Sturm--Liouville Problems

Brown University, Applied Mathematics

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Sturm--Liouville Problems

Consider a periodic function \( f(x) \) with period \( T = e\ell \); for instance, let \( f(x) =x \). We can expand this function into Fourier series:
\[ f(x) \sim \frac{a_0}{2} + \sum_{n\ge 1} \left( a_n \cos \frac{n\pi x}{\ell} + b_n \sin \frac{n\pi x}{\ell} \right) ,\]

where the coefficients are defined according to Euler--Fourier formulas

\[ a_n = \frac{1}{2\ell} \,\int_{-\ell}^{\ell} f(x)\, \cos \frac{n\pi x}{\ell} \, {|text d}x , \qquad n=0,1,2,\ldots , \] \[ b_n = \frac{1}{2\ell} \,\int_{-\ell}^{\ell} f(x)\, \sin \frac{n\pi x}{\ell} \, {|text d}x , \qquad n=0,1,2,\ldots , \]

Let us choose \( f(x) = x^2 \) on the interval [0,2].

f := x^2:
f2 := (x+2)^2:
l := 1:
a_0 := 1/l*(int(f2,x=-1..0)+int(f,x=0..1)):
cs := cos(k*PI*x/l):
sn := sin(k*PI*x/l):
a_k := simplify(1/l*(int(f2*cs,x=-1..0)+int(f*cs,x=0..1))):
b_k := simplify(1/l*(int(f2*sn,x=-1..0)+int(f*sn,x=0..1))):
Fourier := a_0/2 + sum(a_k*cs + b_k*sn,k=1..n):
a := 100:
Cesaro := sum(Fourier,n=1..a)/a:
plot(Cesaro,x=-2..2)

Another approach:

f:=x->x;
L:=2;
a0:=int(f(x),x=-L..L)/(2*L)
assume(n,Type::NonNegInt)
an:=simplify(int(f(x)*cos(n*PI*x/L),x=-L..L)/L)
bn:=simplify(int(f(x)*sin(n*PI*x/L),x=-L..L)/L)
a:=m->subs(an,n=m);
a(0):=a0;
b:=m->subs(bn,n=m);
b(1)

Fourier series with N +1 terms:
S:=N->a(0) + sum(a(n)*cos(n*PI*x/L)+b(n)*sin(n*PI*x/L),n=1..N)
S(4)
plot(f,S(1),S(2),S(3),x=-L..L)
plot(f,S(10),x=-L..L)

Cesaro series:
C:=N->a(0) + (sum(a(n)*(N+1-n)*cos(n*PI*x/L)+b(n)*(N+1-n)*
sin(n*PI*x/L),n=1..N))/(N+1)
plot(f,C(10),x=-L..L)

Step-function on the interval [-1/2, 1/2]:
f:=piecewise([x<=-1/2,0],[x>-1/2 and x<1/2,1],[x>1/2,0])
plot(f,x=-1..1)

Another piecewise continuous function:
f:=piecewise([x<=-1/2,x],[x>-1/2 and x<1/2,x^2],[x>1/2,-x])
plot(f,x=-1..1)

 



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