Fourier Series Solutions

PDEs · Axiom Academy

The separation of variables method for solving the heat equation We start by solving the heat equation with homogeneous boundary conditions. After separation of variables, the spatial part gives us an eigenvalue problem: For a rod of length L with fixed endpoints at zero temperature: These boundary conditions determine which values of λ are allowed (the eigenvalues). Applying the boundary conditions X(0) = 0 and X(L) = 0, we find that the eigenvalues must be: where n = 1, 2, 3, ... These are the only values of λ that satisfy both boundary conditions! 2 Step 2: Finding Eigenfunctions For each eigenvalue λ_n, there is a corresponding eigenfunction X_n(x). These are the basic building blocks of our solution. Each eigenvalue gives us a sine function: These functions automatically satisfy the boundary conditions X_n(0) = 0 and X_n(L) = 0. Each eigenfunction represents a standing wave mode on the rod: n = 1: fundamental mode (half wavelength) n = 2: first harmonic (full wavelength) n = 3: second harmonic (1.5 wavelengths) 3 Step 3: Time Evolution Factors Each eigenfunction has its own time evolution, controlled by an exponential decay factor. Higher modes decay faster! For each mode n, the time evolution is: where k is the thermal diffusivity constant. Notice the exponential decay! Higher frequencies (larger n) decay exponentially faster because the decay rate is proportional to λ_n = (nπ/L)². This means: Mode n = 1 decays slowest (persists longest)

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