Solution to the Glarum model with a finite relaxation rate

The one-dimensional version of the Glarum model of target relaxation by diffusing defects is generalized by considering a finite relaxation rate upon an encounter defect-target. This eliminates the difficulties associated with the instantaneous relaxation of the targets initially occupied by a defect. The diffusion and relaxation processes are described in terms of the master equation for a continuous-time random walk. An exact solution valid for all times and relaxation rates is obtained by the use of the van Kampen-Oppenheim method. The transition from exponential relaxation at short times to fractional exponential behaviour at long times is described in detail.