Re: Q: Weber equation
- To: mathgroup at smc.vnet.net
- Subject: [mg9730] Re: Q: Weber equation
- From: Paul Abbott <paul at physics.uwa.edu.au>
- Date: Tue, 25 Nov 1997 00:06:29 -0500
- Organization: University of Western Australia
- Sender: owner-wri-mathgroup at wolfram.com
Boguslaw Ptaszynski wrote: > I have a problem with a Weber equation and boundary conditions: > b^2 w''[x] - (x^2/4 + b/2+ a) w[x]==0 Weber equation (b,a -constans) > w[x=0]==1 > w'[x=1]+ w[x=1] d (1/2+b/2) == 0 > > I want to solve this problem in Mathematica3,0 under Windows 95 . I have > solved this but in form very complcated series. Maybe is somebody who > know how solve this in simply form? Appended below is a Mathematica Notebook which solves this problem. Cheers, Paul ____________________________________________________________________ Paul Abbott Phone: +61-8-9380-2734 Department of Physics Fax: +61-8-9380-1014 The University of Western Australia Nedlands WA 6907 mailto:paul at physics.uwa.edu.au AUSTRALIA http://www.pd.uwa.edu.au/~paul God IS a weakly left-handed dice player ____________________________________________________________________ Notebook[{ Cell[TextData[{ "After the transformation ", Cell[BoxData[ \(TraditionalForm \`w[x] \[Rule] \[ExponentialE]\^\(-\(x\^2\/\(4\ b\)\)\)\ \(H(x)\)\)]], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ FormBox[ RowBox[{"Simplify", "[", RowBox[{\(\[ExponentialE]\^\(x\^2\/\(4\ b\)\)\), " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{\(b\^2\), " ", RowBox[{ SuperscriptBox["w", "\[DoublePrime]", MultilineFunction->None], "(", "x", ")"}]}], "-", \(\((x\^2\/4 + a + b\/2)\)\ \(w(x)\)\)}], "/.", \(w \[Rule] Function[x, \[ExponentialE]\^\(-\(x\^2\/\(4\ b\)\)\)\ \(H(x)\)]\)}], ")"}]}], "]"}], TraditionalForm]], "Input"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"b", " ", RowBox[{"(", RowBox[{ RowBox[{"b", " ", RowBox[{ SuperscriptBox["H", "\[DoublePrime]", MultilineFunction->None], "(", "x", ")"}]}], "-", RowBox[{"x", " ", RowBox[{ SuperscriptBox["H", "\[Prime]", MultilineFunction->None], "(", "x", ")"}]}]}], ")"}]}], "-", \(\((a + b)\)\ \(H(x)\)\)}], TraditionalForm]], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can solve the differential equation:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(TraditionalForm\`DSolve[% == 0, H(x), x]\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{"{", RowBox[{\(H(x)\), "\[Rule]", RowBox[{ RowBox[{ SubscriptBox[ TagBox["c", C], "1"], " ", TagBox[ RowBox[{\(\(\[ThinSpace]\_1\) F\_1\), "(", RowBox[{ TagBox[\(a\/\(2\ b\) + 1\/2\), (Editable -> True)], ";", TagBox[\(1\/2\), (Editable -> True)], ";", TagBox[\(x\^2\/\(2\ b\)\), (Editable -> True)]}], ")"}], InterpretTemplate[ Hypergeometric1F1[ #, #2, #3]&]]}], "+", RowBox[{"x", " ", SubscriptBox[ TagBox["c", C], "2"], " ", TagBox[ RowBox[{\(\(\[ThinSpace]\_1\) F\_1\), "(", RowBox[{ TagBox[\(a\/\(2\ b\) + 1\), (Editable -> True)], ";", TagBox[\(3\/2\), (Editable -> True)], ";", TagBox[\(x\^2\/\(2\ b\)\), (Editable -> True)]}], ")"}], InterpretTemplate[ Hypergeometric1F1[ #, #2, #3]&]]}]}]}], "}"}], "}"}], TraditionalForm]], "Output"] }, Open ]], Cell["The solution to the original differential equation is", "Text"], Cell[BoxData[ \(TraditionalForm \`\(w(x_) = \[ExponentialE]\^\(-\(x\^2\/\(4\ b\)\)\)\ \(H(x)\) /. 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