This paper provides a solution for two-dimensional (2D) heating over a rectangular region on a homogeneous plate. It has application to verification of numerical conduction codes as well as direct application for heating and cooling of electronic equipment. Additionally, it can be applied as a direct solution for the inverse heat conduction problem, most notably used in thermal protection systems for re-entry vehicles. The solutions used in this work are generated using Green's functions (GFs). Two approaches are used, which provide solutions for either semi-infinite plates or finite plates with isothermal conditions, which are located a long distance from the heating. The methods are both efficient numerically and have extreme accuracy, which can be used to provide additional solution verification. The solutions have components that are shown to have physical significance. The procedure involves the derivation of previously unknown simpler forms for the summations, in some cases by virtue of the use of algebraic components. Also, a mathematical identity given in this paper can be used for a variety of related problems.