Non-linear partial differential equation encountered in problems of wave propagation
An eikonal equation (from Greek εἰκών, image[1][2]) 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
(1)
where lies in an open subset of , is a
positive function, denotes the gradient, and is the Euclidean norm. The function is given and one seeks solutions .
In the context of geometric optics, the function is the refractive index of the medium.
More generally, an eikonal equation is an equation of the form
(2)
where is a function of variables.
Here the function is given, and is the solution.
If , then equation (2) becomes (1).
Eikonal equations naturally arise in the WKB method[3]
and the study of Maxwell's equations.[4] 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.
^The Oxford English Dictionary. 2nd ed. 1989. OED Online. Oxford University Press. 4 April 2000 http://dictionary.oed.com/cgi/entry/00292404
^Evans, L. C. Partial Differential Equations. AMS Graduate Texts in Mathematics. Vol. 19. p. 93.
^Dimassi, Mouez; Sjöstrand, Johannes (1999). Spectral asymptotics in the semi-classical limit. London Math. Society Lecture Notes 268. Cambridge University Press. ISBN 0-521-66544-2.
^Rauch, Jeffrey (2012), Hyperbolic partial differential equations and geometric optics, Graduate Studies in Mathematics, 133, American Mathematical Society, Bibcode:2012hpde.book.....R, ISBN 978-0-8218-7291-8
An eikonalequation (from Greek εἰκών, image) is a non-linear first-order partial differential equation that is encountered in problems of wave propagation...
Eikonal is the German form of the Greek word εἰκών, meaning likeness, icon or image. It can refer to: Eikonalequation, a non-linear partial differential...
physics, the eikonal approximation (Greek εἰκών for likeness, icon or image) is an approximative method useful in wave scattering equations, which occur...
first equation is known as the eikonalequation, which determines the eikonal S ( r ) {\displaystyle S(\mathbf {r} )} is a Hamilton–Jacobi equation, written...
recommendation. Solve the eikonalequation. The first two are only of 1st order approximation (see Orders of approximation). For the eikonalequation, many numerical...
created by James Sethian for solving boundary value problems of the Eikonalequation: | ∇ u ( x ) | = 1 / f ( x ) for x ∈ Ω {\displaystyle |\nabla u(x)|=1/f(x){\text{...
method is a numerical method for solving boundary value problems of the Eikonalequation. | ∇ u ( x ) | = 1 f ( x ) for x ∈ Ω {\displaystyle |\nabla u(\mathbf...
is differentiable almost everywhere, and its gradient satisfies the eikonalequation | ∇ f | = 1. {\displaystyle |\nabla f|=1.} If the boundary of Ω is...
See also Nonlinear partial differential equation, List of partial differential equation topics and List of nonlinear ordinary differential equations....
wave equationEikonalequation in wave propagation Euler–Poisson–Darboux equation in wave theory Helmholtz equation Chua's circuit Liénard equation to model...
\epsilon ^{0}:\quad S_{0}'^{2}=Q(x),} which can be recognized as the eikonalequation, with solution S 0 ( x ) = ± ∫ x 0 x Q ( x ′ ) d x ′ . {\displaystyle...
physical space) of a wave-front issued from a point source, satisfies the Eikonalequation: ‖ ∇ τ ( x ) ‖ = s ( x ) , {\displaystyle \|\nabla \tau (x)\|=s(x)...
u(x_{1}^{0},\dots ,x_{n}^{0})=z^{0}} . Eikonalequation The simplest fully nonlinear equation is the eikonalequation. This has the form | ∇ u | 2 = 1 , {\displaystyle...
the fast marching method for solving boundary value problems of the Eikonalequation, used to model wave propagation. This method finds the times at which...
a more refined argument. The previous boundary value problem is an eikonalequation in a single spatial dimension with f = 1 {\displaystyle f=1} , where...
The above equation is same as Hamilton–Jacobi equation where the eikonal can be considered as the action. Since Hamilton–Jacobi equation is equivalent...
approximation that is often called geometric. It formulates as the Eikonalequation and is only applicable under certain conditions including that the...
is examined by the BKM; in the BKM is also used to solve nonlinear Eikonalequation. Method of fundamental solutions Regularized meshfree method Boundary...
ray tracing which relied on a finite-difference approximation of the eikonalequation and which has been used widely in both earthquake and reflection seismology...
play sports games. Zhao, Hongkai (2005). "A fast sweeping method for Eikonalequations". Mathematics of Computation. 74 (250): 603–627. doi:10.1090/S0025-5718-04-01678-3...
ordered upwind methods for solving static Hamilton–Jacobi equations. In the case of an Eikonalequation, the first method to do so was developed by Jon N. Tsitsiklis...
important "quasi-optical" singularities of the so-called eikonal approximations of many wave equations, namely the "caustics", are resolved into finite peaks...
D. R. (1986). "The eikonal approximation and the asymptotics of the total scattering cross-section for the Schrödinger equation" (PDF). Annales de l'Institut...
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