Heat in a Plate Examples

PDEs · Axiom Academy

Solve the 2D heat equation on a rectangular domain using separation of variables. Excellent work! You've successfully solved the 2D heat equation using separation of variables. Here's what we learned: 2D Separation of Variables: Assuming u(x,y,t) = X(x)Y(y)T(t) converts the 2D PDE into three separate ODEs, extending the 1D technique to higher dimensions. Double Eigenvalue Problem: Each spatial dimension contributes its own eigenvalues: λ from the x-direction and μ from the y-direction, giving λ_nm = (nπ/L)² + (mπ/L)². Product Eigenfunctions: The spatial eigenfunctions are products: X_n(x)Y_m(y) = sin(nπx/L)sin(mπy/L), forming a complete orthogonal basis on the rectangle. Exponential Decay: Each mode decays at rate exp(-kλ_nm t), with higher modes (larger n or m) decaying faster, smoothing the temperature distribution. Double Fourier Series: The general solution is a double sum over all modes, with coefficients c_nm determined by the initial condition u(x,y,0) = f(x,y) using 2D Fourier series. Physical Interpretation: Heat diffuses from hot regions to cold regions, and the zero boundary conditions mean the edges are held at zero temperature, causing all interior heat to eventually dissipate. This technique generalizes to other 2D PDEs (wave equation, Laplace equation) and to other rectangular domains with different boundary conditions. The key is always to separate variables and solve the resulting eigenvalue problems!

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