A new explicit axisymmetric solver for the diffusion of electromagnetic fields in an inhomogeneous medium is described. The proposed method is based on the Krylov subspace (Lanczos) approximation of the solution in frequency domain. The finite-element method (FEM) is employed to discretize Maxwell's equations. It is shown that the spectral Lanczos decomposition method (SLDM) is extrernely fast. Furthermore, the electromagnetic fields at many frequencies can be evaluated by performing the SLDM iterations only at the lowest frequency. Analysis of some practical geophysical problems is included. A comparison is also made with the analytical results for a homogeneous medium.