Analytische Lösungen der Strahlungstransfergleichung und ihrer Approximationen

Abstract

This thesis presents the derivation of analytical expressions of the radiative transfer equation and their approximations in the steady-state and time domains. The diffusion equation, which is the most often used approximation of the radiative transfer equation, is solved analytically for different homogeneous and layered geometries. By applying the integral transform formalism the respective form of the diffusion equation is reduced to an ordinary differential equation. At this stage the solution of the boundary value problem is obtained via standard techniques. The derived solutions were validated against other independent analytical methods found in literature and the Monte Carlo method. Analytical solutions of the simplified spherical harmonics equations are derived for homogeneous media. The Fourier transform is used to convert these coupled diffusion-like equations to a system of ordinary differential equations. The obtained solutions for infinite and semi-infinite media were validated against the finite difference method. The infinite space fluence within the transport theory is derived for an anisotropically scattering medium and different source distributions. Additionally, for the case of isotropic scattering a simplified version is given. Based on results of the infinite space fluence and by applying the eigenvalue method the radiance caused by an isotropic point source in an infinitely extended medium is obtained. The analytical solutions were compared to the Monte Carlo method. Within the stochastic nature of the Monte Carlo simulations an exact agreement was found in the steady-state and time domains.