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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 717232, 19672]*) (*NotebookOutlinePosition[ 731999, 20210]*) (* CellTagsIndexPosition[ 731955, 20206]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[StyleBox["Eigenvalue problems", FontColor->RGBColor[0, 1, 1]]], "Title"], Cell[TextData[{ "This notebook has been written in ", StyleBox["Mathematica ", FontSlant->"Italic"], "by \n\n", StyleBox[ "Mark J. McCready\nProfessor and Chair of Chemical Engineering\nUniversity \ of Notre Dame\nNotre Dame, IN 46556\nUSA", FontSize->14], "\n\n", ButtonBox["mjm@nd.edu", ButtonStyle->None], "\n", ButtonBox["http://www.nd.edu/~mjm/", ButtonData:>{ URL[ "http://www.nd.edu/~mjm/"], None}, ButtonStyle->"Hyperlink"], "\n\n\nIt is copyrighted to the extent allowed by which ever laws pertain \ to the World Wide Web and the Internet.\n\nI would hope that as a \ professional courtesy, this notice remain visible to other users. \nThere is \ no charge for copying and dissemination \n\nVersion: 11/23/98" }], "Text"], Cell[CellGroupData[{ Cell["No resistance at the ends, u[0]=u[1]=0", "Subsection"], Cell[CellGroupData[{ Cell["\<\ We might start solving this problem with a homogenous boundary \ condition for u[0] and u[1]. However, we cannot solve all of this at once \ because all we will get is the trivial solution. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"DSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["u", "\[DoublePrime]", MultilineFunction->None], "[", "t", "]"}], "+", \(\[Lambda]\ u[t]\)}], "==", "0"}], ",", \(u[0] == 0\), ",", " ", \(u[1]\ == 0\)}], "}"}], ",", "u", ",", "t"}], "]"}]], "Input", CellLabel->"In[123]:="], Cell[BoxData[ \(TraditionalForm\`{{u \[Rule] \((0&)\)}}\)], "Output", CellLabel->"Out[123]="] }, Open ]], Cell["\<\ Note that this is a more general solution (although still 0) than \ you get from the similar command:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"DSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["u", "\[DoublePrime]", MultilineFunction->None], "[", "t", "]"}], "+", \(\[Lambda]\ u[t]\)}], "==", "0"}], ",", \(u[0] == 0\), ",", " ", \(u[1]\ == 0\)}], "}"}], ",", \(u[t]\), ",", "t"}], "]"}]], "Input", CellLabel->"In[124]:="], Cell[BoxData[ \(TraditionalForm\`{{u(t) \[Rule] 0}}\)], "Output", CellLabel->"Out[124]="] }, Open ]], Cell[TextData[{ "To get a solution we opt to leave off the far boundary condition. ", StyleBox["Mathematica", FontSlant->"Italic"], " is telling us that we need to adjust \[Lambda] to special values to solve \ the problem. We first solve the ode and the u[0]=0 bondary condition. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"ans1", "=", RowBox[{"DSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["u", "\[DoublePrime]", MultilineFunction->None], "[", "t", "]"}], "+", \(\[Lambda]\ u[t]\)}], "==", "0"}], ",", \(u[0] == 0\)}], "}"}], ",", "u", ",", "t"}], "]"}]}]], "Input", CellLabel->"In[125]:="], Cell[BoxData[ FormBox[ RowBox[{ \(General::"spell1"\), \( : \ \), "\<\"Possible spelling error: new symbol name \ \\\"\\!