A Useful Transformation for Solving the Discrete Beltrami Equation and Reducing a Difference Equation of Second Order to a System of Equations of First Order

Abstract

In the plane elliptic equations of second order with constant coefficients can be transformed into an 2 × 2-equation system of first order by using a coordinate transform. The system can be described by methods of complex analysis. Equations similar to Vekua equations are obtained. From the classical theory it is well known that by using the coordinate transformation the Beltrami equation is fulfilled. The solution of this equation can be represented by the help of the Π -operator. This paper shows how similar results can be obtained in the discrete case. In this case the functions are defined on an uniform lattice with step size h. The aim is not only an approximation of the classical operators. The most important fact is that the discrete operators should preserve the basic properties. Especially a discrete Π -operator is defined which can be used to describe the solution of the discrete Beltrami equation.