situation where one chooses an \
initial condition that is spatially periodic, i.e., ",
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 "(x). If in addition, ",
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 "are chosen to be a ",
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 " for the ODE system (i.e., obtained from the PDE system by ignoring the \
spatial diffusion term), then we will in fact have:   ",
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 "(x+\[Lambda],t) =   ",
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 "(x,t), ",
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 "(x+\[Lambda],t) =   ",
 Cell[BoxData[
  FormBox["w", TraditionalForm]]],
 "(x,t).\n\nIn Question 3, we will set up such an initial condition profile. \
From the PDE solution (with small diffusion), we will see that c(x,t) = c(x+v \
t), where v \[TildeTilde] ",
 Cell[BoxData[
  FormBox[
   RowBox[{
    FractionBox["\[Lambda]", "\[Tau]"], " "}], TraditionalForm]]],
 "with \[Tau]being the period of limit cycle f oscillation or the ODE  \
solution."
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Cell[TextData[{
 StyleBox["Q3 (25 points)",
  FontWeight->"Bold"],
 ": construct 2 sets of initial conditions (for",
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  FormBox[
   RowBox[{"  ", 
    SubscriptBox["c", "0"]}], TraditionalForm]]],
 "(x),  ",
 Cell[BoxData[
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 ") that both lie on the limit cycle of the ODE system "
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     RowBox[{"w", "'"}], "[", "t", "]"}], "\[Equal]", " ", 
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Cell[TextData[{
 "but with the (spatial) periods 25 and 100 respectively (hint: use scaling \
as was done in setting up Q1, via ",
 StyleBox["DesiredWavePeriod", "Input",
  FontColor->RGBColor[1, 0, 1]],
 ").  Then, demonstrate kinetic waves using diffusion coefficient = 0.05. \n\n\
You can either do this on your own, or following the steps suggested below."
}], "Subtitle",
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Cell[CellGroupData[{

Cell["Q3.1", "Subsection",
 CellChangeTimes->{{3.409900624719713*^9, 3.4099006266464853`*^9}, {
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Cell[TextData[{
 "Solve the ODE system above to ",
 StyleBox["tEnd", "Input"],
 "=2000, with initial conditions  w[0] \[Equal] 1.1, c[0] \[Equal] 0.16  "
}], "Text",
 CellChangeTimes->{{3.409900656236801*^9, 3.4099006859679947`*^9}}],

Cell["\<\
If you ignore the initial transient (i.e., say up to t=1500), you should get \
a numerical approximation to the limit cycle looking like below:\
\>", "Text",
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\>"]],

Cell["\<\
One can get a feel for the time-series by superimposing on the nullclines of \
the 2-dimensional ODE system:\
\>", "Text",
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Cell["Q3.2", "Subsection",
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Cell["\<\
We then need to compute the period for the above computed limit cycle \
solution.\
\>", "Text",
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Cell[TextData[{
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times at which the peaks occur.\nWe can use the commands ",
 StyleBox["NMaximize", "Input"],
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Cell[TextData[{
 "Hint: use commands of the form ",
 StyleBox["cMax = NMaximize[{(c[t] /. Flatten[ODESoln]), t <= tEnd, t >= \
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Cell[TextData[{
 "and then do a  ",
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temporal period \[Tau] should be around 45. "
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Cell["Q3.3", "Subsection",
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Cell["\<\
Transform the initial conditions to have spatial period of 25, so that if you \
plot them they look something like this: \
\>", "Text",
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9H/5o/W5/wVqnBTs\
\>"]],

Cell[TextData[{
 "Hint: you may do this using a command that involves the replacement rule:  \
",
 StyleBox["t \[Rule](x + tEnd/4)*\[Tau]/DesiredWavePeriod", "Input"]
}], "Text",
 CellChangeTimes->{{3.4099026026029177`*^9, 3.409902621930903*^9}, {
  3.4099026641549177`*^9, 3.409902680500175*^9}, {3.409902754351166*^9, 
  3.409902758147472*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["Q3.4", "Subsection",
 CellChangeTimes->{{3.409900624719713*^9, 3.4099006266464853`*^9}, {
  3.409900755999432*^9, 3.409900756148443*^9}, {3.409902393018853*^9, 
  3.409902393101275*^9}, {3.409902720084482*^9, 3.409902720182328*^9}, {
  3.409903336539872*^9, 3.4099033366300097`*^9}}],

Cell["\<\
Using the above initial condition, solve the PDE system and do a density plot \
of the solution.
The solution should look translationally invariant.\
\>", "Text",
 CellChangeTimes->{{3.4099028021567173`*^9, 3.409902858766424*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["Q3.5", "Subsection",
 CellChangeTimes->{{3.409900624719713*^9, 3.4099006266464853`*^9}, {
  3.409900755999432*^9, 3.409900756148443*^9}, {3.409902393018853*^9, 
  3.409902393101275*^9}, {3.409902720084482*^9, 3.409902720182328*^9}, {
  3.409902866551786*^9, 3.409902866670162*^9}, {3.409903339402907*^9, 
  3.409903339510854*^9}}],

Cell["\<\
Repeat the process, using the folowing initial condition, with \
DesiredWavePeriod = 100: \
\>", "Text",
 CellChangeTimes->{{3.4099028021567173`*^9, 3.409902929313077*^9}}],

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