\\(TraditionalForm\\`ans1\\)\\\" is similar to existing symbol \\\"\\!\ \\(TraditionalForm\\`ans21\\)\\\".\"\>"}], TraditionalForm]], "Message"], Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{"{", RowBox[{"u", "\[Rule]", RowBox[{"(", RowBox[{ RowBox[{ SubscriptBox[ TagBox["c", C], "1"], " ", \(sin(\@\[Lambda]\ #1)\)}], "&"}], ")"}]}], "}"}], "}"}], TraditionalForm]], "Output", CellLabel->"Out[125]="] }, Open ]], Cell["Construct the boudary condition at the far end", "Text"], Cell[CellGroupData[{ Cell[" ans2= u[1] /.ans1[[1]]", "Input", CellLabel->"In[126]:="], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox[ TagBox["c", C], "1"], " ", \(sin(\@\[Lambda])\)}], TraditionalForm]], "Output",\ CellLabel->"Out[126]="] }, Open ]], Cell[TextData[{ "We need to choose \[Lambda] to make this 0. We know the answer is \ \[Lambda] = ", Cell[BoxData[ \(TraditionalForm\`\((n\ \[Pi])\)\^2\)]], ". (Which is good because ", StyleBox["Mathematica", FontSlant->"Italic"], " does not give a useful answer. " }], "Text"], Cell[CellGroupData[{ Cell["eq1=ans2==0", "Input", CellLabel->"In[127]:="], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox[ TagBox["c", C], "1"], " ", \(sin(\@\[Lambda])\)}], "==", "0"}], TraditionalForm]], "Output", CellLabel->"Out[127]="] }, Open ]], Cell["We could find roots numerically if we had to, ", "Text"], Cell[CellGroupData[{ Cell["root1=ans2/.{C[1]->1}", "Input", CellLabel->"In[128]:="], Cell[BoxData[ \(TraditionalForm\`sin(\@\[Lambda])\)], "Output", CellLabel->"Out[128]="] }, Open ]], Cell[CellGroupData[{ Cell[TextData["FindRoot[root1==0,{\[Lambda],9}]"], "Input", CellLabel->"In[129]:="], Cell[BoxData[ \(TraditionalForm\`{\[Lambda] \[Rule] 9.86959922412076728`}\)], "Output", CellLabel->"Out[129]="] }, Open ]], Cell[CellGroupData[{ Cell[TextData["FindRoot[root1==0,{\[Lambda],36}]"], "Input", CellLabel->"In[130]:="], Cell[BoxData[ \(TraditionalForm\`{\[Lambda] \[Rule] 39.4784164181274288`}\)], "Output", CellLabel->"Out[130]="] }, Open ]], Cell[CellGroupData[{ Cell[TextData["FindRoot[root1==0,{\[Lambda],84}]"], "Input", CellLabel->"In[131]:="], Cell[BoxData[ \(TraditionalForm\`{\[Lambda] \[Rule] 88.8264344585236287`}\)], "Output", CellLabel->"Out[131]="] }, Open ]], Cell["Just to beat the dead horse ... 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"], "Text"], Cell["Check orthogonality", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(int1 = Integrate[Sin[n\ Pi\ x]\ \ Sin[m\ \ Pi\ x], {x, 0, 1}]\)], "Input", CellLabel->"In[133]:="], Cell[BoxData[ \(TraditionalForm \`\(m\ \(sin(m\ \[Pi] - n\ \[Pi])\) + n\ \(sin(m\ \[Pi] - n\ \[Pi])\) - m\ \(sin(\[Pi]\ m + n\ \[Pi])\) + n\ \(sin(\[Pi]\ m + n\ \[Pi])\)\)\/\(2\ \((m - n)\)\ \((m + n)\)\ \[Pi]\)\)], "Output", CellLabel->"Out[133]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(int2 = FullSimplify[%]\)], "Input", CellLabel->"In[134]:="], Cell[BoxData[ \(TraditionalForm \`\(\(sin(\((m - n)\)\ \[Pi])\)\/\(m - n\) - \(sin(\((m + n)\)\ \[Pi])\)\/\(m + n\)\)\/\(2\ \[Pi]\)\)], "Output", CellLabel->"Out[134]="] }, Open ]], Cell[TextData["This is obviously 0 if n\[NotEqual]m. For example, "], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(int2 /. {n -> 3, \ m -> 8}\)], "Input", CellLabel->"In[135]:="], Cell[BoxData[ \(TraditionalForm\`0\)], "Output", CellLabel->"Out[135]="] }, Open ]], Cell["Otherwise, ", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(int3 = Integrate[Sin[n\ Pi\ x]\ \ Sin[n\ \ Pi\ x], {x, 0, 1}]\)], "Input", CellLabel->"In[136]:="], Cell[BoxData[ \(TraditionalForm\`1\/2 - \(sin(2\ n\ \[Pi])\)\/\(4\ n\ \[Pi]\)\)], "Output", CellLabel->"Out[136]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Table[int3, {n, 1, 10, 1}]\)], "Input", CellLabel->"In[137]:="], Cell[BoxData[ \(TraditionalForm \`{1\/2, 1\/2, 1\/2, 1\/2, 1\/2, 1\/2, 1\/2, 1\/2, 1\/2, 1\/2}\)], "Output", CellLabel->"Out[137]="] }, Open ]], Cell["\<\ Thus our eigenfunctions are orthogonal and they can be easily \ normalized. \ \>", "Text"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Now use more general boundary conditions", "Subsection"], Cell[CellGroupData[{ Cell[TextData[{ "Now we try solving this problem with a homogenous boundary condition for \ u'[0] + ", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_1\)]], " u[0] = 0, at the front end and u'[L] + ", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_2\)]], "u[L] =0 at the far end. However, we cannot solve all of this at once \ because all we will get is the trivial solution. " }], "Text"], Cell[TextData[{ "Thus we opt to leave off the far boundary condition. ", StyleBox["Mathematica", FontSlant->"Italic"], " is telling us that we need to adjust \[Lambda] to special values to solve \ the problem. We first solve the ode and the u[0] boundary condition. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"ans21", "=", RowBox[{"DSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["u", "\[DoublePrime]", MultilineFunction->None], "[", "t", "]"}], "+", \(\[Lambda]\ u[t]\)}], "==", "0"}], ",", \(\(-\(u'\)[0]\) + \ \(\(\[Alpha]\ \)\_1\) u[0] == 0\)}], "}"}], ",", "u", ",", "t"}], "]"}]}]], "Input", CellLabel->"In[138]:="], Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{"{", RowBox[{"u", "\[Rule]", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{ SubscriptBox[ TagBox["c", C], "2"], " ", \(cos(\@\[Lambda]\ #1)\)}], "+", FractionBox[ RowBox[{ SubscriptBox[ TagBox["c", C], "2"], " ", \(sin(\@\[Lambda]\ #1)\), " ", \(\[Alpha]\_1\)}], \(\@\[Lambda]\)]}], "&"}], ")"}]}], "}"}], "}"}], TraditionalForm]], "Output", CellLabel->"Out[138]="] }, Open ]], Cell["Construct the boundary condition at the far end", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"ans22", "=", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["u", "\[Prime]", MultilineFunction->None], "[", "L", "]"}], "+", \(\[Alpha]\_2\ u[L]\)}], "/.", \(ans21\[LeftDoubleBracket]1\[RightDoubleBracket]\)}]}]], "Input", CellLabel->"In[139]:="], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(-\@\[Lambda]\), " ", SubscriptBox[ TagBox["c", C], "2"], " ", \(sin(L\ \@\[Lambda])\)}], "+", RowBox[{ SubscriptBox[ TagBox["c", C], "2"], " ", \(cos(L\ \@\[Lambda])\), " ", \(\[Alpha]\_1\)}], "+", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ SubscriptBox[ TagBox["c", C], "2"], " ", \(cos(L\ \@\[Lambda])\)}], "+", FractionBox[ RowBox[{ SubscriptBox[ TagBox["c", C], "2"], " ", \(sin(L\ \@\[Lambda])\), " ", \(\[Alpha]\_1\)}], \(\@\[Lambda]\)]}], ")"}], " ", \(\[Alpha]\_2\)}]}], TraditionalForm]], "Output", CellLabel->"Out[139]="] }, Open ]], Cell[TextData["We need to choose \[Lambda] to make this 0. "], "Text"], Cell[CellGroupData[{ Cell[TextData[ "bc1=Expand[Expand[ans22]/Sin[L Sqrt[\[Lambda]]]/C[2]]"], "Input", CellLabel->"In[140]:="], Cell[BoxData[ \(TraditionalForm \`\(cot(L\ \@\[Lambda])\)\ \[Alpha]\_1 + \(\[Alpha]\_2\ \[Alpha]\_1\)\/\@\[Lambda] + \(cot(L\ \@\[Lambda])\)\ \[Alpha]\_2 - \@\[Lambda]\)], "Output", CellLabel->"Out[140]="] }, Open ]], Cell["\<\ This takes a bit of work to make this into the nice form in V&M. \ \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(bc1x = Cancel[Simplify[bc1[\([2]\)] + bc1[\([3]\)]]/ \((\[Alpha]\_1 + \[Alpha]\_2)\)] + Cancel[Simplify[bc1[\([1]\)] + bc1[\([4]\)]]/ \((\[Alpha]\_1 + \[Alpha]\_2)\)]\)], "Input", CellLabel->"In[141]:="], Cell[BoxData[ \(TraditionalForm \`cot(L\ \@\[Lambda]) + \(\[Alpha]\_1\ \[Alpha]\_2 - \[Lambda]\)\/\(\@\[Lambda]\ \((\[Alpha]\_1 + \[Alpha]\_2)\)\)\)], "Output", CellLabel->"Out[141]="] }, Open ]], Cell[TextData["Make the substitution, \[Lambda]-> s L"], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(fofs = 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Editable->False], TraditionalForm]], "Output", CellLabel->"Out[157]="] }, Open ]] }, Closed]], Cell["\<\ You see that the \"fit\" is of a different form for the more \ complex boundary conditions\ \>", "Text"], Cell["Now verify orthogonality.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Integrate[ \((\((func /. {\@\[Lambda] -> s, 1/\@\[Lambda] -> 1/s})\) /. root4)\)* \ \ \((\((func /. {\@\[Lambda] -> s, 1/\@\[Lambda] -> 1/s})\) /. root1)\)\n, {x, 0, 1}]\)], "Input", CellLabel->"In[158]:="], Cell[BoxData[ \(TraditionalForm \`{\(-5.93384207789225293`*^-10\) + 0.`\ \[ImaginaryI]}\)], "Output", CellLabel->"Out[158]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Integrate[ \((\((func /. {\@\[Lambda] -> s, 1/\@\[Lambda] -> 1/s})\) /. root1)\)* \ \ \((\((func /. {\@\[Lambda] -> s, 1/\@\[Lambda] -> 1/s})\) /. root1)\)\n, {x, 0, 1}]\)], "Input", CellLabel->"In[159]:="], Cell[BoxData[ \(TraditionalForm \`{\(1.37870738206160359`\[InvisibleSpace]\) + 0.