Differential Equations
PDE Boundary Value Problems
Ansys earned 2.3 billion dollars in 2023 by helping Boeing, Tesla, and NASA solve boundary value problems for PDEs. Every finite-element computation is a discretization of the heat equation, wave equation, or Laplace equation with boundary conditions.
- Tesla: battery thermal management via heat equation with spatially varying conductivity
- Dolby: concert hall acoustics via wave equation with absorbing boundary conditions
- Ansys HFSS: electromagnetic simulation via Maxwell equations (a system of Laplace-type PDEs)
Heat Equation
Tesla uses the heat equation to design battery modules: during 100 kW charging, uneven heating reduces lithium-ion cell lifespan by 15-30%. The heat equation describes thermal diffusion across space and time, and forms the basis of battery thermal management systems.
Heat equation u_t = αu_xx on [0,L] with zero boundary conditions. How do high-frequency modes decay compared to low-frequency modes?
Mode n decays as e^{-α(nπ/L)²t}. The exponent is proportional to n² -- high-frequency modes (large n) decay n² times faster. This explains why temperature distributions smooth out over time: fine details disappear first.
Wave Equation
Dolby Laboratories, with a 200 million dollar contract in 2003 for IMAX theaters, used the wave equation to design hall acoustics. Sound propagation in a room obeys u_tt = c² u_xx with boundary conditions at the walls -- the same equation governing vibrating strings and electromagnetic waves.
A string of length L=1 m with fixed ends, wave speed c=340 m/s. What is the fundamental frequency f_1?
Fundamental frequency f_1 = c/(2L) = 340/(2×1) = 170 Hz. The formula f_n = nc/(2L) gives harmonics: 170, 340, 510 Hz. This is the standard calculation for acoustic instruments and organ pipes.
Laplace Equation and Harmonic Functions
Ansys, engineering simulation software with 2.3 billion dollar revenue in 2023, builds most simulations on the Laplace equation. Electrostatics, steady-state heat transfer, potential flow -- all are Laplace problems in domains with complex boundary conditions, solved by FEM discretization.
The Laplace equation Delta(u)=0 in domain Omega with u=5 on the entire boundary. What is u inside?
By the maximum principle, the maximum and minimum of a harmonic function are attained on the boundary. With u=5 everywhere on the boundary, max=min=5, so u=5 inside. This reflects the physical fact: constant temperature on the boundary gives constant temperature at steady state.
Key results
- Heat equation: sine Fourier series solution; high-frequency modes decay faster
- Wave equation: standing waves, d'Alembert formula, frequencies omega_n = nπc/L
- Laplace equation: harmonic functions, maximum principle, separation of variables