Symbolic polynomial approximation of an LDE solution

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Description:

This worksheet illustrates  LdeApprox  package capability of doing symbolic polynomial approximation of an LDE solution. First of all we load LdeApprox package and define an IVP. Then  we use ApproxSol procedure to find 3-rd degree polynomial approximation for the IVP solution on interval x = [-1,1]. After that we find exact solution by Maple™ procedure dsolve . Finally we compare exact and approximate results using Maple™ procedure plot3d .

This loads the package.

restart:
with(LdeApprox):

Example

Initial value problem

ivp:={diff(y(x),x,x,x)+lambda*x*diff(y(x),x)+lambda^2*y(x) = 0,y(0)=0,D(y)(0)=0,`@@`(D,2)(y)(0)=lambda};

Finding polynomial approximation for solution of the IVP.

apr:=ApproxSol(ivp,y(x),x=-1..1, 3);

Finding exact solution of the IVP using Maple™  function dsolve .

sol:=dsolve(ivp,y(x));

Comparing exact and approximate results.

plot3d(subs(sol,y(x))-subs(apr,y(x)),lambda=-1/8..1/8,x=-1..1,axes=boxed,orientation=[45,45]);

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.