There is an approach to solve the heat equation $$ \begin{align*} \frac{\partial u}{\partial t}(x,t) &= \frac{\partial^2 u}{\partial x^2}(x,t) \quad \text{for $(x,t) \in \mathbb{R} \times (0, \infty)$} \\ u(x,0) &= f(x) \quad \text{for $x \in \mathbb{R}$} \end{align*} $$ by applying Fourier Series.
What is the motivation for that? How did Fourier see that his Fourier Series will help to resolve this equation? What makes Fourier Series to special to be applicable to this problem?
Fourier noticed that:
(a) the heat equation is linear, so if $A(x,t)$ and $B(x,t)$ are solutions than so is $rA + sB$ for any constants $r,s$.
(b) the family of functions $\sin(kx)e^{-k^2t}$ and $\cos(kx)e^{-k^2t}$ are solutions of the heat equation.
So if we can express $u(x,0)$ as the sum of sine and cosine functions $\sin(k_ix), \cos(k_ix)$ then each of these components will decay over time in a known way and we can write down an expression for $u(x,t)$ as the sum of these decaying sine and cosine components.
The "frequencies" $k_i$ are typically multiples of a "natural" frequency which is determined by the boundary conditions - for example, if we know that $u(0,t)=u(L,t)=0 \space \forall t$ then we want to express $u(x,0)$ as the sum of functions of the form $\sin(\frac{n\pi x}{L})$.
So Fourier's challenge was to find a way to express a general function as an infinite series of sines and cosines.