`\ \[ImaginaryI]}\)], "Output", CellLabel->"Out[159]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Integrate[ \((\((func /. {\@\[Lambda] -> s, 1/\@\[Lambda] -> 1/s})\) /. root4)\)* \ \ \((\((func /. {\@\[Lambda] -> s, 1/\@\[Lambda] -> 1/s})\) /. root3)\)\n, {x, 0, 1}]\)], "Input", CellLabel->"In[160]:="], Cell[BoxData[ \(TraditionalForm\`{9.50542011238596273`*^-10 + 0.`\ \[ImaginaryI]}\)], "Output", CellLabel->"Out[160]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Integrate[ \((\((func /. {\@\[Lambda] -> s, 1/\@\[Lambda] -> 1/s})\) /. root3)\)* \ \ \((\((func /. {\@\[Lambda] -> s, 1/\@\[Lambda] -> 1/s})\) /. root3)\)\n, {x, 0, 1}]\)], "Input", CellLabel->"In[161]:="], Cell[BoxData[ \(TraditionalForm \`{\(0.534596313180830318`\[InvisibleSpace]\) + 0.`\ \[ImaginaryI]}\)], "Output", CellLabel->"Out[161]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Integrate[ \((\((func /. {\@\[Lambda] -> s, 1/\@\[Lambda] -> 1/s})\) /. root4)\)* \ \ \((\((func /. {\@\[Lambda] -> s, 1/\@\[Lambda] -> 1/s})\) /. root4)\)\n, {x, 0, 1}]\)], "Input", CellLabel->"In[162]:="], Cell[BoxData[ \(TraditionalForm \`{\(0.516169133403697344`\[InvisibleSpace]\) + 0.`\ \[ImaginaryI]}\)], "Output", CellLabel->"Out[162]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Integrate[\ \((func[\([1]\)] /. {\[Lambda] -> \[Lambda]\_1})\)*\ \ \((func[\([1]\)] /. {\[Lambda] -> \[Lambda]\_2})\), {x, 0, 1}]\)], "Input", CellLabel->"In[163]:="], Cell[BoxData[ \(TraditionalForm \`Integrate::"gener" \( : \ \) "Unable to check convergence"\)], "Message"], Cell[BoxData[ \(TraditionalForm \`\((\((cos(\@\[Lambda]\_1) + \(sin(\@\[Lambda]\_1)\)\/\@\[Lambda]\_1)\)\ \((cos(\@\[Lambda]\_2) + \(sin(\@\[Lambda]\_2)\)\/\@\[Lambda]\_2) \)\ \((\@\[Lambda]\_1\ \[Lambda]\_2\ \(sin(\@\[Lambda]\_1 - \@\[Lambda]\_2)\) + \@\[Lambda]\_1\ \(sin(\@\[Lambda]\_1 - \@\[Lambda]\_2)\) + \[Lambda]\_1\ \@\[Lambda]\_2\ \(sin(\@\[Lambda]\_1 - \@\[Lambda]\_2)\) + \@\[Lambda]\_2\ \(sin(\@\[Lambda]\_1 - \@\[Lambda]\_2)\) + \(cos(\@\[Lambda]\_1 - \@\[Lambda]\_2)\)\ \[Lambda]\_1 - \(cos(\@\[Lambda]\_1 + \@\[Lambda]\_2)\)\ \[Lambda]\_1 - \(cos(\@\[Lambda]\_1 - \@\[Lambda]\_2)\)\ \[Lambda]\_2 + \(cos(\@\[Lambda]\_1 + \@\[Lambda]\_2)\)\ \[Lambda]\_2 - \(sin(\@\[Lambda]\_1 + \@\[Lambda]\_2)\)\ \@\[Lambda]\_1\ \[Lambda]\_2 - \(sin(\@\[Lambda]\_1 + \@\[Lambda]\_2)\)\ \@\[Lambda]\_1 + \(sin(\@\[Lambda]\_1 + \@\[Lambda]\_2)\)\ \@\[Lambda]\_2 + \(sin(\@\[Lambda]\_1 + \@\[Lambda]\_2)\)\ \[Lambda]\_1\ \@\[Lambda]\_2)\))\)/ \((2\ \((\@\[Lambda]\_1\ \(cos(\@\[Lambda]\_1)\) + sin(\@\[Lambda]\_1)) \)\ \((\@\[Lambda]\_1 - \@\[Lambda]\_2)\)\ \((\@\[Lambda]\_1 + \@\[Lambda]\_2)\)\ \((\@\[Lambda]\_2\ \(cos(\@\[Lambda]\_2)\) + sin(\@\[Lambda]\_2))\)) \)\)], "Output", CellLabel->"Out[163]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(FullSimplify[%]\)], "Input", CellLabel->"In[164]:="], Cell[BoxData[ \(TraditionalForm \`\(\(\(sin(\@\[Lambda]\_1)\)\ \@\[Lambda]\_1\ \((\@\[Lambda]\_2\ \(cos(\@\[Lambda]\_2)\) + sin(\@\[Lambda]\_2)) 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