Why can we assume that these eignenfunctions are known, in the Sturm-Liouville problem?

Question

Consider the following heat equation PDE, with non-constant coefficients. That is $c = c(x)$, $\rho = \rho(x)$, and $K_0 = K_0(x)$

\begin{align} \begin{cases} c \rho u_t = (K_0 u_x)_x \\ u(x,0) = f(x) \\ u(0,t) = u_x(L,t) = 0 \\ t\geq 0, \,\, 0\leq x\leq L \end{cases} \end{align}

Separation of variables gives rise to the Sturm-Liouville ODE for the spatial function $\phi(x)$ \begin{align} (K_0 \phi')' + \lambda c \rho \phi = 0 \end{align}

Here my textbook states that while the eigenfunctions $\phi_n$ might be difficult to calculate (and will most likely be done numerically), that nevertheless they can be calculated. So we can continue our solution analysis assuming the eigenfunctions $\phi_n$ are known.

That's where my question comes in; how would we calculate the eigenfunctions given that our eigenvalues are still unknown? Rayleigh's quotient for calculating eigenvalues wouldn't work because $\phi_n$ is yet unknown. How can I be confident that $\phi_n(x)$ is known?

Answer 1

how would we calculate the eigenfunctions given that our eigenvalues are still unknown?

By replacing the eigenfunction problem with its discretization: finite difference method, or finite element method, and solving the resulting linear system. There are books written about numerical methods for differential equations, including eigenproblems; check out some. There is also Wikipedia.