• Homogeneous BVP example of the AnalyticalApproximations`LdeApprox` package

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    This notebook illustrates AnalyticalApproximations`LdeApprox` package capability of working with BVPs. First of all we load LdeApprox package and define a BVP. Then we use ApproxSol procedure to find symbolic 3-rd degree polynomial approximation and numeric 5-rd degree polynomial approximation for the BVP solution on interval x = [0, Pi]. After that we find exact solution by Mathematica™ function DSolve. Finally we compare exact and approximate results using Mathematica™ functions Plot and Plot3D.

  • This loads the package.

  • [Graphics:Mathematica/LdeApprox/hombvp/images/index_gr_1.gif]
  • This is simple homogenious BVP:

  • [Graphics:Mathematica/LdeApprox/hombvp/images/index_gr_2.gif]
  • Using ApproxSol function to find 3-rd degree symbolic polynomial approximation of solutions of the BVP (Exact option is True by default).

  • [Graphics:Mathematica/LdeApprox/hombvp/images/index_gr_4.gif]
  • Using ApproxSol function to find 5-th degree numeric polynomial approximation of solutions of the BVP (Exact -> False). As corresponding eigenvalue variable is not specified then it is determined automatically. Only 6 values of eigenvalue variable are available for 5-degree approximation.

  • [Graphics:Mathematica/LdeApprox/hombvp/images/index_gr_6.gif]
  • One has to normalize the result to compare it with the exact solution.

  • [Graphics:Mathematica/LdeApprox/hombvp/images/index_gr_10.gif]
  • The exact solution is as follows.

  • [Graphics:Mathematica/LdeApprox/hombvp/images/index_gr_12.gif]
  • This is the normalized form of the exact solution.

  • [Graphics:Mathematica/LdeApprox/hombvp/images/index_gr_14.gif]
  • Now one can compare the results.

  • For λ = 1

  • [Graphics:Mathematica/LdeApprox/hombvp/images/index_gr_16.gif]


  • For λ = 4

  • [Graphics:Mathematica/LdeApprox/hombvp/images/index_gr_21.gif]




    The method applied in the package is numerically - analytical one. It means that you can use symbolic expressions as boundary conditions, interval of approximation etc. However these kind of examples leads to huge output so its not for Web. This reason force us to introduce simple example with one parameter only. You can try more complex examples in your computer.