(* 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[ 16028, 538] NotebookOptionsPosition[ 14778, 490] NotebookOutlinePosition[ 15145, 506] CellTagsIndexPosition[ 15102, 503] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell["\<\ Note that c2 really stands for c2/c1 in the problem definition\ \>", "Text", CellChangeTimes->{{3.5310657587352457`*^9, 3.531065768688498*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"Clear", "[", "x", "]"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"DSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"x", "'"}], "[", "t", "]"}], "\[Equal]", RowBox[{ RowBox[{ FractionBox[ RowBox[{"-", "1"}], "T"], RowBox[{"(", RowBox[{ RowBox[{"x", "[", "t", "]"}], "-", "xa"}], ")"}]}], "-", RowBox[{ FractionBox[ SuperscriptBox["k1", "2"], "c2"], "EPD", " ", SuperscriptBox["E", FractionBox[ RowBox[{"(", RowBox[{"t", "-", "tf"}], ")"}], "T"]]}]}]}], ",", RowBox[{ RowBox[{"x", "[", "t0", "]"}], "==", "x0"}]}], "}"}], ",", RowBox[{"x", "[", "t", "]"}], ",", "t"}], "]"}], "//", "Simplify"}]}], "Input", CellChangeTimes->{{3.5309691914900894`*^9, 3.5309692376469297`*^9}, { 3.530969438274498*^9, 3.530969446196474*^9}, {3.5309694776031265`*^9, 3.5309694888063946`*^9}, 3.531065703468913*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{"x", "[", "t", "]"}], "\[Rule]", FractionBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ RowBox[{"t", "+", "tf"}], "T"]}]], " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", SuperscriptBox["\[ExponentialE]", FractionBox[ RowBox[{"2", " ", "t"}], "T"]]}], " ", "EPD", " ", SuperscriptBox["k1", "2"], " ", "T"}], "+", RowBox[{ SuperscriptBox["\[ExponentialE]", FractionBox[ RowBox[{"2", " ", "t0"}], "T"]], " ", "EPD", " ", SuperscriptBox["k1", "2"], " ", "T"}], "+", RowBox[{"2", " ", "c2", " ", SuperscriptBox["\[ExponentialE]", FractionBox[ RowBox[{"t0", "+", "tf"}], "T"]], " ", RowBox[{"(", RowBox[{"x0", "-", "xa"}], ")"}]}], "+", RowBox[{"2", " ", "c2", " ", SuperscriptBox["\[ExponentialE]", FractionBox[ RowBox[{"t", "+", "tf"}], "T"]], " ", "xa"}]}], ")"}]}], RowBox[{"2", " ", "c2"}]]}], "}"}], "}"}]], "Output", CellChangeTimes->{ 3.530969325304302*^9, 3.5309694469152336`*^9, {3.530969481696929*^9, 3.5309694893220263`*^9}, 3.5310658119546766`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Expand", "[", FractionBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ RowBox[{"t", "+", "tf"}], "T"]}]], " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", SuperscriptBox["\[ExponentialE]", FractionBox[ RowBox[{"2", " ", "t"}], "T"]]}], " ", "EPD", " ", SuperscriptBox["k1", "2"], " ", "T"}], "+", RowBox[{ SuperscriptBox["\[ExponentialE]", FractionBox[ RowBox[{"2", " ", "t0"}], "T"]], " ", "EPD", " ", SuperscriptBox["k1", "2"], " ", "T"}], "+", RowBox[{"2", " ", "c2", " ", SuperscriptBox["\[ExponentialE]", FractionBox[ RowBox[{"t0", "+", "tf"}], "T"]], " ", RowBox[{"(", RowBox[{"x0", "-", "xa"}], ")"}]}], "+", RowBox[{"2", " ", "c2", " ", SuperscriptBox["\[ExponentialE]", FractionBox[ RowBox[{"t", "+", "tf"}], "T"]], " ", "xa"}]}], ")"}]}], RowBox[{"2", " ", "c2"}]], "]"}]], "Input", CellChangeTimes->{{3.5309695215099382`*^9, 