Partial Differential Equations
Partial differential equations relate how quantities change across space and time, and they describe nearly all continuum physics.
What they express
A partial differential equation (PDE) links the rates at which a quantity changes with respect to several variables at once — typically space and time. Heat diffusion, fluid flow, electromagnetism, and plasma transport are all governed by PDEs.
Three archetypes
- Elliptic (e.g. Laplace’s equation): steady-state balance, like an equilibrium field.
- Parabolic (e.g. the heat equation): smooth diffusion over time.
- Hyperbolic (e.g. the wave equation): signals and disturbances that propagate.
Boundary and initial conditions
A PDE alone does not determine a unique answer. It must be paired with boundary conditions describing behavior at the edges of the domain and, for time-dependent problems, initial conditions at the start. Getting these wrong is a common source of confidently incorrect results.
Solving them
Except in simple cases, PDEs have no closed-form solution and are solved numerically — by finite element, finite volume, or finite difference methods on a mesh. The choice of method depends on the equation type, the geometry, and the accuracy required.
In fusion modeling
Plasma equilibrium, magnetic field configuration, neutron transport, and component heating are all PDE problems. The magnetohydrodynamic equilibrium that sets a tokamak’s shape — including the negative triangularity of about −0.30 chosen for the breeder Hyperion — is found by solving an elliptic PDE for the magnetic flux.