FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2006, VOLUME 12, NUMBER 1, PAGES 205-236

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

L. B. Pereyaslavskaya

Abstract

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We consider an approximate construction of the surface $S$ being the graph of a $C2$-smooth solution $z\; =\; z(x,y)$ of the parabolic Monge--Ampère equation

of a special form with the initial conditions

where $a\; =\; a(y)$ and $b\; =\; b(y)$ are given functions. In the method proposed, the desired solution is approximated by a sequence of $C1$-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 $n\; \to$¥ are presented; this allows one to construct a numerical solution of the problem with any accuracy given in advance.

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Last modified: July 8, 2006