heatsimilar

Heatsimilar is a term used in the study of heat conduction and diffusion processes to describe systems, solutions, or profiles that exhibit scale-invariant behavior under the natural parabolic scaling of the heat equation.

The heat equation ∂t u = κ Δu on R^n is invariant under the scaling u(x,t) → uλ(x,t) = u(λx, λ^2 t). A function is heatsimilar if, up to an overall time-dependent factor, its spatial profile depends only on x/√t. Self-similar solutions have the form u(x,t) = t^{-n/2} F( x/√t ) with F determined by an ordinary differential equation.

The Gaussian heat kernel u(x,t) = (4πκ t)^{-n/2} exp( - |x|^2 / (4κ t) ) is the canonical heatsimilar solution. Other examples include Barenblatt-type solutions for nonlinear diffusion equations, which also exhibit heat-like self-similarity under appropriate scaling.

In practice, heatsimilarity is used to analyze long-time behavior, to reduce partial differential equations to ordinary differential equations using similarity variables, and to compare diffusion in heterogeneous materials by rescaling experiments. The term is not as standard as “self-similar,” but it appears in some texts exploring the connection between diffusion and scaling.

See also: heat equation, self-similar solution, scaling invariance, parabolic dilation, diffusion. References: standard texts on partial differential equations for background on self-similarity and diffusion, such as introductory PDE manuals.