Heat Equation on [0,L]

Differential Equations · Axiom Academy

EXAMPLE Solving the Heat Equation on [0,L] Use separation of variables to find eigenvalues, eigenfunctions, and build the series solution. Solve the heat equation on the interval [0,L] with both ends held at zero temperature: Excellent work! You've solved the heat equation using separation of variables. Here's what we learned: Separation of variables: Assuming u(x,t) = X(x)T(t) converts the PDE into two ODEs, making the problem tractable. Eigenvalue problem: The boundary conditions give the Sturm–Liouville problem with X(0) = X(L) = 0 , yielding eigenvalues . Eigenfunctions: Each eigenvalue corresponds to an eigenfunction , forming an orthogonal basis on [0,L] . Exponential decay: The time component shows higher modes (larger n ) decay faster, smoothing the solution over time. Series solution: The general solution is a superposition of all modes, with coefficients c_n determined from the initial condition via a Fourier sine series. This same technique applies to many other PDEs with different boundary conditions — the key is identifying the right eigenvalue problem for your specific boundaries.

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