(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 16021, 499] NotebookOptionsPosition[ 14697, 448] NotebookOutlinePosition[ 15061, 464] CellTagsIndexPosition[ 15018, 461] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{ "solve", " ", "for", " ", "the", " ", "steady", " ", "state", " ", "uncertainty", " ", "pInfinity"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"pInfRoots", "=", RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{ FractionBox[ RowBox[{"-", SuperscriptBox["b", "2"]}], "r"], SuperscriptBox["p", "2"]}], "+", RowBox[{"2", " ", "a", " ", "p"}], "+", "q"}], "\[Equal]", "0"}], ",", "p"}], "]"}]}]}]], "Input", CellChangeTimes->{{3.484044279986908*^9, 3.484044296002943*^9}, { 3.484044706029725*^9, 3.484044707951661*^9}, {3.48404557152617*^9, 3.484045589714252*^9}, {3.48404699307955*^9, 3.4840469953764834`*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"p", "\[Rule]", FractionBox[ RowBox[{ RowBox[{"a", " ", "r"}], "-", SqrtBox[ RowBox[{ RowBox[{ SuperscriptBox["b", "2"], " ", "q", " ", "r"}], "+", RowBox[{ SuperscriptBox["a", "2"], " ", SuperscriptBox["r", "2"]}]}]]}], SuperscriptBox["b", "2"]]}], "}"}], ",", RowBox[{"{", RowBox[{"p", "\[Rule]", FractionBox[ RowBox[{ RowBox[{"a", " ", "r"}], "+", SqrtBox[ RowBox[{ RowBox[{ SuperscriptBox["b", "2"], " ", "q", " ", "r"}], "+", RowBox[{ SuperscriptBox["a", "2"], " ", SuperscriptBox["r", "2"]}]}]]}], SuperscriptBox["b", "2"]]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.4840442966904607`*^9, 3.484044708467303*^9, 3.4840455934331207`*^9, 3.4840470092205877`*^9, 3.484048364485394*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{ "we", " ", "must", " ", "take", " ", "the", " ", "postive", " ", "root", " ", "since", " ", "p_", RowBox[{"{", "Infty", "}"}], " ", "must", " ", "be", " ", "postive"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"pInf", "=", RowBox[{"pInfRoots", "[", RowBox[{"[", RowBox[{"2", ",", "1", ",", "2"}], "]"}], "]"}]}]}]], "Input", CellChangeTimes->{{3.4840446957950225`*^9, 3.4840447569844804`*^9}, { 3.484045566057245*^9, 3.484045608605481*^9}}], Cell[BoxData[ FractionBox[ RowBox[{ RowBox[{"a", " ", "r"}], "+", SqrtBox[ RowBox[{ RowBox[{ SuperscriptBox["b", "2"], " ", "q", " ", "r"}], "+", RowBox[{ SuperscriptBox["a", "2"], " ", SuperscriptBox["r", "2"]}]}]]}], SuperscriptBox["b", "2"]]], "Output", CellChangeTimes->{{3.484044723905297*^9, 3.4840447574063687`*^9}, 3.4840451097457685`*^9, 3.48404560933988*^9, 3.484047010720626*^9, 3.4840483665323086`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{ RowBox[{"the", " ", "transformation", " ", "z"}], " ", "=", " ", RowBox[{ RowBox[{"1", "/", RowBox[{"(", RowBox[{"p", "-", "pInf"}], ")"}]}], " ", "gives", " ", "a", " ", "coefficient", " ", "of", " ", RowBox[{"z", "[", "t", "]"}], " ", "given", " ", "by"}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"zCoef", "=", RowBox[{"-", RowBox[{"Simplify", "[", RowBox[{ RowBox[{"2", " ", "a"}], " ", "-", " ", RowBox[{"2", " ", FractionBox[ SuperscriptBox["b", "2"], "r"], "pInf"}]}], "]"}]}]}]}]], "Input", CellChangeTimes->{{3.484044760906481*^9, 3.4840447721724663`*^9}, { 3.4840448028921995`*^9, 3.484044812908145*^9}, {3.484044956897127*^9, 3.484044960709749*^9}, {3.4840451058237677`*^9, 3.484045106855051*^9}, { 3.4840456128712425`*^9, 3.484045652403758*^9}}], Cell[BoxData[ FractionBox[ RowBox[{"2", " ", SqrtBox[ RowBox[{"r", " ", RowBox[{"(", RowBox[{ RowBox[{ SuperscriptBox["b", "2"], " ", "q"}], "+", RowBox[{ SuperscriptBox["a", "2"], " ", "r"}]}], ")"}]}]]}], "r"]], "Output", CellChangeTimes->{3.4840447724849763`*^9, 3.48404481400193*^9, 3.4840449618504105`*^9, 3.484045111386446*^9, 3.484047020595879*^9, 3.484048368126089*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{"solve", " ", "for", " ", RowBox[{"z", "[", "t", "]"}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"zSol", "=", RowBox[{ RowBox[{"DSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"z", "'"}], "[", "t", "]"}], "\[Equal]", RowBox[{ RowBox[{"2", " ", "beta", " ", RowBox[{"z", "[", "t", "]"}]}], "+", FractionBox[ SuperscriptBox["b", "2"], "r"]}]}], ",", RowBox[{ RowBox[{"z", "[", "0", "]"}], "\[Equal]", "z0"}]}], "}"}], ",", RowBox[{"z", "[", "t", "]"}], ",", "t"}], "]"}], "[", RowBox[{"[", RowBox[{"1", ",", "1", ",", "2"}], "]"}], "]"}]}]}]], "Input", CellChangeTimes->{{3.4840447847041173`*^9, 3.4840448692693233`*^9}, 3.4840449036766744`*^9, {3.484044968944388*^9, 3.4840449836167326`*^9}, { 3.4840451435906014`*^9, 3.48404518307624*^9}, {3.484045661372795*^9, 3.484045676623283*^9}, {3.484047057628077*^9, 3.4840470632688465`*^9}}], Cell[BoxData[ FractionBox[ RowBox[{ RowBox[{"-", SuperscriptBox["b", "2"]}], "+", RowBox[{ SuperscriptBox["b", "2"], " ", SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", "beta", " ", "t"}]]}], "+", RowBox[{"2", " ", "beta", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", "beta", " ", "t"}]], " ", "r", " ", "z0"}]}], RowBox[{"2", " ", "beta", " ", "r"}]]], "Output", CellChangeTimes->{{3.4840448344088326`*^9, 3.484044870784997*^9}, 3.4840449042704434`*^9, {3.4840449709600773`*^9, 3.484044983976119*^9}, 3.484045118542925*^9, {3.4840451590910974`*^9, 3.4840451836075068`*^9}, 3.484045678701474*^9, 3.4840470655814056`*^9, 3.4840483700011253`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{"from", " ", RowBox[{"z", "[", "t", "]"}], " ", "compute", " ", "what", " ", "the", " ", "solution", " ", "for", " ", RowBox[{"p", "[", "t", "]"}], " ", "is"}], " ", "*)"}], " ", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Clear", "[", "pInf", "]"}], ";"}], "\[IndentingNewLine]", RowBox[{"pSol", "=", RowBox[{"Simplify", "[", RowBox[{"pInf", "+", FractionBox["1", "zSol"]}], "]"}]}]}]}]], "Input", CellChangeTimes->{{3.4840451279651012`*^9, 3.484045140277995*^9}, { 3.4840451914983845`*^9, 3.4840452474533*^9}, {3.484045681717196*^9, 3.4840456982802258`*^9}, {3.484045796377115*^9, 3.4840458001272345`*^9}}], Cell[BoxData[ RowBox[{"pInf", "+", FractionBox[ RowBox[{"2", " ", "beta", " ", "r"}], RowBox[{ RowBox[{ SuperscriptBox["b", "2"], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", "beta", " ", "t"}]]}], ")"}]}], "+", RowBox[{"2", " ", "beta", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", "beta", " ", "t"}]], " ", "r", " ", "z0"}]}]]}]], "Output", CellChangeTimes->{{3.4840451939984646`*^9, 3.484045247797061*^9}, 3.4840458008303823`*^9, 3.4840470727378387`*^9, 3.4840483718136597`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{ "find", " ", "out", " ", "what", " ", "the", " ", "value", " ", "of", " ", "z0", " ", "is", " ", "in", " ", "terms", " ", "of", " ", "p0", " ", "the", " ", "initial", " ", "uncertainty", " ", "in", " ", "the", " ", "state"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"p0Eval", "=", RowBox[{"pSol", "/.", RowBox[{"{", RowBox[{"t", "\[Rule]", "0"}], "}"}]}]}], "\[IndentingNewLine]", RowBox[{"z0InTermsOfP0", "=", RowBox[{ RowBox[{"Solve", "[", RowBox[{ RowBox[{"p0Eval", "\[Equal]", "p0"}], ",", "z0"}], "]"}], "//", "Simplify"}]}]}]}]], "Input", CellChangeTimes->{{3.4840452491252284`*^9, 3.484045308767762*^9}, { 3.484045713452586*^9, 3.484045738562765*^9}, {3.4840470896132708`*^9, 3.484047093300865*^9}}], Cell[BoxData[ RowBox[{"pInf", "+", FractionBox["1", "z0"]}]], "Output", CellChangeTimes->{{3.4840452533128624`*^9, 3.484045309611539*^9}, { 3.484045723468532*^9, 3.4840457395002947`*^9}, 3.484045804533626*^9, { 3.484047077894221*^9, 3.4840470936915*^9}, 3.484048373751197*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"z0", "\[Rule]", FractionBox["1", RowBox[{"p0", "-", "pInf"}]]}], "}"}], "}"}]], "Output", CellChangeTimes->{{3.4840452533128624`*^9, 3.484045309611539*^9}, { 3.484045723468532*^9, 3.4840457395002947`*^9}, 3.484045804533626*^9, { 3.484047077894221*^9, 3.4840470936915*^9}, 3.4840483737668223`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{ "put", " ", "this", " ", "value", " ", "back", " ", "into", " ", "the", " ", "solution", " ", "for", " ", RowBox[{"p", "[", "t", "]"}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"pSol2", "=", RowBox[{ RowBox[{"pSol", "/.", "z0InTermsOfP0"}], "//", "Simplify"}]}]}]], "Input", CellChangeTimes->{{3.4840458296125536`*^9, 3.4840458707013807`*^9}, { 3.484047098222866*^9, 3.484047100863559*^9}, {3.4840482666241403`*^9, 3.484048267764787*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"pInf", "+", FractionBox[ RowBox[{"2", " ", "beta", " ", "r"}], RowBox[{ RowBox[{ SuperscriptBox["b", "2"], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", "beta", " ", "t"}]]}], ")"}]}], "+", FractionBox[ RowBox[{"2", " ", "beta", " ", SuperscriptBox["\[ExponentialE]", RowBox[{"2", " ", "beta", " ", "t"}]], " ", "r"}], RowBox[{"p0", "-", "pInf"}]]}]]}], "}"}]], "Output", CellChangeTimes->{{3.4840458633788843`*^9, 3.4840458710448666`*^9}, 3.4840471012698193`*^9, 3.484048268217921*^9, 3.484048377970028*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{"convert", " ", "to", " ", "hyperbolic", " ", "functions"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"pSol3", "=", RowBox[{ RowBox[{ RowBox[{"ExpToTrig", "[", RowBox[{"pSol2", "[", RowBox[{"[", "1", "]"}], "]"}], "]"}], "//", "Together"}], "//", "Simplify"}]}]}]], "Input", CellChangeTimes->{{3.4840475221868443`*^9, 3.484047550281314*^9}, { 3.484048273483647*^9, 3.4840483010779266`*^9}}], Cell[BoxData[ FractionBox[ RowBox[{ RowBox[{"beta", " ", "p0", " ", "r", " ", RowBox[{"Cosh", "[", RowBox[{"beta", " ", "t"}], "]"}]}], "+", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ SuperscriptBox["b", "2"], " ", RowBox[{"(", RowBox[{"p0", "-", "pInf"}], ")"}], " ", "pInf"}], "-", RowBox[{"beta", " ", RowBox[{"(", RowBox[{"p0", "-", RowBox[{"2", " ", "pInf"}]}], ")"}], " ", "r"}]}], ")"}], " ", RowBox[{"Sinh", "[", RowBox[{"beta", " ", "t"}], "]"}]}]}], RowBox[{ RowBox[{"beta", " ", "r", " ", RowBox[{"Cosh", "[", RowBox[{"beta", " ", "t"}], "]"}]}], "+", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ SuperscriptBox["b", "2"], " ", RowBox[{"(", RowBox[{"p0", "-", "pInf"}], ")"}]}], "+", RowBox[{"beta", " ", "r"}]}], ")"}], " ", RowBox[{"Sinh", "[", RowBox[{"beta", " ", "t"}], "]"}]}]}]]], "Output", CellChangeTimes->{{3.484047532171475*^9, 