Heat in a Rod Examples

PDEs · Axiom Academy

Solving the heat equation using separation of variables Solve the heat equation on the interval [0,π] with the following conditions: Physical interpretation: A rod of length π has both ends held at temperature zero (ice baths), and starts with an initial temperature distribution sin(x). Excellent work! You've successfully solved the heat equation using separation of variables. Here are the essential steps: Separation of Variables: We assume u(x,t) = X(x)T(t), which transforms the PDE into two ODEs. This only works when boundary conditions are homogeneous (equal to zero). Eigenvalue Problem: The spatial equation X'' + λX = 0 with X(0) = X(π) = 0 is a Sturm-Liouville problem. The eigenvalues are λ n = n² and eigenfunctions are X n (x) = sin(nx). Temporal Decay: Each eigenvalue gives a temporal solution T n (t) = e -n²t . Higher modes (larger n) decay faster, which is physically intuitive. Superposition Principle: The general solution is u(x,t) = Σ c n sin(nx)e -n²t . We use the initial condition to determine the coefficients c n . Special Case: When the initial condition exactly matches one eigenfunction (sin(x) here), only that mode is present. For general f(x), we'd need Fourier series expansion. Physical Interpretation: The solution shows exponential decay to the equilibrium state (u = 0 everywhere). The temperature profile maintains its sine shape but decreases in amplitude as heat diffuses out through the boundaries.

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