Approximation methods for LDE solutions 

Copyright © 2004-2012  DigiArea, Inc. . All rights reserved.

Description:

This worksheet illustrates ©LdeApprox ©package capability of doing polynomial approximation of solution of simple LDE. First of all we load LdeApprox package and define simple IVP. Then we use ApproxSol procedure to find 7-th degree polynomial approximation for the IVP solution on interval x = [-h,h]. After that we find exact solution by Maple™ procedure dsolve . Finally we compare exact and approximate results using Maple™ procedures plot ©and plot3d .

This loads the package.

restart:
with(LdeApprox):

Examples:

This is simple IVP with parameter .

ivp:={diff(y(x),x,x)+xi*y(x)=0,y(0)=0,D(y)(0)=1};

Finding polynomial approximation of the IVP solution on interval [-h, h].

apr:=ApproxSol(ivp,y(x),x=-h..h,7);

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

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

Comparing exact and approximate results for ©and

plot(subs({h=Pi/8,xi=1/4},subs(sol,y(x))-subs(apr,y(x))),x=-Pi/8..Pi/8);

Comparing exact and approximate results for ©using Maple™ ©function plot3d

plot3d(subs({h=Pi/8},subs(sol,y(x))-subs(apr,y(x))),xi=1/16..2,x=-Pi/8..Pi/8,axes=boxed);

Comparing exact and approximate results for ©©using Maple™ ©function plot3d .

plot3d(subs({xi=1/4},subs(sol,y(x))-subs(apr,y(x))),x=-h..h,h=-Pi/4..Pi/4,axes=boxed);

Note:

This trivial example is just for Web. You can try more complex examples in your computer.