Heat Equation in 2D

PDEs · Axiom Academy

Diffusion on a plate: from point sources to separable solutions 1 The Equation: Heat Flow in Two Dimensions In one dimension, heat flows along a rod. In two dimensions, heat spreads across a thin plate. The 2D heat equation describes how temperature evolves over time on a surface. u(x,y,t) = temperature at position (x,y) and time t k = thermal diffusivity constant (material property) ²u = Laplacian = u xx + u yy (sum of second spatial derivatives) 2 Separation of Variables in a Rectangle For a rectangular domain with homogeneous boundary conditions, we can separate the solution into independent functions of x, y, and t. Substituting into the heat equation and dividing by XYT: Since each term depends on different independent variables, each must equal a constant: This gives us three ordinary differential equations: 3 Double Fourier Series Solutions For a rectangle [0,L x ] × [0,L y ] with zero boundary conditions, the solution is a double Fourier series. Spatial modes: sin(n x/L x )sin(m y/L y ) are the eigenfunctions Eigenvalues: n ² = (n /L x )², m ² = (m /L y )² Time decay: Each mode decays exponentially with rate k( n ² + m ²) Coefficients A nm : Determined by initial conditions u(x,y,0) 4 Visualizing Heat Diffusion on a Plate Watch how heat spreads from an initial point source or pattern on a square plate. The animation shows the temperature distribution evolving over time. 5 Circular Domains and Bessel Functions (Preview)

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