SOURCE RECORD · wikipedia
Eikonal equation
An eikonal equation (from Greek εἰκών, image) is a non-linear first-order partial differential equation that is encountered in problems of wave propagation. The classical eikonal equation in geometric optics is a differential equation of the form where x {\displaystyle x} lies in an open subset of R n {\displaystyle \mathbb {R} ^{n}} , n ( x ) {\displaystyle n(x)} is a positive function, ∇ {\displaystyle \nabla } denotes the gradient, and | ⋅ | {\displaystyle |\cdot |} is the Euclidean norm. The function n {\displaystyle n} is given and one seeks solutions u {\displaystyle u} . In the context of geometric optics, the function n {\displaystyle n} is the refractive index of the medium. More generally, an eikonal equation is an equation of the form where H {\displaystyle H} is a function of 2 n {\displaystyle 2n} variables. Here the function H {\displaystyle H} is given, and u {\displaystyle u} is the solution. If H ( x , y ) = | y | − n ( x ) {\displaystyle H(x,y)=|y|-n(x)} , then equation (2) becomes (1). Eikonal equations naturally arise in the WKB method and the study of Maxwell's equations. Eikonal equations provide a link between physical (wave) optics and geometric (ray) optics. One fast computational algorithm to approximate the solution to the eikonal equation is the fast marching method.