(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 2041129, 34059] NotebookOptionsPosition[ 2037389, 33943] NotebookOutlinePosition[ 2038278, 33975] CellTagsIndexPosition[ 2038235, 33972] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Assignment 3", "Title", CellChangeTimes->{{3.433567408527817*^9, 3.433567415725386*^9}, { 3.437123188014092*^9, 3.437123188075443*^9}, {3.441962243810688*^9, 3.441962243903389*^9}}], Cell["Due: Monday 16 February, 2008", "Subtitle", CellChangeTimes->{{3.43356747830616*^9, 3.433567490982802*^9}, { 3.437123153109909*^9, 3.437123167817243*^9}, {3.441962257602857*^9, 3.441962263793868*^9}, {3.441962345071497*^9, 3.441962348286543*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ "SetDirectory", "[", "\"\<~/Teaching/MathModelBioSciences_Winter08/Ex2\>\"", "]"}], ";"}]], "Input", CellOpen->False, CellGroupingRules->{GroupTogetherGrouping, 10000.}, CellChangeTimes->{{3.433567727678491*^9, 3.433567749540262*^9}, { 3.437123811017949*^9, 3.437123813136998*^9}, 3.437208667416116*^9}], Cell[TextData[{ StyleBox["Question (30 points)", FontColor->GrayLevel[0], Background->RGBColor[1, 1, 0]], ": we have seen in lecthre that for the calcium signaling model discussed, \ there is possibility for generating kinetic waves under appropriate \ conditions on the parameters. These waves have the particular property that \ the apparent wavespeed has dependence on the initial condition profile. In \ particular, we will now look at the situation where one chooses an initial \ condition that is spatially periodic, i.e., ", Cell[BoxData[ FormBox[ SubscriptBox["c", "0"], TraditionalForm]]], "(x+\[Lambda]) = ", Cell[BoxData[ FormBox[ RowBox[{" ", SubscriptBox["c", "0"]}], TraditionalForm]]], "(x) and ", Cell[BoxData[ FormBox[ SubscriptBox["w", "0"], TraditionalForm]]], "(x+\[Lambda]) = ", Cell[BoxData[ FormBox[ RowBox[{" ", SubscriptBox["w", "0"]}], TraditionalForm]]], "(x). If in addition, ", Cell[BoxData[ FormBox[ SubscriptBox["c", "0"], TraditionalForm]]], "(x) and ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["w", "o"], "(", "x", ")"}], " "}], TraditionalForm]]], "are chosen to be a ", StyleBox["limit cycle solution", FontSlant->"Italic"], " for the ODE system (i.e., obtained from the PDE system by ignoring the \ spatial diffusion term), then we will in fact have: ", Cell[BoxData[ FormBox["c", TraditionalForm]]], "(x+\[Lambda],t) = ", Cell[BoxData[ FormBox["c", TraditionalForm]]], "(x,t), ", Cell[BoxData[ FormBox["w", TraditionalForm]]], "(x+\[Lambda],t) = ", Cell[BoxData[ FormBox["w", TraditionalForm]]], "(x,t). In this question, we will set up such an initial condition \ profile. From the PDE solution, we will see that c(x,t) ", StyleBox["\[TildeTilde]", FontSize->24], " c(x+v t), where", StyleBox[" v \[TildeTilde] ", FontSize->24], Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ FractionBox["\[Lambda]", "\[Tau]"], FontWeight->"Bold"], " "}], TraditionalForm]], FontSize->24], "with ", StyleBox["\[Tau]", FontSize->24], " being the period of limit cycle oscillation or the ODE solution.\n" }], "Item1", CellGroupingRules->{GroupTogetherGrouping, 10001.}, CellChangeTimes->{{3.433567500149698*^9, 3.433567548592442*^9}, { 3.437123898287451*^9, 3.437123968229729*^9}, {3.437124016957098*^9, 3.437124065054027*^9}, 3.43712409580995*^9, {3.437124317335202*^9, 3.437124374321207*^9}, {3.437124949642591*^9, 3.437124951331112*^9}, 3.437125034209606*^9, {3.437126631097504*^9, 3.437126632281896*^9}, { 3.43720251911145*^9, 3.437202527025999*^9}, {3.43720866741642*^9, 3.437208676891072*^9}, {3.441962294103612*^9, 3.441962312419731*^9}, { 3.441962463162*^9, 3.441962502905883*^9}, 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