3.530969563729229*^9}, { 3.531065703484538*^9, 3.531065703515789*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"-", FractionBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{ FractionBox[ RowBox[{"2", " ", "t"}], "T"], "-", FractionBox[ RowBox[{"t", "+", "tf"}], "T"]}]], " ", "EPD", " ", SuperscriptBox["k1", "2"], " ", "T"}], RowBox[{"2", " ", "c2"}]]}], "+", FractionBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{ FractionBox[ RowBox[{"2", " ", "t0"}], "T"], "-", FractionBox[ RowBox[{"t", "+", "tf"}], "T"]}]], " ", "EPD", " ", SuperscriptBox["k1", "2"], " ", "T"}], RowBox[{"2", " ", "c2"}]], "+", RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"-", FractionBox[ RowBox[{"t", "+", "tf"}], "T"]}], "+", FractionBox[ RowBox[{"t0", "+", "tf"}], "T"]}]], " ", "x0"}], "+", "xa", "-", RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"-", FractionBox[ RowBox[{"t", "+", "tf"}], "T"]}], "+", FractionBox[ RowBox[{"t0", "+", "tf"}], "T"]}]], " ", "xa"}]}]], "Output", CellChangeTimes->{{3.5309695239474697`*^9, 3.530969564041733*^9}, 3.5310658120796785`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"-", FractionBox[ RowBox[{"t", "+", "tf"}], "T"]}], "+", FractionBox[ RowBox[{"t0", "+", "tf"}], "T"]}]], " ", "x0"}], "-", RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"-", FractionBox[ RowBox[{"t", "+", "tf"}], "T"]}], "+", FractionBox[ RowBox[{"t0", "+", "tf"}], "T"]}]], " ", "xa"}]}], "//", "Simplify"}]], "Input", CellChangeTimes->{{3.5309696181674256`*^9, 3.530969621917474*^9}}], Cell[BoxData[ RowBox[{ SuperscriptBox["\[ExponentialE]", FractionBox[ RowBox[{ RowBox[{"-", "t"}], "+", "t0"}], "T"]], " ", RowBox[{"(", RowBox[{"x0", "-", "xa"}], ")"}]}]], "Output", CellChangeTimes->{3.5309696222456026`*^9, 3.5310658121578045`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"-", FractionBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{ FractionBox[ RowBox[{"2", " ", "t"}], "T"], "-", FractionBox[ RowBox[{"t", "+", "tf"}], "T"]}]], " ", "EPD", " ", SuperscriptBox["k1", "2"], " ", "T"}], RowBox[{"2", " ", "c2"}]]}], "+", FractionBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{ FractionBox[ RowBox[{"2", " ", "t0"}], "T"], "-", FractionBox[ RowBox[{"t", "+", "tf"}], "T"]}]], " ", "EPD", " ", SuperscriptBox["k1", "2"], " ", "T"}], RowBox[{"2", " ", "c2"}]]}], "//", "Simplify"}]], "Input", CellChangeTimes->{{3.5309696720899906`*^9, 3.5309696735431347`*^9}, 3.530970335332855*^9, {3.531065703547039*^9, 3.531065703547039*^9}}], Cell[BoxData[ RowBox[{"-", FractionBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ RowBox[{"t", "+", "tf"}], "T"]}]], " ", RowBox[{"(", RowBox[{ SuperscriptBox["\[ExponentialE]", FractionBox[ RowBox[{"2", " ", "t"}], "T"]], "-", SuperscriptBox["\[ExponentialE]", FractionBox[ RowBox[{"2", " ", "t0"}], "T"]]}], ")"}], " ", "EPD", " ", SuperscriptBox["k1", "2"], " ", "T"}], RowBox[{"2", " ", "c2"}]]}]], "Output", CellChangeTimes->{3.530969673824388*^9, 3.5309703362078667`*^9, 3.531065812282806*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ RowBox[{"t", "+", "tf"}], "T"]}]], " ", RowBox[{"(", RowBox[{ SuperscriptBox["\[ExponentialE]", FractionBox[ RowBox[{"2", " ", "t"}], "T"]], "+", SuperscriptBox["\[ExponentialE]", FractionBox[ RowBox[{"2", " ", "t0"}], "T"]]}], ")"}]}], "//", "Simplify"}]], "Input", CellChangeTimes->{{3.5309697222312574`*^9, 3.5309697458565598`*^9}}], Cell[BoxData[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ RowBox[{"t", "+", "tf"}], "T"]}]], " ", RowBox[{"(", RowBox[{ SuperscriptBox["\[ExponentialE]", FractionBox[ RowBox[{"2", " ", "t"}], "T"]], "+", SuperscriptBox["\[ExponentialE]", FractionBox[ RowBox[{"2", " ", "t0"}], "T"]]}], ")"}]}]], "Output", CellChangeTimes->{{3.530969727450074*^9, 3.5309697461534386`*^9}, 3.5309703544737253`*^9, 3.5310658123609324`*^9}] }, Open ]], Cell["Lets check this solution : ", "Text", CellChangeTimes->{{3.530970073813883*^9, 3.530970078517068*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"x", "[", "t_", "]"}], ":=", RowBox[{"xa", "+", RowBox[{ RowBox[{"(", RowBox[{"x0", "-", "xa"}], ")"}], " ", SuperscriptBox["E", FractionBox[ RowBox[{"-", RowBox[{"(", RowBox[{"t", "+", "t0"}], ")"}]}], "T"]]}], "-", RowBox[{"EPD", " ", SuperscriptBox["k1", "2"], FractionBox["T", RowBox[{"2", " ", "c2"}]], RowBox[{"(", " ", RowBox[{ SuperscriptBox["E", FractionBox[ RowBox[{"(", RowBox[{"t", "-", "tf"}], ")"}], "T"]], "+", SuperscriptBox["E", FractionBox[ RowBox[{"(", RowBox[{ RowBox[{"2", " ", "t0"}], "-", "t", "-", "tf"}], ")"}], "T"]]}], ")"}]}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"D", "[", RowBox[{ RowBox[{"x", "[", "t", "]"}], ",", "t"}], "]"}], "+", RowBox[{ FractionBox["1", "T"], RowBox[{"(", RowBox[{ RowBox[{"x", "[", "t", "]"}], "-", "xa"}], ")"}]}], "+", RowBox[{ FractionBox[ SuperscriptBox["k1", "2"], "c2"], "EPD", " ", SuperscriptBox["E", FractionBox[ RowBox[{"(", RowBox[{"t", "-", "tf"}], ")"}], "T"]]}]}], "//", "Simplify"}]}], "Input", CellChangeTimes->{{3.530970091876614*^9, 3.530970227737728*^9}, { 3.530970393458599*^9, 3.530970396568014*^9}, {3.530970450068699*^9, 3.5309704518968472`*^9}, {3.5310657035782895`*^9, 3.531065703593915*^9}}], Cell[BoxData["0"], "Output", CellChangeTimes->{ 3.5309701544555397`*^9, {3.5309701984092274`*^9, 3.5309702283158607`*^9}, 3.5309703971461463`*^9, 3.5309704536468697`*^9, 3.5310658124546833`*^9}] }, Open ]], Cell["Evaluate the above at tf and we get ", "Text", CellChangeTimes->{{3.5309704312715836`*^9, 3.530970438959182*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"x", "[", "tf", "]"}], "//", "Simplify"}]], "Input", CellChangeTimes->{{3.530970446240525*^9, 3.5309704629282384`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"-", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"1", "+", SuperscriptBox["\[ExponentialE]", FractionBox[ RowBox[{"2", " ", RowBox[{"(", RowBox[{"t0", "-", "tf"}], ")"}]}], "T"]]}], ")"}], " ", "EPD", " ", SuperscriptBox["k1", "2"], " ", "T"}], RowBox[{"2", " ", "c2"}]]}], "+", RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ RowBox[{"t0", "+", "tf"}], "T"]}]], " ", RowBox[{"(", RowBox[{"x0", "-", "xa"}], ")"}]}], "+", "xa"}]], "Output", CellChangeTimes->{{3.5309704584438057`*^9, 3.5309704633188686`*^9}, 