1. The heat equation is a partial differential equation that describes how heat diffuses through a given region over time. 2. The standard form of the heat equation in one dimension is: $$\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$$ where $u(x,t)$ is the temperature at position $x$ and time $t$, and $\alpha$ is the thermal diffusivity constant. 3. Important rules: - The equation models the flow of heat from regions of higher temperature to lower temperature. - Boundary and initial conditions are necessary to solve the equation uniquely. 4. To solve, one typically uses methods like separation of variables, Fourier series, or numerical methods depending on the problem setup. 5. Example: If the initial temperature distribution is $u(x,0) = f(x)$ and the ends are held at zero temperature, the solution can be expressed as a Fourier sine series. 6. This equation is fundamental in physics and engineering for modeling heat transfer and diffusion processes.