  The homogeneous dynamo effect is at the root of cosmic magnetic field
generation. With only a very few exceptions, the numerical treatment of
homogeneous dynamos is carried out in the framework of the differential
equation approach. The present paper tries to facilitate the use of integral
equations in dynamo research. Apart from the pedagogical value to illustrate
dynamo action within the well-known picture of Biot-Savart's law, the integral
equation approach has a number of practical advantages. The first advantage is
its proven numerical robustness and stability. The second and perhaps most
important advantage is its applicability to dynamos in arbitrary geometries.
The third advantage is its intimate connection to inverse problems relevant not
only for dynamos but also for technical applications of magnetohydrodynamics.
The paper provides the first general formulation and application of the
integral equation approach to time-dependent kinematic dynamos in finite
domains. For the spherically symmetric alpha^2 dynamo model it is shown how the
general formulation is reduced to a coupled system of two radial integral
equations for the defining scalars of the poloidal and the toroidal field
components. The integral equation formulation for spherical dynamos with
general velocity fields is also derived. Two numerical examples, the alpha^2
dynamo model with radially varying alpha, and the Bullard-Gellman model
illustrate the equivalence of the approach with the usual differential equation
method.
