The heat equation is the partial differential equation $$\frac{\partial f(x,t)}{\partial t} = D \frac{\partial^2 f(x,t)}{\partial x^2},$$ where $D$ is a positive constant. This equation applied both to heat flow and to diffusion, and Fourier series was invented to help solve it. If we apply the dynamic picture, viewing $f(x,t)$ as a function of $x$ that evolves in time, then the equation becomes $$ \frac{d {\bf f}}{dt} = A{\bf f},$$ where $A= D d^2/dx^2$. This is just like Section 5.2 and the equation $\dot {\bf x} = A {\bf x}$, only now $A$ is an operator on a very large vector space instead of an $n\times n$ matrix. But the form of the solution is exactly the same as before: $${\bf f}(t) = \sum_n c_n \xi_n e^{\lambda_n t},$$ where each $\xi_n$ is an eigenvector of $A$ with eigenvalue $\lambda_n$. Since we already diagonalized $A$ (eigenvalues $-n^2\pi^2 D/L^2$, eigenvectors $\xi_n(x) = \sin(n\pi x/L)$), we can just write down the solution: $$f(x,t) = \sum_n c_n e^{-n^2 \pi^2 Dt/L^2} \sin(n\pi x/L),$$ the the constants $c_n$ are just the Fourier coefficients of $f(x,0)$.

The wave equation $$\frac{\partial^2 f(x,t)}{\partial t^2} = c^2 \frac{\partial^2 f(x,t)}{\partial x^2}$$ is treated almost exactly the same way as the Section 5.3 problem $\ddot {\bf x} = A {\bf x}$, and very similarly to the heat equation. The general solution to $\ddot {\bf x} = A {\bf x}$ is $${\bf x}(t) = \sum_j {\bf \xi}_j [a_j \cos(\omega_j t) + b_n \sin(\omega_j t)]$$ if all the eigenvalues of $A$ are negative (with $\omega_j =\sqrt{-\lambda_j}$), and with $\cosh$ and $\sinh$ terms if some of the eigenvalues are positive. The coefficients $a_j$ and $b_j$ are related to the decompositions of ${\bf x}(0)$ and $\dot{\bf x}(0)$ in the basis $\{ \xi_n \}$ of eigenvectors. For the wave equations, if we have boundary condition $f(0,t)=f(L,t)=0$, then the general solution is $$ f(x,t) = \sum_n \sin(n\pi x/L) [a_n \cos(\omega_n t) + b_n \sin(\omega_n t)], $$ where $\omega_n = n \pi c/L$. Each term is called a standing wave because it keeps its shape and just goes up and down. The $a_n$'s are the Fourier coefficients of the initial "position" $f_0(x) = f(x,0)$, and the $b_n$'s are closely related to the Fourier coefficients of the initial "velocity" $g_0(x) = \partial_t f(x,0)$.