Steady currents in composite conductors, eigenfunction expansions, and exactly solvable linear integral equations

Abstract

A novel connection is uncovered between the simple physics of steady current flow in a composite conductor and the theory of integral equations. With a judicious choice of eigenfunction expansions, exploitation of the physical continuity of current flow across a chosen interface in a composite conductor is shown to yield an infinite class of integral equations with exact closed-form solutions. The mathematical derivation of this class is based on the elementary (but also new) notion of formally equating two different eigenfunction expansions of a given arbitrary function. The new class contains as special cases the celebrated Abel integral equation of classical mechanics and the Kramers–Kronig relations of electromagnetic scattering. But it also contains new integral equations (with exact solutions), some with the Cauchy-singularity 1/(x−y) in their kernels, and a new summation equation. These new equations are in themselves intriguing and their exact solutions do not appear to be derivable by the known methods for solving integral equations. An application of the new class of integral equations is given in the context of a particular composite conductor, which consists of a semi-infinite strip imbedded in an otherwise homogeneous whole space conductor (containing a uniform current flow parallel to the strip). The coefficient in the eigenfunction expansion of the potential in the strip satisfies a one-dimensional singular integral equation with a Cauchy-singularity. This singularity is regularized by the application of an integral equation and its exact solution from the new class, resulting in an integral equation with a smooth kernel. This equation together with the eigenfunction expansion provides an elegant representation for the potential in the strip. (The only known exact solutions are for the cases of elliptic-cylinder and ellipsoid geometries in two and three dimensions, respectively.) The new class of integral equations yields the first examples of singular kernels which possess a bilinear expansion in terms of two different complete sets of eigenfunctions, with only the diagonal terms (i.e., those terms in which the summation indices or integration variables are equal) in the expansion being nonzero. Such an expansion for square-integrable kernels (as opposed to singular kernels) is well known in the Hilbert–Schmidt theory of Hermitian operators and in Schmidt’s extension to the non-Hermitian case, and it forms the basis for a method of solving Fredholm integral equations. None of these theories, however, yields the bilinear expansions for the singular kernels of our new class.