Methods of the kind obtain the viscosity solution of the eikonal equation
through numerical analysis.
It describes SIMD, shared memory, and distributed memory machine models; decomposition as a fundamental activity in parallel algorithmic design; key programming models; key concepts of performance analysis and optimization; and three case studies applying these concepts to the Single Shortest Path Problem, the Eikonal equation
, and computation of the two-dimensional convex hull.
An approximate solution which is valid for short wavelengths is the eikonal equation
(see, e.g., [22, 23]).
Equation (20) constitutes the eikonal equation
[16,17] as well as the dispersion equation for propagation of waves in an inhomogeneous anisotropic media with [??] = [??] and may be written as:
As regards the scientific review, the eikonal equation
is expressed, and some complex-valued solutions are defined corresponding to complex rays and caustics.
The phase of the wave packet can be obtained from the corresponding eikonal equation
, which can be solved numerically on the characteristics (rays).
Expressing the solution U(r) of wave equation in terms of well known Luneberg-Kline series yields Eikonal equation
for phase s(r) and transport equation for amplitude.
It consists of iteratively solving the Eikonal equation
(1982) (29) proved that the mild-slope equation includes the eikonal equation
with diffraction effect and the energy conservation equation between two wave rays.
In , helix submanifolds in Euclidean space were studied by solving the Eikonal equation
. The applications of constant angle surfaces in the theory of liquid crystals and of layered fluids were considered by P.
The eikonal equation
. Waves move through [k.sub.i][dx.sub.i] = -d[PHI], where [k.sub.i] = - [delta][PHI]/[delta][x.sup.i] = (w/c , k), w is the frequency, k is the wavevector and [PHI] is called the eikonal.