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

Abstract

We consider an approximate construction of the surface S that is 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 special form with the initial conditions

$$z(x,0) = \varphi (x),q(x,0) = \psi (x),$$

, 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 that are curved lines in general are approximated by characteristic projections of the surfaces S _n that are 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 → ∞ are presented; this allows one to construct a numerical solution of the problem with any accuracy given in advance.