Forward and Inverse Problems: An Exploration of Differential Equations

This thesis has two parts. The first discusses the Laplacian and how to solve it in arbitrary domains. It will cover Dirichlet's Principle, the Fundamental Solution, and Green's Function, as well as examples on the half space and ball. The second will cover the inverse problem of detecting corrosion in a plate of metal using heat. It will discuss the general solution method, regularization of the solution, and analyze the results.