• Numerical example of the AnalyticalApproximations`LdeApprox` package

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    Description:

    This notebook illustrates AnalyticalApproximations`LdeApprox` package capability of doing numerical polynomial approximation of an LDE solution.First of all we load the package and define an IVP. Then we use ApproxSol procedure to find 5-th degree polynomial approximation for the IVP 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™ function Plot.
     

  • This loads the package.

  • [Graphics:Mathematica/LdeApprox/numerical/images/index_gr_1.gif]
  • Initial value problem.

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

  • [Graphics:Mathematica/LdeApprox/numerical/images/index_gr_4.gif]
    [Graphics:Mathematica/LdeApprox/numerical/images/index_gr_5.gif]
  • UnfortunatelyDSolvecan not find exact solution of the IVP.

  • [Graphics:Mathematica/LdeApprox/numerical/images/index_gr_6.gif]
    [Graphics:Mathematica/LdeApprox/numerical/images/index_gr_7.gif]
  • Nevertheless the exact solution is as follows.

  • [Graphics:Mathematica/LdeApprox/numerical/images/index_gr_8.gif]
    [Graphics:Mathematica/LdeApprox/numerical/images/index_gr_9.gif]
  • Comparing exact and approximate results using Mathematica™ function Plot.

  • [Graphics:Mathematica/LdeApprox/numerical/images/index_gr_10.gif]

    [Graphics:Mathematica/LdeApprox/numerical/images/index_gr_11.gif]

    [Graphics:Mathematica/LdeApprox/numerical/images/index_gr_12.gif]

    Note:
    The example is quite simple (just for Web). You can try more complex examples in your computer.