the Fourier transform method
Suppose you want to solve a PDE on the whole infinite line, where there are no walls and no boundaries to bounce off. Separation of variables and Fourier series are built for a finite box; on an infinite domain the natural tool is the Fourier transform. The idea is to rewrite an arbitrary function not as a sum of discrete sine waves (a series) but as a continuous blend of pure waves e^(i*xi*x) of every possible frequency xi, and to track what the equation does to each frequency on its own.
Concretely, the Fourier transform of a function f(x) is the new function f-hat(xi) = integral from -infinity to infinity of f(x) e^(-i*xi*x) dx. It records 'how much of frequency xi is present in f'. The reason this is powerful for PDEs is the differentiation rule: transforming a derivative df/dx just multiplies f-hat by i*xi. So when you apply the transform in the x-variable to a PDE like the heat equation u_t = k u_xx, every x-derivative turns into a factor of i*xi, and the PDE collapses into an ordinary differential equation in time for each fixed xi: d/dt of u-hat(xi, t) = k (i*xi)^2 u-hat = -k*xi^2 u-hat. That is a one-line ODE whose solution is u-hat(xi, t) = u-hat(xi, 0) e^(-k*xi^2 t).
The method is always three steps: transform the PDE in space to get an easy ODE in the transform variable, solve that ODE using the transformed initial data, then invert the transform to come back to physical x. It is the cleanest way to derive the heat kernel and d'Alembert's formula, and the factor that multiplies u-hat in the middle step (here -k*xi^2) is exactly the symbol of the differential operator. The honest catch: the transform integral needs f to decay (or at least be tempered) for the integral to make sense, so this clean machinery lives most comfortably on infinite domains for data that dies off at infinity.
For u_t = k u_xx on the whole line with u(x,0) = f(x): transform in x to get u-hat_t = -k*xi^2 u-hat, solve to get u-hat(xi,t) = f-hat(xi) e^(-k*xi^2 t), then invert. Since e^(-k*xi^2 t) is itself the transform of a Gaussian, the inverse turns the product into a convolution with that Gaussian — and that Gaussian is the heat kernel.
Transform, solve a tiny ODE, invert — and the heat kernel falls out for free.
The transform sends a hard PDE to an easy ODE, but the work moves to the final step: inverting the transform can be the genuinely difficult part, often requiring the convolution theorem, a table of known pairs, or contour integration.