Review / Part IV — PDEs & Fourier Series / Ch 21–22

Solving PDEs via SoV + Fourier Series

Ch 21–22 — Part IV — PDEs & Fourier Series

Combining separation of variables with Fourier series solves a wide class of linear PDEs on bounded domains with homogeneous BCs.

Heat equation with general IC

For \(u_t = \alpha u_{xx}\) on \([0,L]\), Dirichlet zero BCs, \(u(x,0) = f(x)\): $$u(x,t) = \sum_{n=1}^\infty b_n \sin\!\left(\frac{n\pi x}{L}\right) e^{-\alpha(n\pi/L)^2 t},$$ $$b_n = \frac{2}{L}\int_0^L f(x) \sin\!\frac{n\pi x}{L}\,dx.$$

Wave equation on a string

For \(u_{tt} = c^2 u_{xx}\) on \([0,L]\), \(u(0,t)=u(L,t)=0\), \(u(x,0)=f(x)\), \(u_t(x,0)=g(x)\): $$u(x,t) = \sum_n \sin\!\frac{n\pi x}{L}\left[b_n \cos\omega_n t + d_n \sin\omega_n t\right],\quad \omega_n = \frac{n\pi c}{L}.$$

Inhomogeneous BCs

Split \(u = u_s + w\) where \(u_s\) is a steady-state satisfying the BCs (and, for heat, Laplace's equation); \(w\) then solves a homogeneous-BC problem.

Practice Problems

Problem 1medium
Solve \(u_t = u_{xx},\; u(0,t)=u(1,t)=0,\; u(x,0) = 2\sin(\pi x) - \sin(4\pi x)\).
Hint
Read off coefficients directly.
Solution

\(u(x,t) = 2\sin(\pi x) e^{-\pi^2 t} - \sin(4\pi x) e^{-16\pi^2 t}\).

Answer: See solution.

Problem 2medium
Solve the wave equation \(u_{tt} = u_{xx}\) on \([0, \pi]\) with \(u=0\) at endpoints, \(u(x,0) = 0,\; u_t(x,0) = \sin(2x)\).
Hint
Only \(d_2\) nonzero.
Solution

\(\omega_n = n\). \(d_n = (2/n)\cdot\) (sine coefficient of \(u_t(x,0)\)). Only \(n=2\) mode: \(d_2 = 1/2\). So \(u(x,t) = \tfrac{1}{2}\sin(2x)\sin(2t)\).

Answer: \(u = \tfrac{1}{2}\sin(2x)\sin(2t).\)

Problem 3medium
How do you handle \(u(0,t)=0,\; u(L,t)=T_1\) for the heat equation?
Hint
Subtract a linear steady state.
Solution

Let \(u_s(x) = T_1 x / L\) (solves steady state and BCs). Then \(w = u - u_s\) satisfies heat equation with zero BCs and initial data \(f(x) - u_s(x)\).

Answer: Split off steady state.