Parabolic partial differential equation

A partial differential equation whose principal part has parabolic structure, typically describing diffusive evolution over time. It includes the heat equation and many models of diffusion.

Heat equation: The equation that models heat diffusion, with time evolution proportional to spatial curvature. It is the canonical linear example of parabolic evolution.
Partial differential equation: An equation involving an unknown function and its partial derivatives with respect to multiple variables. Parabolic equations are one structural class of partial differential equations.
Elliptic partial differential equation: A partial differential equation whose principal symbol is definite, commonly governing spatial equilibrium. Unlike parabolic equations, elliptic equations usually impose constraints across space rather than evolve in time.
Advection–diffusion equation: An equation combining transport by a flow with spreading by diffusion. Its diffusion term gives the equation parabolic structure in common formulations.
Fundamental solution: A solution that generates solutions for a linear differential equation through superposition or integration. For the heat equation, it describes how a point source spreads over time.
Principal symbol: The polynomial formed from the highest-order derivative terms of a differential operator. Its degeneracy and signature identify the equation's principal structure.
Hyperbolic partial differential equation: A partial differential equation whose principal part supports wave-like propagation with finite-speed signals. Its wave propagation differs from the smoothing and spreading typical of parabolic equations.
Fokker–Planck equation: An equation for the time evolution of a probability density under drift and random fluctuations. It describes density evolution using drift and diffusion terms.
Maximum principle: A theorem constraining where extrema of solutions to certain differential equations can occur. It captures how parabolic evolution prevents new interior maxima under suitable conditions.
Second-order partial differential equation: A partial differential equation whose highest derivatives have order two. The familiar parabolic classification is based on second-order principal parts.

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