4.5.5   Sturm-Liouville Problem for equation :    X''- X=0,   case 2)    -X'(0)+HX(0)=0,  X(L)=0    (Robin-Dirichlet)

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restart;

>	L:=2;H:=3;

Characteristic equation:

>	w(x):=x*cos(x*L)+H*sin(x*L);

>	plot(w(x),x=0..20);

Eigenvalues:

>	lambda:=array(1..100);

>	n:=1: for m from 1 to 100 do z:=fsolve(w(x)=0,x=m*1..(m+1)*1): if type(z,float) then lambda[n]:=z: n:=n+1 fi od:

>	for i to n-1 do lambda[i] od;

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N:=n-1;

>	n:='n':i:='i':

Eigenfunctions:

>	X[n]:=sin(lambda[n]*(x-L));

Squared norm:

>	NX[n]:=int(X[n]^2,x=0..L);

GENERALIZED FOURIER SERIES

Function:

>	f(x):=exp(x/2);

Fourier coefficients:

>	a[n]:=int(f(x)*X[n],x=0..L)/NX[n];

Generalized Fourier series:

>	u(x):=sum(a[n]*X[n],n=1..5):

>	plot({f(x),u(x)},x=0..L,axes=boxed);

>	u(x):=sum(a[n]*X[n],n=1..20):

>	plot({f(x),u(x)},x=0..L,axes=boxed);

>	u(x):=sum(a[n]*X[n],n=1..N):

>	plot({f(x),u(x)},x=0..L,axes=boxed);

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