Geodesic
Numerical-analytic construction of a~generalized solution to the eikonal equation in the plane case
Sbornik. Mathematics, Tome 215 (2024) no. 9, pp. 1224-1248

Voir la notice de l'article provenant de la source Math-Net.Ru

In the 1970s Kruzhkov introduced the concept of a generalized solution to the eikonal equation, and, for a medium of constant refractive index, indicated a class of functions containing the generalized solution to the Dirichlet boundary value problem. We present constructive methods for finding such a solution in the plane case. Nonsmooth singularities are known to develop in generalized solutions owing to the presence of pseudovertices, which are singular points of the boundary of the boundary set. Identifying such pseudovertices is related to finding fixed points of mappings formed in a local reparameterization of this boundary. We obtain necessary conditions for the existence of pseudovertices in the case when the curvature of the parametrically defined boundary of boundary sets is not smooth. These conditions are written as an equation for the pseudovertex marker (a numerical measure of the local nonconvexity of the boundary set). This equation, which has the typical structure of constructions involving fixed points, can be reduced to an algebraic equation. The solution of this equation (the marker) is found analytically if the curvature of the boundary of the boundary set has a nonsmooth extremum at a pseudovertex. We also give an example of a numerical-analytic construction of a generalized solution to the boundary value problem, with indication of the singular set and the evolution of wave fronts. Bibliography: 29 titles.
Keywords: generalized solution, singular set, pseudovertex, fixed point.
Mots-clés : eikonal equation 
@article{SM_2024_215_9_a4,
     author = {P. D. Lebedev and A. A. Uspenskii},
     title = {Numerical-analytic construction of a~generalized solution to the eikonal equation in the plane case},
     journal = {Sbornik. Mathematics},
     pages = {1224--1248},
     publisher = {mathdoc},
     volume = {215},
     number = {9},
     year = {2024},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2024_215_9_a4/}
}
TY  - JOUR
AU  - P. D. Lebedev
AU  - A. A. Uspenskii
TI  - Numerical-analytic construction of a~generalized solution to the eikonal equation in the plane case
JO  - Sbornik. Mathematics
PY  - 2024
SP  - 1224
EP  - 1248
VL  - 215
IS  - 9
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SM_2024_215_9_a4/
LA  - en
ID  - SM_2024_215_9_a4
ER  - 
%0 Journal Article
%A P. D. Lebedev
%A A. A. Uspenskii
%T Numerical-analytic construction of a~generalized solution to the eikonal equation in the plane case
%J Sbornik. Mathematics
%D 2024
%P 1224-1248
%V 215
%N 9
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SM_2024_215_9_a4/
%G en
%F SM_2024_215_9_a4
P. D. Lebedev; A. A. Uspenskii. Numerical-analytic construction of a~generalized solution to the eikonal equation in the plane case. Sbornik. Mathematics, Tome 215 (2024) no. 9, pp. 1224-1248. http://geodesic.mathdoc.fr/item/SM_2024_215_9_a4/