Partial Differential Equations glossary

Clear, one-line definitions of the Partial Differential Equations terms used across the OgbonLab textbooks. Each entry links to the interactive sections where the idea is taught.

antinode

A point on a standing wave where the oscillation amplitude is maximum; located midway between consecutive nodes.

dalembert formula

The solution u(x,t) = ½(f(x+ct) + f(x-ct)) + (1/2c)∫_{x-ct}^{x+ct} g(s) ds of the 1D wave equation with initial position f and velocity g.

dirichlet problem

A boundary-value problem for Laplace's equation Δu = 0 on a domain with prescribed values of u on the boundary.

heat equation

The parabolic PDE ∂u/∂t = α Δu describing diffusion of heat (or any quantity) with diffusivity α.

See: The Heat Equation

laplace equation

The second-order PDE Δu = ∂²u/∂x² + ∂²u/∂y² + ... = 0; its solutions are called harmonic functions.

natural frequency

A frequency at which an undamped system oscillates freely after a disturbance; for a vibrating string, an integer multiple of c/2L.

node

A point on a standing wave where the displacement is always zero; in graph theory, a synonym for vertex.

principal directions

The orthogonal eigenvector directions of a symmetric tensor (e.g. stress, strain); along these, the tensor acts as pure scaling.

standing wave

A wave pattern u(x,t) = X(x)T(t) with fixed nodes that does not propagate; produced by superposition of equal-amplitude waves traveling in opposite directions.

superposition principle

For a linear equation, any linear combination of solutions is again a solution; the structural foundation of Fourier methods.

thermal diffusivity

The constant α = k/(ρc) in the heat equation ∂u/∂t = α Δu, governing how quickly temperature differences spread through a material.

wave equation

The hyperbolic PDE ∂²u/∂t² = c² Δu describing propagation of waves at speed c through a medium.

See: The Wave Equation, Deriving the Wave Equation

wave speed

The constant c in the wave equation ∂²u/∂t² = c² ∂²u/∂x²; the rate at which a disturbance propagates through the medium.