An integral equation approach to unsteady kinematic dynamos

Natural and laboratory homogeneous dynamos can be modeled by supposing that the electrically conducting fluid fills a finite volume D surrounded by an insulating infinite region D'. Our considerations are restricted to the kinematic dynamo regime in which the back-reaction of the self-excited magnetic field on the flow is negligible. We also assume the velocity field to be steady. Even for such kinematic dynamo problems an analytical treatment is usually impossible, hence numerical methods have to be utilized.
A notorious problem with the usual differential equation approach is the handling of the non-local boundary conditions for the magnetic field. The integral equation approach provides a solution to the problem of non-local boundary conditions. The integral equation approach has a number of interesting features. The most important one is that it avoids the discretization of the infinite region D'. The second one is its intimate connection to inverse problems relevant not only for dynamos but also for technical applications of magnetohydrodynamics. The third one is its proven numerical robustness and stability.
We examine the integral equation approach by a few numerical examples, including the alpha^2 dynamo model with radially varying alpha, and the Bullard-Gellman model, and demonstrate its equivalence with the differential equation approach.