Exact and approximate analytical solutions of the two- und three-dimensional radiative transfer equation

辐射传递方程（RTE）是描述光在生物介质、塑料、涂料、石头、土壤、大气或星际空间等介观和宏观尺度散射介质中的传播的基本方程。通常，RTE是通过蒙特卡罗方法等数值方法求解的。然而，最近我们成功地推导了不同几何的RTE及其一些扩展的解析解，例如广义RTE。该项目的目的是推导出所有空间频率域和所有空间域的各种几何形状的RTE的进一步重要解析解，同时考虑散射相互作用离散数的解。此外，将推导出相关RTE和扩散方程的解析解，扩散方程是RTE的常用近似值。新的解析解将与蒙特卡罗模拟进行验证或比较。RTE的解析解将与热传导方程的解结合起来，对于不同的几何形状，也将推导出解析解。此外，新的分析解决方案将得到有效实施，首先，研究不同技术和医学领域的重要应用，其次，使解决方案可供感兴趣的用户使用。

The radiative transfer equation (RTE) is the fundamental equation for describing light propagation in scattering media in the mesoscopic and macroscopic scales such as in biological media, in plastics, in paints, in stones, in soils, in the atmosphere or in the interstellar space. Usually, the RTE is solved by numerical approaches like the Monte Carlo method. Recently, however we succeeded in deriving analytical solutions to the RTE for different geometries and to some of its extensions, e.g. for the generalized RTE. The aim of the project is to derive further important analytical solutions to the RTE for a variety of geometries in all spatial frequency domains and in all spatial domains considering also the solutions for discrete numbers of scattering interactions. Furthermore, analytical solutions will be derived for the correlation RTE and the diffusion equation, an often used approximation of the RTE. The new analytical solutions will be validated with or compared to Monte Carlo simulations. Analytical solutions of the RTE will be combined to solutions of the heat conduction equation, to which also analytical solutions will be derived for different geometries. In addition, the new analytical solutions will be efficiently implemented, first, to study important applications in different technical and medical fields and, second, to make the solutions available to the interested user.