(* 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[ 1869435, 31650] NotebookOptionsPosition[ 1863552, 31471] NotebookOutlinePosition[ 1864036, 31490] CellTagsIndexPosition[ 1863993, 31487] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell["", "Text", CellChangeTimes->{{3.4336588778627996`*^9, 3.4336588805257998`*^9}}], Cell[CellGroupData[{ Cell["Assignment 1", "Title", CellChangeTimes->{{3.433567408527817*^9, 3.433567415725386*^9}}, TextAlignment->Center], Cell["Based on the Solution of Michael Kolmbauer", "Text", CellChangeTimes->{{3.4336614852408*^9, 3.4336614969438*^9}, { 3.436581650975928*^9, 3.436581660865187*^9}, {3.4366008172044783`*^9, 3.436600833146834*^9}}, TextAlignment->Center, FontSize->18, Background->RGBColor[ 0.7137254901960784, 0.8235294117647058, 0.8784313725490196]], Cell["In this exercise we are looking at the following equation", "Text", CellChangeTimes->{{3.4336588979778*^9, 3.4336589238118*^9}, { 3.433772559042*^9, 3.433772572856*^9}}, FontSize->18], Cell[BoxData[ RowBox[{ RowBox[{"D", RowBox[{"(", RowBox[{"(", RowBox[{"x", ",", "y"}], ")"}], ")"}]}], "=", RowBox[{"f", RowBox[{"(", RowBox[{"x", ",", "y"}], ")"}]}]}]], "DisplayFormulaNumbered", CellChangeTimes->{{3.4337712459630003`*^9, 3.433771265797*^9}, { 3.4337713590559998`*^9, 3.433771496373*^9}, {3.433771536635*^9, 3.433771543185*^9}, {3.4337715743059998`*^9, 3.433771611674*^9}, { 3.4337718730299997`*^9, 3.433771875031*^9}, {3.433772461467*^9, 3.433772463861*^9}, {3.433772542178*^9, 3.433772551689*^9}}, TextAlignment->Center, FontSize->14], Cell[TextData[{ "where ", StyleBox["D", FontSlant->"Italic"], " is a given Differential Operator and ", StyleBox["f", FontSlant->"Italic"], " is a given righthandside." }], "Text", CellChangeTimes->{{3.4336588979778*^9, 3.4336589238118*^9}, { 3.433772559042*^9, 3.433772642237*^9}, 3.4347217834584*^9}, FontSize->18], Cell[BoxData[ RowBox[{ RowBox[{"D", RowBox[{"(", RowBox[{"(", RowBox[{"x", ",", "y"}], ")"}], ")"}]}], "=", RowBox[{"(", "\[NoBreak]", GridBox[{ { RowBox[{ RowBox[{ FractionBox["\[PartialD]", RowBox[{"\[PartialD]", "t"}]], "x"}], "-", RowBox[{ SuperscriptBox["\[Del]", "2"], "x"}]}]}, { RowBox[{ RowBox[{ FractionBox["\[PartialD]", RowBox[{"\[PartialD]", "t"}]], "y"}], "-", RowBox[{"\[Delta]", " ", "c", RowBox[{ SuperscriptBox["\[Del]", "2"], "y"}]}]}]} }], "\[NoBreak]", ")"}]}]], "DisplayFormulaNumbered", CellChangeTimes->{{3.4337712459630003`*^9, 3.433771265797*^9}, { 3.4337713590559998`*^9, 3.433771496373*^9}, {3.433771536635*^9, 3.433771543185*^9}, {3.4337715743059998`*^9, 3.433771611674*^9}, { 3.4337718730299997`*^9, 3.433771875031*^9}, {3.433772461467*^9, 3.433772463861*^9}, {3.433772542178*^9, 3.433772551689*^9}, { 3.4347219151104*^9, 3.4347220518943996`*^9}, {3.4347221120534*^9, 3.4347221159224*^9}, {3.4347228538884*^9, 3.4347228599994*^9}}, TextAlignment->Center, FontSize->14], Cell[BoxData[ RowBox[{ RowBox[{"f", RowBox[{"(", RowBox[{"(", RowBox[{"x", ",", "y"}], ")"}], ")"}]}], "=", RowBox[{"(", "\[NoBreak]", GridBox[{ { RowBox[{ RowBox[{"a", "-", "x", "-", FractionBox[ RowBox[{"4", "x", " ", "y"}], RowBox[{"1", "+", SuperscriptBox["x", "2"]}]]}], ")"}]}, { RowBox[{"\[Delta]", " ", "b", " ", RowBox[{"(", RowBox[{"x", "-", FractionBox[ RowBox[{"x", " ", "y"}], RowBox[{"1", "+", SuperscriptBox["x", "2"]}]]}], ")"}]}]} }], ")"}]}]], "DisplayFormulaNumbered", CellChangeTimes->{{3.4337712459630003`*^9, 3.433771265797*^9}, { 3.4337713590559998`*^9, 3.433771496373*^9}, {3.433771536635*^9, 3.433771543185*^9}, {3.4337715743059998`*^9, 3.433771611674*^9}, { 3.4337718730299997`*^9, 3.433771875031*^9}, {3.433772461467*^9, 3.433772463861*^9}, {3.433772542178*^9, 3.433772551689*^9}, { 3.4347219151104*^9, 3.4347221041084003`*^9}, {3.4347221997254*^9, 3.4347222437153997`*^9}, {3.4347228628914003`*^9, 3.4347228958034*^9}}, TextAlignment->Center, FontSize->14], Cell["", "Text", CellChangeTimes->{{3.436594045196377*^9, 3.4365940481155987`*^9}}], Cell["We note that X is an activator and Y is an inhibitor. 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