Heat Equation

Fourier Analysis · Axiom Academy

Understanding heat diffusion through partial differential equations and Fourier series solutions The heat equation is a partial differential equation that describes the distribution of heat (or temperature variation) in a given region over time. u(x,t) is the temperature at position x and time t α (alpha) is the thermal diffusivity constant ∂u/∂t represents how temperature changes with time ∂²u/∂x² represents the spatial curvature of temperature 2. Physical Interpretation: Heat Diffusion The heat equation models the physical process of heat diffusion. Heat naturally flows from hotter regions to cooler regions, and the second derivative captures this flow. If ∂²u/∂x² > 0 (concave up), heat flows INTO the point → temperature increases If ∂²u/∂x² < 0 (concave down), heat flows OUT of the point → temperature decreases If ∂²u/∂x² = 0 (linear), no net heat flow → temperature stays constant To solve the heat equation, we need to specify boundary conditions. A common scenario is a rod of length L with fixed temperatures at both ends. u(0,t) = 0 : Left end maintained at temperature 0 u(L,t) = 0 : Right end maintained at temperature 0 u(x,0) = f(x) : Initial temperature distribution These boundary conditions mean we're looking for solutions where the ends are held at zero temperature, and we know the initial temperature distribution along the rod. 4. Separation of Variables Approach

This is the written version of the interactive lesson above. See the full Fourier Analysis course.