A Non-Self-Adjoint Problem in Heat Conduction

Linear boundary value problems in heat conduction have been investigated extensively for a diverse variety of boundary conditions, but almost invariably these have been of the type that produce self-adjoint problems. The property of selfadjointness makes it possible to obtain readily the solution of steady and unsteady-state heat conduction problems in the form of a series expansion of orthogonal eigenfunctions. Fortunately, in a good many situations, the boundary conditions fit the physics naturally. For example, in onedimensional heat conduction problems the boundary conditions at the ends are always "unmixed" in the sense that each boundary condition involves only one or the other endpoint. Boundary value problems of this type are always selfadjoint. A situation, in which, self-adjointness arises from boundary conditions featuring both endpoints is that which employs the familiar 'periodicity' criteria characteristic of circular domains. In general, however, "mixed" boundary conditions produce non-self-adjoint problems, for which orthogonal eigenfunctions do not exist so that the solution to the problem cannot be obtained by the technique applicable to the self-adjoint problems. Frequently, the non-self-adjoint problem defines an adjoint problem with a common set of eigenvalues and sets of eigenvectors which are biorthogonal. If each of the above sets of biorthogonal vectors is complete, then a solution to the non-self-adjoint problem is possible in the form of a series solution. Completeness can certainly not be taken for granted since non-self-adjoint problems can display very strange behavior in regard to their eigenvalues and eigenvectors. It is sometimes possible to test for completeness of the eigenvectors by rather powerful theorems (see Chapter V of [1]).