1. The diffusion equation, often written as $$\frac{\partial u}{\partial t} = D \frac{\partial^2 u}{\partial x^2},$$ describes how a substance diffuses over time along one spatial dimension. 2. Here, $u(x,t)$ is the concentration at position $x$ and time $t$, and $D$ is the diffusion coefficient. 3. To solve or analyze this equation, typical steps include specifying initial concentration $u(x,0)$ and boundary conditions. 4. The equation models the rate of change of concentration over time as proportional to its spatial curvature, meaning areas with steep concentration gradients cause faster diffusion. 5. This is a fundamental partial differential equation in physics and engineering for processes involving heat, particles, or chemicals spreading out. Final note: please specify if you want a solution method, particular initial/boundary conditions, or numerical examples.