3.4840475506563234`*^9}, { 3.484048274858673*^9, 3.4840483020779457`*^9}, 3.4840483797044363`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{ "put", " ", "in", " ", "pInf", " ", "in", " ", "terms", " ", "of", " ", "beta", " ", "and", " ", "simlify"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"pSol3", "/.", RowBox[{"{", RowBox[{"pInf", "\[Rule]", RowBox[{ FractionBox["r", SuperscriptBox["b", "2"]], RowBox[{"(", RowBox[{"beta", " ", "+", " ", "a"}], ")"}]}]}], "}"}]}], "//", "Simplify"}]}]], "Input", CellChangeTimes->{{3.4840478124747953`*^9, 3.4840478391159325`*^9}, { 3.484048283374462*^9, 3.4840483180313773`*^9}}], Cell[BoxData[ FractionBox[ RowBox[{"r", " ", RowBox[{"(", RowBox[{ RowBox[{ SuperscriptBox["b", "2"], " ", "beta", " ", "p0", " ", RowBox[{"Cosh", "[", RowBox[{"beta", " ", "t"}], "]"}]}], "+", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"a", " ", SuperscriptBox["b", "2"], " ", "p0"}], "-", RowBox[{ SuperscriptBox["a", "2"], " ", "r"}], "+", RowBox[{ SuperscriptBox["beta", "2"], " ", "r"}]}], ")"}], " ", RowBox[{"Sinh", "[", RowBox[{"beta", " ", "t"}], "]"}]}]}], ")"}]}], RowBox[{ SuperscriptBox["b", "2"], " ", RowBox[{"(", RowBox[{ RowBox[{"beta", " ", "r", " ", RowBox[{"Cosh", "[", RowBox[{"beta", " ", "t"}], "]"}]}], "+", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ SuperscriptBox["b", "2"], " ", "p0"}], "-", RowBox[{"a", " ", "r"}]}], ")"}], " ", RowBox[{"Sinh", "[", RowBox[{"beta", " ", "t"}], "]"}]}]}], ")"}]}]]], "Output", CellChangeTimes->{{3.4840478349439774`*^9, 3.484047839490939*^9}, 3.484048286858904*^9, 3.484048318765766*^9, 3.484048381251341*^9}] }, Open ]], Cell[BoxData[ RowBox[{"(*", " ", RowBox[{ "this", " ", "last", " ", "expression", " ", "can", " ", "be", " ", "manipulated", " ", "into", " ", "the", " ", "form", " ", "claimed", " ", "in", " ", "the", " ", "book"}], " ", "*)"}]], "Input", CellChangeTimes->{{3.484048323890865*^9, 3.4840483395474153`*^9}}] }, WindowSize->{894, 750}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, ShowSelection->True, FrontEndVersion->"7.0 for Microsoft Windows (32-bit) (February 18, 2009)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[567, 22, 770, 20, 70, "Input"], Cell[1340, 44, 960, 30, 56, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[2337, 79, 539, 13, 52, "Input"], Cell[2879, 94, 470, 14, 56, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[3386, 113, 919, 23, 70, "Input"], Cell[4308, 138, 439, 13, 56, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[4784, 156, 1084, 27, 70, "Input"], Cell[5871, 185, 722, 16, 51, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[6630, 206, 727, 17, 87, "Input"], Cell[7360, 225, 638, 17, 52, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[8035, 247, 859, 21, 72, "Input"], Cell[8897, 270, 287, 5, 47, "Output"], Cell[9187, 277, 371, 8, 49, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[9595, 290, 531, 12, 52, "Input"], Cell[10129, 304, 713, 19, 62, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[10879, 328, 493, 13, 52, "Input"], Cell[11375, 343, 1129, 34, 56, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[12541, 382, 621, 17, 65, "Input"], Cell[13165, 401, 1192, 36, 56, "Output"] }, Open ]], Cell[14372, 440, 321, 6, 31, "Input"] } ] *) (* End of internal cache information *)