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


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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

(zxx + a)(zyy + b) - zxy2 = 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 {Sn} 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 Sn 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 Sn 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.

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Last modified: July 8, 2006