LDE with regular singular point

Copyright © 2004-2009  DigiArea Group . All rights reserved.

Description:

This worksheet illustrates  LdeApprox  package capability of working wi th LDEs with regular singular points. First of all we load LdeApprox package and define a BVP with regular singular point. 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 Maple™ procedure dsolve . Finally we compare exact and approximate results using Maple™ procedures plot .

This loads the package.

restart:
with(LdeApprox):

Example

Boundary value problem.

bvp:={2*x*y(x)+3*diff(y(x),x)+2*x*diff(y(x),`$`(x,2))=0,y(0)=1/3};

Finding polynomial approximation for solution of the BVP.

apr:=ApproxSol(bvp, y(x), x=0..1, 3);

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

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

Comparing exact and approximate results.

plot(subs(sol,y(x)-subs(apr,y(x))),x=0..1);

Note

The method applied in the package is a 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 no parameters. You can try more complex examples in your computer.