Introduction to the LdeApprox  package

Copyright © 2004-2008  DigiArea Group All rights reserved.  

Description:

• LdeApprox  package provides some procedures for finding polynomial approximation of solutions of a linear differential equation (LDE). The main procedure ApproxSol allows one to find polynomial approximation for solutions of an LDE with polynomial coefficients on a given interval. The main algorithm applied is so called a-method  (see LdeApprox[references] ) which has the following special properties.
• The method is numerically analytical  one. It means that LDE coefficients, boundary or initial conditions and parameters of ApproxSol  procedure (except degree of approximation polynomial) can be either symbolical or numerical expressions (see symbolic examples ).
• The method gives the polynomial approximation which is very close to the uniform  one. It means that the result is such a polynomial that minimizes the maximum value of the absolute error between the solution and a polynomial of the given degree over the interval under consideration (see numerical examples ).
• The method is fast saturated . It means that the required precision can be reached very fast by increasing the degree of approximation polynomial.
• The method can be applied to solve either initial value problems  (IVP) or boundary value problems  (BVP) (see BVP examples ). To receive normalized results of the solution of a homogeneous BVP one can use the package procedure Normalize .
• The method can be applied to solve IVP or BVP even if the LDE has non-polynomial coefficients . To do this one can use the package procedure ToRatCoeffs  (which gives rational approximation of the LDE coefficients) and then apply ApproxSol procedure to the result.
• The method can be applied to solve IVP or BVP with regular singular points (RSP) (see RSP examples ).
• Before using routines of the LDEapprox  package, you must load the package with one of the commands with(LDEapprox) , with(LDEapprox,[]), with(LDEapprox,<function>)  or LDEapprox[<function>] . See with  for details.
• The procedures available are:

ApproxSol    Normalize    ToRatCoeffs

Examples:

This loads the package.
restart:
with(LdeApprox):

This is a 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);

See Also:

LdeApprox[references]