Spectral properties and finite range elementary solutions of the linear Boltzmann equation

The property of the operator (zδx + 1) to transform a definite class of functions ψ(x,z ) depending on two variables to functions ψ(x) depending only on one variable allows the algebraic determination of the spectral properties of the Boltzmann equation in a finite domain of R1. A number of theorems have been proved regarding the spectrum of characteristic values. In the case of homogeneous boundary conditions the spectral property [formule] has been established, where [formule] satisfy [formule]. The eigenfunctions are under certain conditions invariant against parity, scaling and translation transformation.Syros C. Spectral properties and finite range elementary solutions of the linear Boltzmann equation. In: Bulletin de la Classe des sciences, tome 59, 1973. pp. 926-947

Structural properties and finite range elementary solutions of the linear Boltzmann equation

The natural solutions of the time-and energy-independent linear Boltzmann equation with isotropic scattering have been found on A ⊗ B° where A is the interval [a,b] a,b > 0 and B° = [— 1,0[⋃]0,1]. These functions are regular everywhere on A ⊗ B° and they satisfy homogeneous or inhomogeneous boundary conditions. The solutions become uniformly continuous functions on A ⊗ B and have bounded derivatives of any order if the convention for taking limits, [formule], is used. The construction of the solutions is based on a number of structural properties established by corresponding new theorems. They exhibit invariance properties with respect to translation, partity and scaling transformations.Syros C. Structural properties and finite range elementary solutions of the linear Boltzmann equation. In: Bulletin de la Classe des sciences, tome 59, 1973. pp. 500-518