Characteristic value problems posed by differential equations arising in hydrodynamics and hydromagnetics

AbstractA study is made of the general eigenvalue problem posed by a differential equation whose solutions must satisfy certain specified boundary conditions. It is shown that an adjoint eigenvalue problem can be defined and that, in general, it can be represented by a differential equation of the same order as that of the original system together with boundary conditions as numerous as those originally specified. This adjoint system generally differs from the original system but has the same eigenvalues and, moreover, each of its eigensolutions is orthogonal to every eigensolution of the original set except the one(s) belonging to the same eigen-value. It is shown that the representation of this adjoint system is not, in general, unique, and, if it can be chosen to be identical with the original system, the problem is self-adjoint. A variational method is given for determining the eigenvalues.To illustrate the application of the method, three problems are considered which are not self-adjoint: the stability of Couette flow, the onset of convection in a rotating sphere heated within, and self-maintained dynamo action in a conducting fluid of infinite extent