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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

State Academy of Consumer Services

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.

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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
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\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}


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  • Фундаментальная и прикладная математика
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