Classical analysis is not able to treat functions whose domain is fractal. We present an introduction to analysis on a particular class of fractals known as post-critically finite (PCF) self-similar sets that is suitable for the undergraduate reader. We develop discrete approximations of PCF self-similar sets, and construct discrete Dirichlet forms and corresponding discrete Laplacians that both preserve self-similarity and are compatible with a notion of harmonic functions that is analogous to a classical setting. By taking the limit of these discrete Laplacians, we construct continuous Laplacians on PCF self-similar sets. With respect to this continuous Laplacian, we also construct a Green's function that can be used to find solutions to the Dirichlet problem for Poisson's equation.
"An Introduction to Fractal Analysis,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 22:
1, Article 2.
Available at: https://scholar.rose-hulman.edu/rhumj/vol22/iss1/2