Separation of variables and Fourier series
Separation of variables splits a PDE into ODEs, and Fourier series assemble the pieces into a solution that matches any reasonable initial condition.
Separating variables
For \(u_t = ku_{xx}\) on \(0 < x < L\) with \(u(0, t) = u(L, t) = 0\), try \(u = X(x)T(t)\). Substituting gives \(\frac{T'}{kT} = \frac{X''}{X}\). The left depends only on \(t\) and the right only on \(x\), so both equal a constant \(-\lambda\). This yields two ODEs: \(X'' + \lambda X = 0\) and \(T' = -k\lambda T\).
The spatial eigenvalue problem
With \(X(0) = X(L) = 0\), the earlier boundary value problem gives eigenvalues \(\lambda_n = \left(\frac{n\pi}{L}\right)^2\) and eigenfunctions \(X_n = \sin\frac{n\pi x}{L}\). The time equation then gives \(T_n = e^{-k\lambda_nt}\). Each product is a decaying sine mode.
Building the solution
By superposition, \(u(x, t) = \sum_{n=1}^{\infty} b_n\sin\frac{n\pi x}{L}e^{-k(n\pi/L)^2t}\). The coefficients \(b_n\) are chosen so that \(u(x, 0) = f(x)\), the initial temperature. Higher modes decay much faster, so the solution quickly becomes smooth and dominated by the first mode.
u = \sum b_n \sin\tfrac{n\pi x}{L}\, e^{-k(n\pi/L)^2 t}Fourier sine series
To match \(f(x) = \sum b_n\sin\frac{n\pi x}{L}\), use orthogonality: multiply by \(\sin\frac{m\pi x}{L}\) and integrate. Every term except \(n = m\) vanishes, giving \(b_n = \frac{2}{L}\int_0^L f(x)\sin\frac{n\pi x}{L}\,dx\). Joseph Fourier introduced this idea in 1807 to study heat, and it revolutionised mathematics.
b_n = \frac{2}{L}\int_0^L f(x)\sin\frac{n\pi x}{L}\,dxWorked example
For \(f(x) = 1\) on \([0, \pi]\) (with \(L = \pi\)), \(b_n = \frac{2}{\pi}\int_0^\pi \sin nx\,dx = \frac{2}{\pi}\cdot\frac{1 - \cos n\pi}{n}\), which is \(\frac{4}{n\pi}\) for odd \(n\) and 0 for even \(n\). So \(1 = \frac{4}{\pi}\left(\sin x + \frac{\sin 3x}{3} + \frac{\sin 5x}{5} + \cdots\right)\) on \((0, \pi)\).
The vibrating string
For \(u_{tt} = c^2u_{xx}\) with fixed ends, separation gives modes \(\sin\frac{n\pi x}{L}\left(a_n\cos\frac{n\pi ct}{L} + b_n\sin\frac{n\pi ct}{L}\right)\). Instead of decaying, each mode oscillates at frequency \(\frac{nc}{2L}\) cycles per unit time: the fundamental and its harmonics. The mix of \(a_n\) and \(b_n\) gives an instrument its timbre.
Full Fourier series and convergence
On \([-L, L]\), a function can be expanded in both sines and cosines. For piecewise smooth functions, the series converges to the function at points of continuity and to the average of the left and right limits at jumps. Near jumps, partial sums overshoot by about 9% (the Gibbs phenomenon), no matter how many terms are used.
Common mistakes
Forgetting that both sides of the separated equation equal the same constant. Using the wrong interval or normalisation in the Fourier coefficient formula. Ignoring even \(n\) terms that are zero. Expecting a Fourier series to converge to the function value exactly at a jump.
Key takeaways
Separation of variables \(u = X(x)T(t)\) turns a PDE into ODEs linked by a separation constant. Boundary conditions give eigenfunctions; superposition gives a series solution. Fourier coefficients come from orthogonality. Heat modes decay, string modes oscillate at harmonic frequencies, and series converge to averages at jumps.
Flashcard
u = X(x)T(t)
Check your understanding
In separation of variables for the heat equation, why must X''/X and T'/(kT) equal the same constant?