RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Journal history Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Fundam. Prikl. Mat.: Year: Volume: Issue: Page: Find

 Fundam. Prikl. Mat., 2006, Volume 12, Issue 1, Pages 205–236 (Mi fpm928)

Approximation of solutions of the Monge–Ampère equations by surfaces reduced to developable surfaces

L. B. Pereyaslavskaya

Abstract: We consider an approximate construction of the surface $S$ being the graph of a $C^2$-smooth solution $z=z(x,y)$ of the parabolic Monge–Ampère equation
$$(z_{xx}+a)(z_{yy}+b)-z_{xy}^2=0$$
of a special form with the initial conditions
$$z(x,0)=\varphi(x),\quad q(x,0)=\psi(x),$$
where $a=a(y)$ and $b=b(y)$ are given functions. In the method proposed, the desired solution is approximated by a sequence of $C^1$-smooth surfaces $\{S_n\}$ each of which consists of parts of surfaces reduced to developable surfaces. In this case, the projections of characteristics of the surface $S$ being curved lines in general are approximated by characteristic projections of the surfaces $S_{n}$ being polygonal lines composed of $n$ links. The results of these constructions are formulated in the theorem. Sufficient conditions for the convergence of the family of surfaces $S_{n}$ to the surface $S$ as $n\to\infty$ are presented; this allows one to construct a numerical solution of the problem with any accuracy given in advance.

Full text: PDF file (274 kB)
References: PDF file   HTML file

English version:
Journal of Mathematical Sciences (New York), 2008, 149:1, 996–1020

Bibliographic databases:

UDC: 517.956

Citation: L. B. Pereyaslavskaya, “Approximation of solutions of the Monge–Ampère equations by surfaces reduced to developable surfaces”, Fundam. Prikl. Mat., 12:1 (2006), 205–236; J. Math. Sci., 149:1 (2008), 996–1020

Citation in format AMSBIB
\Bibitem{Per06} \by L.~B.~Pereyaslavskaya \paper Approximation of solutions of the Monge--Amp\ere equations by surfaces reduced to developable surfaces \jour Fundam. Prikl. Mat. \yr 2006 \vol 12 \issue 1 \pages 205--236 \mathnet{http://mi.mathnet.ru/fpm928} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2249685} \zmath{https://zbmath.org/?q=an:1153.35326} \elib{http://elibrary.ru/item.asp?id=9166890} \transl \jour J. Math. Sci. \yr 2008 \vol 149 \issue 1 \pages 996--1020 \crossref{https://doi.org/10.1007/s10958-008-0039-7} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-38349127670} `