CMMExamples.nb

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Implementation using the Compound Matrix Method to calculate the Evans \
function, in order to solve boundary value eigenvalue problems.\
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The following command will install the most recent version from the \
PacletServer:\
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We need to load the package each time, so this is set as a initialization \
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Cell["Example 1: Second order equation with constant coefficients ", \
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Example 2: Second order equation with complex constant coefficients \
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We can plot the real and imaginary parts in the complex plane, where \
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Cell["\<\
We can use the winding number to find how many roots are inside a contour (as \
the Evans function is analytic). 
This is implemented for both circular and semicircular contours \
(incorporating the positive half-plane).\
\>", "Text",
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Cell["\<\
Here plot the system with a contour of radius 1 (the contour discretization \
is automatically set to be r/50), which does not include a root.
We can see that the contour does not go round the origin in the D(\[Lambda]) \
plane. Note that the contour overlaps itself here (which is why the lines \
look jagged).\
\>", "Text",
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If we move the circle, the Evans function contour now goes round the origin, \
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         0.029208934075158843` I, -0.0820005351513391 - 0.014860913681193647` 
         I, -0.06304240479117182 - 0.05447613295036802 
         I, -0.027846399496360384` - 0.07850047818063963 I, 
         0.012344244599502806` - 0.08239884347458887 I, 0.04133539937774587 - 
         0.06051894040487224 I, 0.028443514052604563` - 0.018352632936423745` 
         I, 0.015902783382089905` - 0.004035232481695461 I, 
         0.009079756858489094 - 0.00041399154126322657` I, 
         0.0070506174361986374`}, CompoundMatrixMethod`WindingNumber[
         PatternTest[
          Pattern[CompoundMatrixMethod`Private`EvansPoints, 
           Blank[]], MatrixQ[#, NumericQ]& ], 
         Optional[
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           Blank[]], {0, 0}]] := 
       Module[{}, 
         If[Length[CompoundMatrixMethod`Private`EvansPoints] == 0, Message[
             MessageName[
             CompoundMatrixMethod`WindingNumber, "noPointsGiven"]]; 
           Return[$Failed]]; 
         If[Length[CompoundMatrixMethod`Private`Centre] == 0, 
           CompoundMatrixMethod`Private`Centre = {0, 0}]; 
         Graphics`PolygonUtils`PointWindingNumber[
          CompoundMatrixMethod`Private`EvansPoints, 
           CompoundMatrixMethod`Private`Centre]], 
       CompoundMatrixMethod`WindingNumber[
         Pattern[CompoundMatrixMethod`Private`EvansPoints, 
          Blank[]], 
         Condition[
          Pattern[CompoundMatrixMethod`Private`Centre, 
           Blank[]], Im[CompoundMatrixMethod`Private`Centre] != 0]] := 
       CompoundMatrixMethod`WindingNumber[
        CompoundMatrixMethod`Private`EvansPoints, 
         ReIm[CompoundMatrixMethod`Private`Centre]], 
       CompoundMatrixMethod`WindingNumber[
         Pattern[CompoundMatrixMethod`Private`EvansPoints, 
          Blank[]], 
         Optional[
          Pattern[CompoundMatrixMethod`Private`Centre, 
           Blank[]], {0, 0}]] := 
       Module[{}, 
         If[Length[CompoundMatrixMethod`Private`EvansPoints] == 0, Message[
             MessageName[
             CompoundMatrixMethod`WindingNumber, "noPointsGiven"]]; 
           Return[$Failed]]; 
         If[Length[CompoundMatrixMethod`Private`Centre] == 0, 
           CompoundMatrixMethod`Private`Centre = {0, 0}]; 
         Graphics`PolygonUtils`PointWindingNumber[
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           CompoundMatrixMethod`Private`Centre]], 
       TagSet[CompoundMatrixMethod`WindingNumber, 
        MessageName[CompoundMatrixMethod`WindingNumber, "noPointsGiven"], 
        "No contour points given to calculate the winding number from."], 
       TagSet[CompoundMatrixMethod`WindingNumber, 
        MessageName[CompoundMatrixMethod`WindingNumber, "usage"], 
        "WindingNumber[pts] calculates the Winding Number of the set of \
points which make a closed contour in the complex plane, around the origin.\n\
WindingNumber[pts,{x0,y0}] calculates the Winding Number around centre \
{x0,y0}."]}; Typeset`initDone$$ = True),
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Cell["\<\
A semicircular contour centred at the origin (and including the imaginary \
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\>", "Text",
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If the discretization of the contour is not sufficient, the winding number \
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Each point takes longer to evaluate to evaluate for a large system like this, \
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Example 6: Fourth order equation with non-constant coefficients and infinite \
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The method also works very well for systems with two domains connected by an \
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We can find zeroes of the determinant of this matrix using FindRoot: \
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This determinant method struggles for more complicated systems.
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Now given a test value of \[Sigma], we can find the Evans function there: \
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And then plot the Evans function as we change the value of \[Sigma], and we \
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Cell["Example 10: Interface with non-constant coefficients", "Subchapter",
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Cell["\<\
This example is part of the paper \[OpenCurlyDoubleQuote]Curvature delays \
growth-induced wrinkling\[CloseCurlyDoubleQuote] published in Physical Review \
E by Jia, Pearce and Goriely: \
(https://link.aps.org/doi/10.1103/PhysRevE.98.033003). Here a thin growing \
surface is attached to an elastic cylinder.\
\>", "Text",
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Example 11: Fourth order, eigenvalue in the integration range\
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This example comes from \
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