• BVP example of the AnalyticalApproximations`LdeApprox` package

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    Description:

    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 3-rd degree polynomial approximation for the BVP solution on interval x = [0,1]. 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/bvp/images/index_gr_1.gif]
  • The boundary value problem.

  • [Graphics:Mathematica/LdeApprox/bvp/images/index_gr_2.gif]
    [Graphics:Mathematica/LdeApprox/bvp/images/index_gr_3.gif]
  • Finding polynomial approximation for solution of the BVP.

  • [Graphics:Mathematica/LdeApprox/bvp/images/index_gr_4.gif]
    [Graphics:Mathematica/LdeApprox/bvp/images/index_gr_5.gif]
  • Using Mathematica™ function DSolve to get exact solution of the BVP.

  • [Graphics:Mathematica/LdeApprox/bvp/images/index_gr_6.gif]
    [Graphics:Mathematica/LdeApprox/bvp/images/index_gr_7.gif]
  • Comparing exact and approximate results using Mathematica™ function Plot3D.

  • [Graphics:Mathematica/LdeApprox/bvp/images/index_gr_8.gif]

    [Graphics:Mathematica/LdeApprox/bvp/images/index_gr_9.gif]

    [Graphics:Mathematica/LdeApprox/bvp/images/index_gr_10.gif]

    Note:

    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.