Method of variable directions for the Cauchy — Riemann equation system

Abstract

Methods of variable directions bave become widely known in recent years for the numerical integration of parabolic and elliptic differential equations with several space variables (see, e.g. [1]–[4]). An attempt is made in this paper to transfer these methods to elliptic equation systems by the simplest example of the Hilbert problem, which involves the determination of two functions u( z, y) and v( x, y) satisfying the Cauchy — Rlemann system du dx − dv dy = 0, du du + dv dx = 0 (1) and the limiting condition α ( s) ( s) u + β ( s) v = ( s) ( α 2 + β 2 ≠ 0). (2)