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

辐射传输方程(RTE)是描述光在介观和宏观尺度的散射介质中传播的基本方程，如生物介质、塑料、涂料、石头、土壤、大气或星际空间。通常用蒙特卡罗方法等数值方法来求解RTE。然而，最近我们成功地导出了不同几何形状的RTE的解析解，以及它的一些扩展，例如广义RTE。该项目的目的是在所有空间频率域和所有空间域中进一步推导各种几何形状的RTE的重要解析解，同时考虑离散数量的散射相互作用的解。此外，还将得到相关RTE和扩散方程的解析解，这是RTE的一个常用近似。新的解析解将通过蒙特卡罗模拟进行验证或与之进行比较。RTE的解析解将被合并为热传导方程的解，对其也将推导出不同几何形状的解析解。此外，新的分析解决方案将得到有效实施，第一，研究不同技术和医疗领域的重要应用，第二，向感兴趣的用户提供解决方案。

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.