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Sturm-Liouville problem

  • aka S.L.P

  • definition

\[ \begin{gather} (ry')' + (q+\lambda p)y = 0 \\\\ p>0 \text{ and } r>0 \quad \text{on } (a, b) \end{gather} \]

  • eigenvalue \(\lambda\)

    • S.L.P has a nontrivial solution for \(\lambda\), then the \(\lambda\) is an eigenvalue

  • eigenfunction

    • the solution for the corresponding S.L.P is called eigenfunction

## Regular SLP

  • boundary condition
\[ \begin{align} &A_1 y(a) + A_2 y'(a) = 0 \\\\ &B_1 y(b) + B_2 y'(b) = 0 \end{align} \]

in which,

\[ \begin{align} &A_1^2 +A_2^2 \neq 0 \\\\ &B_1^2 + B_2^2 \neq 0 \end{align} \]

Periodic SLP

  • define on \([a, b]\)
  • \(r(a) = r(b)\)
  • \(y(a) = y(b)\), \(y'(a) = y'(b)\)

Singular SLP

  • \(r(a) = 0\) , or \(r(b)=0\), (or both)

SL Theorem

  1. if \(\phi\) is an eigenfunction, then \(c\ \phi\) is also an eigenfunction \(\forall c \in ℝ\)
  2. \(\lambda_n \neq \lambda_m\)
  3. let SLP define on \((a, b)\), then
\[ \begin{gather} (\phi_n, \phi_m)=\int_a^b{p(x)\ \phi_n(x)\ \phi_m(x)\ dx} = 0 \end{gather} \]

in which \(p\) is weighted function

Eigenfunction expansion

With series solution, we can write any function in

\[ \begin{align} f(x) = \sum_{n =0}^{\infty}{a_n x^n} \end{align} \]

likewise, we can also write

\[ \begin{align} f(x) = \sum_{m=0}^{\infty}{a_m y_m(x)} \end{align} \]

in which \(y\) is an eigenfunction, multiply this equation by \(p(x)y_{n}\)

\[ \begin{gather} p(x)\ f(x)\ y_{n} = \sum_{m=0}^{\infty}{a_{m}\ y_{n}\ y_{m}} \end{gather} \]

and integrate to get

\[ \begin{align} (f, y_m) &= \sum_{n=1}^{\infty}{a_n (y_n, y_m)} \\\\ &=a_m\ (y_m, y_m) \\\\ a_m &= \frac{(f, y_m)}{(y_m, y_m)}=\frac{(f, y_m)}{||y_m||^2} \end{align} \]