School of Science Department of Mathematics 49 Numerical Methods for Solving PDEs on Surfaces Supervisor: LEUNG Shing Yu / MATH Student: MAK Pak Ho / MAEC Course: UROP 1100, Fall Wave equations are an important class of partial differential equations (PDEs) involving the Laplacian differential operator. This report compared different existing approximation methods of the surface Laplacian operator under the framework of the embedding approach and applied them to solve wave equations on a closed curve in two-dimensional Euclidean space. These methods allow the curve to be represented implicitly as the zero-level set of a function, which offers greater flexibility in representing complicated curves. This report implemented the projection method, the modified projection method, the closest-point method, and the push-forward method and conducted numerical analysis on their solutions for wave equations. The advantages and shortcomings of these methods were investigated, and the findings may improve existing methods or develop new embedding methods to solve wave equations. Random Walks and Percolation on Graphs Supervisor: NITZSCHNER Maximilian Alexander / MATH Student: CHEN Yuyang / DSCT Course: UROP 1000, Summer This report studies the recurrence or transience of random walks on infinite graphs. We utilize a remarkable connection between the random walk (as an object of probability theory) and notions pertaining to electric circuits in physics. We review how probabilistic notions can be transferred to voltages and currents by viewing (weighted) graphs as electric networks. The application of standard laws from the theory of electric circuits enables us to infer properties of random walks on infinite graph. As one application, we outline a proof of a classical theorem due to Pólya, stating that random walks on infinite Euclidean lattices are recurrent if and only if the dimension is less than or equal to 2, and are transient otherwise.
RkJQdWJsaXNoZXIy NDk5Njg=