Heat Equation with Initial Temperature

Differential Equations · Axiom Academy

EXAMPLE Heat Equation with Initial Temperature Apply separation of variables and a Fourier sine series to solve the heat equation for a given initial temperature. Solve the heat equation on the rod 0 < x < L , t > 0 , subject to the boundary conditions u(0,t) = 0 and u(L,t) = 0 and the initial temperature . Excellent work! You've solved the heat equation using separation of variables and a Fourier sine series. Here's what we learned: Separation of Variables: Assuming u(x,t) = X(x)T(t) splits the PDE into two ordinary differential equations. Boundary Conditions Determine Eigenvalues: u(0,t) = u(L,t) = 0 forces eigenvalues with eigenfunctions . Fourier Series Representation: The initial condition fixes the coefficients B_n through a Fourier sine series expansion. Orthogonality is Key: The integral formula for B_n comes from the orthogonality of sine functions on [0,L] . Physical Interpretation: Each mode decays exponentially at rate — higher modes ( n larger) decay faster, so a rod's temperature profile smooths out over time. This method extends to any initial temperature: separate variables, apply the boundary conditions, then match the initial condition using a Fourier series!

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