(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 $z = z\left(x,y\right)$ of the parabolic Monge--Ampère equation

$\left(z$xx + a)(zyy + b) - zxy2 = 0

of a special form with the initial conditions

$z\left(x,0\right) =$f(x),    q(x,0) = y(x),

where $a = a\left(y\right)$ and $b = b\left(y\right)$ are given functions. In the method proposed, the desired solution is approximated by a sequence of $C1$-smooth surfaces $\left\{S$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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