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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[
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                    RowBox[{
                      TagBox[\(a\/\(2\ b\) + 1\),
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                      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\/\(4\ b\)\)\)\ \(H(x)\) /. First[%];
\)\)], 
  "Input"],

Cell["Applying the boundary conditions we find", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    FormBox[
      RowBox[{"Solve", "[", 
        RowBox[{\(w(0) == 1\), ",", 
          SubscriptBox[
            TagBox["c",
              C], "1"]}], "]"}], TraditionalForm]], "Input"],

Cell[BoxData[
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      RowBox[{"{", 
        RowBox[{"{", 
          RowBox[{
            SubscriptBox[
              TagBox["c",
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TraditionalForm]], 
  "Output"]
}, Open  ]],

Cell["and", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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          RowBox[{
            RowBox[{
              RowBox[{
                RowBox[{
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            \(First[%]\)}], ",", 
          SubscriptBox[
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                                    TagBox[\(1\/\(2\ b\)\),
                                      (Editable -> True)]}], ")"}],
                                InterpretTemplate[ 
                                Hypergeometric1F1[ #, #2, #3]&]]}], 
                            \(3\ b\)]}], ")"}]}]}], ")"}]}]}]}], "}"}], 
        "}"}], TraditionalForm]], "Output"] }, Open  ]]
}
]


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