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

$(z$_{xx} + a)(z_{yy} + b) -
z_{xy}^{2} = 0
of a special form with the initial conditions

$z(x,0)\; =$f(x), q(x,0) = y(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 $C1$-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.

Location: http://mech.math.msu.su/~fpm/eng/k06/k061/k06107h.htm

Last modified: July 8, 2006