(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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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[ 1021826, 27847]*) (*NotebookOutlinePosition[ 1037054, 28403]*) (* CellTagsIndexPosition[ 1036881, 28394]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ StyleBox["Green's", FontColor->RGBColor[0.0199893, 0.609995, 0.00999466]], " Functions" }], "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: 11/25/98" }], "Text"], Cell[CellGroupData[{ Cell["Summary of important points", "Subtitle"], Cell[TextData[{ "This notebook is intended to give some insight into ", StyleBox["what", FontSlant->"Italic"], " is happening when the Green's function approach is used to solve \ inhomogeneous ODE's. I think that you will see (or have seen) the following: \ \n\n1. No matter how the original problem is motivated, you end up with a \ problem of the form, L[g] = \[Delta][x-t]\n2. Solving this problem amounts \ to solving L[u] = 0, it produces functions that look like the solution of \ L[u] = 0, but there is a \"kink\" at ", StyleBox["x=t", FontSlant->"Italic"], " that results from the singular \[Delta] function at this point. \n3. To \ match the kink, you may produce a function that you are not real familiar \ with.\n4. Since you have already solved L[u] = 0, resolving with \ \[Delta][x-t] on the right side should be easier than solving the \ nonhomgenous problem (at least this is nice claim!). Then you just need to \ evaluate an integral to get the answer. This integral also will involve some \ of these new \"noncommon\" functions. Note that this could be a book-keeping \ problem, but fortunately ", StyleBox["Mathematica", FontSlant->"Italic"], " takes care of it for you. \n" }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell["Initialization", "Subtitle"], Cell["\<\ This notebook is a bit rough in format (you need to do it order as \ some of the names will get written over) but it is an exploration of the \ Green's function and its use to solve inhomogeneous ODE's. You will have to \ work through it to get much benefit. \ \>", "Text"], Cell["\<\ We will need this add-on package which defines the delta function \ and the Heaviside function -- which is called the UnitStep function\ \>", "Text"], Cell[BoxData[ \(<< Calculus`DiracDelta`\)], "Input", CellLabel->"In[1]:="], Cell["On some setups, you may need to use a \"Needs\" command. 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Cell[BoxData[ FormBox[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False], TraditionalForm]], "Output", CellLabel->"Out[28]="] }, Open ]], Cell["\<\ We see that uniform cooling gives the greatest deviation and is, \ not surprisingly, symmetric. A higher power of x is more asymmetric but the \ effect is less (is this obvious?). \ \>", "Text"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Find the Green's function for ", Cell[BoxData[ \(TraditionalForm\`\[PartialD]\^2\/\[PartialD]x\^2\ + \ k\^2\)]], ". " }], "Section"], Cell[CellGroupData[{ Cell[TextData[{ "Now we can solve the problem where the operator is ", Cell[BoxData[ \(TraditionalForm \`\(\[PartialD]\^2\ \)\/\(\((\[PartialD]x)\)\^2\ \)\)]], "+ ", Cell[BoxData[ \(TraditionalForm\`k\^2\)]], " and the boundary conditions are homogeneous. " }], "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(eq2 = D[g[t], {t, 2}]\ + k^2\ g[t]\ - DiracDelta[x - t]\)], "Input", CellLabel->"In[29]:="], Cell[BoxData[ FormBox[ RowBox[{\(\(g(t)\)\ k\^2\), "-", \(DiracDelta(x - t)\), "+", RowBox[{ SuperscriptBox["g", "\[DoublePrime]", MultilineFunction->None], "(", "t", ")"}]}], TraditionalForm]], "Output", CellLabel->"Out[29]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ans2 = DSolve[{eq2 == 0, g[0] == 0, g[1] == 0}, g[t], t]\)], "Input", CellLabel->"In[30]:="], Cell[BoxData[ FormBox[ RowBox[{ \(General::"spell1"\), \( : \ \), "\<\"Possible spelling error: new symbol name \ \\\"\\!\\(TraditionalForm\\`ans2\\)\\\" is similar to existing symbol \\\"\\!\ \\(TraditionalForm\\`ans12\\)\\\".\"\>"}], TraditionalForm]], "Message"], Cell[BoxData[ \(TraditionalForm \`{{g(t) \[Rule] \(1\/k\(( \(sin(k\ t - k\ x)\)\ \(UnitStep(t - x)\) + \(cos(k\ t)\)\ \(sin(k\ x)\)\ \(UnitStep(\(-x\))\) - \(sin(k\ t)\)\ \((\(csc(k)\)\ \(sin(k - k\ x)\)\ \(UnitStep(1 - x)\) + \(cot(k)\)\ \(sin(k\ x)\)\ \(UnitStep(\(-x\))\))\))\)\)}} \)], "Output", CellLabel->"Out[30]="] }, Open ]], Cell["\<\ At this point, g[ ] is effectively now g[x,t] as we would like. \ \ \>", "Text"], Cell[TextData[{ StyleBox["Note that (as expected), the Green's function consists of", FontColor->RGBColor[0.499992, 0, 0]], StyleBox[" sin/cos", FontSlant->"Italic", FontColor->RGBColor[0.499992, 0, 0]], StyleBox[" type terms as required by the operator. 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