PDEs in science and engineering
From cooling electronics to medical imaging and finance, PDEs are everywhere. This lesson surveys how they are applied and solved in practice.
Heat in two and three dimensions
The heat equation generalises to \(u_t = k\nabla^2u\), where \(\nabla^2u = u_{xx} + u_{yy} + u_{zz}\) is the Laplacian. Engineers use it to design heat sinks for computer chips, insulation for buildings, and cooling for batteries. Steady states satisfy Laplace's equation \(\nabla^2u = 0\).
u_t = k\nabla^2 uWaves in higher dimensions
The wave equation \(u_{tt} = c^2\nabla^2u\) describes sound in rooms, ripples on ponds, earthquakes, and electromagnetic waves. On a circular drum, separation in polar coordinates produces Bessel functions, linking this unit to series solutions.
Schrodinger's equation
Quantum mechanics is governed by Schrodinger's equation \(i\hbar\psi_t = -\frac{\hbar^2}{2m}\nabla^2\psi + V\psi\), a PDE for the wavefunction \(\psi\). Separation of variables gives energy levels as eigenvalues: the discrete energies of electrons in atoms come from boundary value problems.
Diffusion in biology and finance
Diffusion equations model oxygen spreading into tissue, pattern formation in animal coats (reaction-diffusion), and option prices in finance. The Black-Scholes equation for option pricing can be transformed into the heat equation, which is how its famous solution was found.
Fluid flow
The Navier-Stokes equations, nonlinear PDEs for fluid velocity and pressure, describe air over wings, blood in arteries, and ocean currents. Whether smooth solutions always exist in three dimensions is one of the Clay Millennium Prize Problems, with a \$1 million reward.
Numerical solution of PDEs
Most real PDEs are solved numerically. Finite difference methods replace derivatives with differences on a grid; finite element methods divide a region into small elements and are standard in structural engineering. Stability conditions matter: for the explicit heat scheme, the time step must satisfy \(\Delta t \le \frac{(\Delta x)^2}{2k}\).
Medical imaging
CT and MRI reconstruction rely on transforms closely related to Fourier analysis. MRI machines measure signals that are essentially Fourier coefficients of tissue properties, and computers invert them to build images. Fourier's 1807 ideas now save lives daily.
Common mistakes
Choosing numerical time steps that violate stability conditions, causing results to explode. Ignoring boundary conditions when modelling real objects. Assuming linear models when the physics is nonlinear, as with turbulent flow.
Key takeaways
The heat, wave, and Laplace equations generalise to higher dimensions with the Laplacian. PDEs underlie quantum mechanics, finance, biology, fluids, and imaging. Real problems are solved with finite difference and finite element methods, which must respect stability conditions.
Flashcard
Laplacian
Check your understanding
Which famous PDE problem carries a one million dollar prize?