| Description |
The Laplacian operator plays a ubiquitous role in the differential equations that describe many physical systems. These include, for example, vibrating membranes, fluid flow, heat flow, and solutions to the Schrödinger equation. In this paper, we investigate a method to find the eigenfunctions of the Laplacian operator and extend it to other surfaces: spherical and flat tori. We code this method and give various examples and applications of the computation results. We develop a novel robust and fast method to solve for the Laplacian eigenfunctions on various surfaces - called the partitioned expansion method - which we describe in full and give examples of computational results. Lastly, we utilize the expansion method to study the dependence of the spectra of domains on some physical parameter. |
