NUMERICAL CONFORMAL MAPPINGS AND THEIR APPLICATIONS

Abstract

Conformal mapping has been a familiar tool of science and engineering for generations. The practical limitations has always been that only for certain special domains exact conformal maps are known, while for the rest, they must be computed numerically. It plays an important role in many two dimensional problems in classical electrostatics, electric current flow, and non-viscous hydro dynamics. Conformal mapping helps to transfer the problem of the calculation of electric potential and temperature etc. from the given body of an arbitrary form (any profile of a section) to a simplest form for which we already know how to solve the problem. Another application where conformal mappings are used is in cartography for making geographical maps. The methods of numerical conformal mapping divide broadly into those which construct the map from a standard domain such as the unit disc onto the problem domain, and those which construct the map in the reverse direction. Although there are alternative approaches, the most commonly used methods are derived from integral equations involving the 'boundary corres-pondence function' that relates the two boundaries pointwise. Typically the boundaries are discretized into n-points, so that the integral equation reduces to an algebraic system. In mapping from problem domain to standard domain, the integral equations are generally linear. The standard methods of this type are perhaps those based on Symm's equation, a singular linear integral equation of the first kind derived from a representation of the conformal map in terms of a single layer potential. Szego kernel method was not~in the vogue in—numerical—conformal_mapping which was due to the difficulty faced in obtaining Szego kernel numerically. Kerzman and Trummer have introduced a second kind integral equation whose solution is Szego kernel which has avoided the ortho-normalization and hence the difficulty faced earlier.