Inhomogeneous

Inhomogeneous is an adjective describing something that is not uniform in composition, structure, or properties across space or time. The term is widely used in science and engineering to denote spatial or temporal variation within a system.

In mathematics and physics, an inhomogeneous (or nonhomogeneous) equation contains terms that are not solely functions of the unknown variable. In differential equations, a nonzero forcing term makes the equation inhomogeneous. For example, in the ordinary differential equation y'' + y = sin x, the right-hand side is nonzero, so the equation is inhomogeneous; the homogeneous counterpart would be y'' + y = 0. In partial differential equations, an equation such as ∇^2 u = f(x) is inhomogeneous when f(x) ≠ 0.

In physical media, an inhomogeneous material or region has properties that vary with position, such as density, stiffness, refractive index, or conductivity. Examples include graded-index optical fibers, composites with spatially varying composition, and geological formations. Such variations affect wave propagation, diffusion, and mechanical stresses. Methods like homogenization or effective medium theory are used to approximate an inhomogeneous medium by an equivalent homogeneous one for analysis.

In probability and statistics, an inhomogeneous (or nonhomogeneous) Poisson process has a rate that depends on location or time, unlike a homogeneous Poisson process with a constant rate.

Etymologywise, the word combines the prefix in- (not) with homogeneous (of the same kind). In practice, inhomogeneity contrasts with homogeneity and, to some extent, with heterogeneity, though the latter emphasizes distinct components rather than smooth variation.