RT Journal Article
SR Electronic
T1 Relaxation methods applied to engineering problems. VIII. Plane-potential problems involving specified normal gradients
JF Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences
JO Proc R Soc Lond A Math Phys Sci
FD The Royal Society
SP 129
OP 151
DO 10.1098/rspa.1943.0027
VO 182
IS 989
A1
A1
YR 1943
UL http://rspa.royalsocietypublishing.org/content/182/989/129.abstract
AB Since every plane-harmonic function is associated with a conjugate, problems in which normal gradients are specified on the boundary can be transformed into problems in which boundary values are specified. There then remains, however, the problem of deducing a function ψ from its conjugate ϕ, and this, when the conjugate has been determined only approximately, entails uncertainties which were exemplified in Part V. To minimize the errors of approximate computation ψ and ϕ should be determined severally and independently, consequently a method of direct attack is still needed on problems in which normal gradients are specified. Recent applications have, moreover, presented cases in which the boundary conditions are ‘mixed’, i.e. values are specified at some parts of the boundary, gradients at others. Here, two methods are propounded for the satisfaction of mixed boundary conditions, the first applicable also to cases in which normal gradients alone are specified. Test examples indicate that the wanted extension of method is now available.