Context : We consider the radiative transfer problem in a plane-parallel slab of thermal electrons in the presence of an ultra-strong magnetic field ( B \gtrsim B _ { c } \approx 4.4 \times 10 ^ { 13 } G ) .
Under such conditions , the magnetic field behaves as a birefringent medium for the propagating photons , and the electromagnetic radiation is split into two polarization modes , ordinary and extraordinary , having different cross-sections .
When the optical depth of the slab is large , the ordinary-mode photons are strongly Comptonized and the photon field is dominated by an isotropic component .

Aims : The radiative transfer problem in strong magnetic fields presents many mathematical issues and analytical or numerical solutions can be obtained only under some given approximations .
We investigate this problem both from the analytical and numerical point of view , providing a test of the previous analytical estimates and extending these results introducing numerical techniques .

Methods : We consider here the case of low temperature blackbody photons propagating in a sub-relativistic temperature plasma , which allows us to deal with a semi Fokker-Planck approximation of the radiative transfer equation .
The problem can be treated then with the variable separation method , and we use a numerical technique for finding solutions of the eigenvalue problem in the case of singular kernel of the space operator .
The singularity of the space kernel is the result of the strong angular dependence of the electron cross-section in the presence of a strong magnetic field .

Results : We report the numerical solution obtained for eigenvalues and eigenfunctions of the space operator , and the emerging Comptonization spectrum of the ordinary-mode photons for any eigenvalue of the space equation and for energies significantly less than the cyclotron energy , which is of the order of MeV for the intensity of the magnetic field here considered .

Conclusions : We derived the specific intensity of the ordinary photons , under the approximation of large angle and large optical depth .
These assumptions allow the equation to be treated using a diffusion-like approximation .