The Diffusion Equation for a Two-Layered Cylinder

doi:10.1117/3.2624517.ch11

Book: Light Propagation through Biological Tissue and Other Diffusive Media: Theory, Solutions, and Validation, Second Edition

Author(s): Fabrizio Martelli, Tiziano Binzoni, André Liemert, Samuele Del Bianco, Alwin Kienle

Published: 2022

https://doi.org/10.1117/3.2624517.ch11

Author Affiliations +

Abstract

The method of images (or mirror images) has been largely used to derive analytical solutions of the diffusion equation for various geometries. An alternative approach to the method of images is the eigenfunction method, which can only be applied for finite-bounded geometries. The eigenfunction method has been introduced in Sec. 5.4 to obtain a DE solution for the slab geometry with respect to the finite dimension of the slab. From here onward, the expression eigenfunction method will be used to denote the method based on an eigenfunction expansion to solve the initial and boundary value problem for parabolic equations. In this chapter, after reviewing some of the literature devoted to photon migration through diffusive layered media, the eigenfunction method will be applied to solve the DE in a bounded, layered medium. In particular, the solution for a two-layer medium will be described. Following the same approach, a solution for a three-layer medium has been obtained elsewhere.