Approximation of solutions of the Monge--Amp\`ere equations by surfaces reduced to developable surfaces

L. B. Pereyaslavskaya

Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 1, pp. 205-236

Résumé

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.