Eikonal equation

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The eikonal equation is a non-linear partial differential equation of the form

$| \nabla u(x)|=F(x), \ x\in \Omega$

subject to $u|_{\partial \Omega}=0$ , where $Ω$ is an open set in $\mathbb{R}^n$ with well-behaved boundary, $F (x)$ is a function with positive values, $\nabla$ ; denotes the gradient and |·| is the Euclidean norm. Here, the right-hand side $F (x)$ is typically supplied as known input. Physically, the solution $u (x)$ is the shortest time needed to travel from the boundary $\partial \Omega$ to $x$ inside $Ω,$ with $F (x)$ being the time cost (not speed) at $x$ .

A fast computational algorithm to approximate the solution to the eikonal equation is the fast marching method. In the special case when $F = 1$ , the solution gives the signed distance from $\partial \Omega$ .

The eikonal equation is encountered in problems of wave propagation, when the wave equation is approximated using the WKB theory.

The eikonal equation is derivable from Maxwell's equations of electromagnetics. It is the link between physical (wave) optics and geometric (ray) optics.

[edit] Physical interpretation

The physical meaning of the eikonal equation is related to the formula

$E = -\nabla \Omega$

where $E$ is the electric field intensity and $Ω$ is the electric potential. There is a similar equation for velocity potential in fluid flow and temperature in heat transfer. The physical meaning of this equation in the electromagnetic example is that any charge occurring in the region is pushed outward at a right angle from lines of constant potential and this charge travels along lines of constant force given by the field of the E vector. Corresponding variables occur in thermodynamics and fluid flow. Ray optics and electromagnetics are related by the fact that the eikonal equation gives a second electromagnetic formula of the same form as the potential equation above where the line of constant potential has been replaced by a line of constant phase and the force lines have been replaced by normal vectors coming out of the constant phase line at right angles. The magnitude of these normal vectors is given by the square root of the relative permittivity. The line of constant phase can be considered the edge of one of the advancing light waves. The normal vectors are the rays the light is traveling down in ray optics. This explanation is in the RMKS system of units used by electrical engineers.