The numerical solution of elliptic differential equations when the boundary conditions involve a derivative

Abstract

The numerical solution of differential equations involves the replacement of derivatives by finitedifference equivalents. The idea of using approximate equations and subsequently correcting for the higher differences, already applied to second-order equations with specified boundary values, is here extended to the case where conditions at the boundary involve a derivative. The method is applied with examples to second- and fourth-order equations. The more difficult problems associated with curved boundaries are discussed, with particular reference to problems of stretching of flat elastic plates. An alternative but more laborious method of obtaining accurate solutions, the method of ‘the deferred approach to the lim it’, is illustrated by examples.