3.5310658126421857`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"1", "+", SuperscriptBox["\[ExponentialE]", FractionBox[ RowBox[{"2", " ", RowBox[{"(", RowBox[{"t0", "-", "tf"}], ")"}]}], "T"]]}], ")"}], " ", RowBox[{"(", RowBox[{"x1f", "-", "xd"}], ")"}], " ", SuperscriptBox["k1", "2"], " ", "T"}], RowBox[{"2", " ", "c2"}]]}], "+", RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ RowBox[{"t0", "+", "tf"}], "T"]}]], " ", RowBox[{"(", RowBox[{"x0", "-", "xa"}], ")"}]}], "+", "xa"}], "\[Equal]", "x1f"}], ",", "x1f"}], "]"}], "//", "Simplify"}]], "Input", CellChangeTimes->{{3.530970690884281*^9, 3.530970723290946*^9}, 3.531065703625165*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"x1f", "\[Rule]", FractionBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox["t0", "T"]}]], " ", RowBox[{"(", RowBox[{ RowBox[{"2", " ", "c2", " ", SuperscriptBox["\[ExponentialE]", FractionBox["tf", "T"]], " ", RowBox[{"(", RowBox[{"x0", "+", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SuperscriptBox["\[ExponentialE]", FractionBox[ RowBox[{"t0", "+", "tf"}], "T"]]}], ")"}], " ", "xa"}]}], ")"}]}], "+", RowBox[{ SuperscriptBox["\[ExponentialE]", FractionBox["t0", "T"]], " ", RowBox[{"(", RowBox[{ SuperscriptBox["\[ExponentialE]", FractionBox[ RowBox[{"2", " ", "t0"}], "T"]], "+", SuperscriptBox["\[ExponentialE]", FractionBox[ RowBox[{"2", " ", "tf"}], "T"]]}], ")"}], " ", SuperscriptBox["k1", "2"], " ", "T", " ", "xd"}]}], ")"}]}], RowBox[{ RowBox[{"2", " ", "c2", " ", SuperscriptBox["\[ExponentialE]", FractionBox[ RowBox[{"2", " ", "tf"}], "T"]]}], "+", RowBox[{ RowBox[{"(", RowBox[{ SuperscriptBox["\[ExponentialE]", FractionBox[ RowBox[{"2", " ", "t0"}], "T"]], "+", SuperscriptBox["\[ExponentialE]", FractionBox[ RowBox[{"2", " ", "tf"}], "T"]]}], ")"}], " ", SuperscriptBox["k1", "2"], " ", "T"}]}]]}], "}"}], "}"}]], "Output", CellChangeTimes->{{3.530970718540885*^9, 3.5309707237909527`*^9}, 3.531065812845313*^9}] }, Open ]] }, WindowSize->{790, 935}, WindowMargins->{{43, Automatic}, {Automatic, 137}}, 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[545, 20, 154, 3, 29, "Text"], Cell[CellGroupData[{ Cell[724, 27, 1066, 31, 70, "Input"], Cell[1793, 60, 1338, 37, 66, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[3168, 102, 1176, 33, 66, "Input"], Cell[4347, 137, 1222, 40, 59, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[5606, 182, 599, 20, 39, "Input"], Cell[6208, 204, 274, 8, 38, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[6519, 217, 856, 25, 59, "Input"], Cell[7378, 244, 626, 19, 66, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[8041, 268, 502, 16, 47, "Input"], Cell[8546, 286, 507, 15, 47, "Output"] }, Open ]], Cell[9068, 304, 109, 1, 29, "Text"], Cell[CellGroupData[{ Cell[9202, 309, 1519, 49, 88, "Input"], Cell[10724, 360, 201, 3, 30, "Output"] }, Open ]], Cell[10940, 366, 120, 1, 29, "Text"], Cell[CellGroupData[{ Cell[11085, 371, 154, 3, 31, "Input"], Cell[11242, 376, 710, 23, 66, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[11989, 404, 966, 29, 66, "Input"], Cell[12958, 435, 1804, 52, 84, "Output"] }, Open ]] } ] *) (* End of internal cache information *)