
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.
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 MongeAmp\`ere equations by surfaces reduced to developable surfaces
\jour Fundam. Prikl. Mat.
\yr 2006
\vol 12
\issue 1
\pages 205236
\mathnet{http://mi.mathnet.ru/fpm928}
\mathscinet{http://www.ams.org/mathscinetgetitem?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 9961020
\crossref{https://doi.org/10.1007/s1095800800397}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2s2.038349127670}
Linking options:
http://mi.mathnet.ru/eng/fpm928 http://mi.mathnet.ru/eng/fpm/v12/i1/p205
Citing articles on Google Scholar:
Russian citations,
English citations
Related articles on Google Scholar:
Russian articles,
English articles

Number of views: 
This page:  184  Full text:  65  References:  23  First page:  1 
