Let us consider the thermal radiation transfer in a plane-parallel layer of absorbing and scattering medium between walls having temperatures *T*_{w1}, *T*_{w2} and spectral emissivities ε_{w1}, ε_{w2}. The temperature profile in the medium layer, *T*(*x*), is assumed to be known.

According to the diffusion approximation, the spectral radiation energy density can be obtained by solving the following boundary-value problem:

(1a) |

(1b) |

(1c) |

(1d) |

Here, *S*_{B0} = 4π*S*_{B} = 4α_{λ}π*B*_{λ}(*T*) (one-temperature medium) and *S*_{wj} = 4π*B*_{λ}(*T _{wj}*). The spectral radiation fluxes toward the walls are

(2a) |

(2b) |

and the profile of the heat generation in the medium can be obtained by integrating over the spectral range:

(3) |

In the case of arbitrary profiles α_{λ}(*z*), *D*_{λ}(*z*), and *S*_{λ}(*z*), boundary-value problem (2) cannot be solved analytically and needs a numerical solution. The usual finite-difference approximation from Eq. (1a) is

(4) |

where

(5) |

For a better approximation of the boundary conditions one should use the expansion of function *I*_{λ}(*z*) in Taylor’s series in the neighborhood of the boundary nodes of the mesh:

(6a) |

(6b) |

where (*d*^{2}*I*_{λ}^{0}/*dz*^{2}) is determined through *I*_{λ}^{0} and (*d**I*_{λ}^{0}/*dz*) is determined according to differential Eq. (1a). After transformations, the finite-difference equation and boundary conditions can be written as follows:

(7a) |

(7b) |

In accordance with the factorization method, the relation between the neighboring nodal values of the function to be found is

(8) |

where φ_{i} and ψ_{i} are the factorization coefficients. From the finite-difference form of the boundary condition at *z* = 0 we find

(9) |

The formulas of upward factorization

(10) |

allow obtaining all of the values of φ_{i} and ψ_{i} up to *i* = *n*. After that, one can calculate the value of *I*^{0}_{λ,n+1} from the boundary condition at *z* = *d*:

(11) |

and all of the other *I*^{0}_{λ,i} are determined by Eq. (8).

A similar algorithm can also be developed with the use of the *DP*_{1} approximation. For the plane-parallel layer of a medium, boundary-value problems (1a), (1b), (2a), and (2b) from the article The Simplest Approximations of Double Spherical Harmonics have the following vector form:

(12a) |

(12b) |

Here,

The other designations are evident from the comparison of Eqs. (12a) and (12b) with Eqs. (1a), (1b), (2a), and (2b) from the article The Simplest Approximations of Double Spherical Harmonics. Let us rewrite Eqs. (12a) and (12b) in the form of the boundary-value problem for function :

(13a) |

(13b) |

(13c) |

The form of these equations coincides with that of Eq. (7). Therefore, the algorithm of matrix factorization for problems (13a)-(13c) is quite similar to the above-described scalar factorization for the diffusion approximation.