Fourier series method

Definition

Let $L > 0$ and let $A$ be a linear differential operator in $x$, with boundary conditions at $0$ and $L$. Consider the linear evolution equation $u_t = A u$ for $u \colon [0, L] \times [0, \infty) \to \R$. Assume that $L^2(0, L)$ has an orthonormal basis $(\varphi_k)_{k \ge 1}$ of functions that satisfy the boundary conditions and $A \varphi_k = -\mu_k \varphi_k$ with $\mu_k \in \R$. Typically the $\varphi_k$ are sines or cosines, so that expansions in them are Fourier series. The Fourier series method for the initial value $u(\cdot, 0) = f$ consists of three steps.

1. Expand $f = \sum_k b_k \varphi_k$ with $b_k = \int_0^L f \varphi_k \, dx$.
2. Seek $u(x, t) = \sum_k c_k(t) \varphi_k(x)$, which turns the equation into the ordinary differential equations $c_k' = -\mu_k c_k$ with $c_k(0) = b_k$.
3. Solve them, which gives $u(x, t) = \sum_k b_k e^{-\mu_k t} \varphi_k(x)$.

The same steps apply to second-order equations in time, such as the wave equation $u_{tt} = A u$, with two initial values.