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.
UDC:
517.956