LdeApprox[Normalize]  - gives normalized form of an expression over given interval

Calling Sequence:

     Normalize(expr, x = a..b)

Parameters:

     expr - any expression

     x = a..b - interval of the normalization (a, b can be symbolic or numeric)

Description:

• Normalize  procedure gives normalized form of an expression over given interval. It means that for any f(x)  one gets
                                                  Normalize:     
• Normalize makes effect only on such parts of expression which depend on x. The procedure is normally used to get "normalized" approximate solution of a BVP or IVP in some cases (especially in a case of homogeneous BVP).

Examples:

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

This is a simple homogeneous BVP.
bvp:={diff(y(x),x,x)+lambda*y(x)=0,y(0)=0,y(Pi)=0};

Using ApproxSol procedure to find polynomial approximation of solutions of the BVP. As corresponding eigen value variable is not specified then it is determined automatically.
apr:=ApproxSol(bvp,y(x),x=0..Pi,5,exact=false);

Normalizing the result to compare it with the exact solution.
apr1:=Normalize(apr,x = 0..Pi);

The exact solution is as fillows:
sol:=dsolve({diff(y(x),x,x)+lambda*y(x)=0,y(0)=0},y(x));

This is the normalized exact solution.
sol1:=radsimp(Normalize(sol,x = 0..Pi));

Now one can compare the results.

 For
plot(subs({lambda=1},subs(sol1,y(x))+subs(apr1[1],y(x))),x=0..Pi);

delta[lambda]=1-subs(apr1[1],lambda);

For
plot(subs({lambda=4},subs(sol1,y(x))-subs(apr1[2],y(x))),x=0..Pi);

delta[lambda]=4-subs(apr1[2],lambda);

See Also:

LdeApprox