School of Science Department of Mathematics 50 Integro-differential Equations: Theory and Applications Supervisor: JIN Tianling / MATH Student: SZETO Chun / MATH-IRE Course: UROP1100, Spring It is a well-known fact that any positive solution to the problem on is radially symmetric (D denotes the unit ball). This paper includes a brief review of its proof using moving planes. Curiously, the analogous conclusion for the fractional Laplacian is also true, in the sense that radial symmetry is also observed in positive solutions of the problem . Organizing arguments made by existing papers, the proof will be presented in details, and the differences between the local and non-local case will be discussed. Efficient Algorithms for Visualizing Dynamical Systems Supervisor: LEUNG Shing Yu / MATH Student: CHAU Wai Ming / MATH-SFM Course: UROP2100, Fall UROP3100, Spring We propose a k-means clustering approach for detecting the Lagrangian coherent structure(LCS) in the continuous dynamic system. The idea is to cluster the whole particle trajectory over a finite time interval for many different initial conditions and use the mean square error from each cluster to quantify the deformation for each cluster. The advantage of the proposed approach is that the method tries to use the whole particle’s trajectory to measure the rate of separation between the adjacent particles instead of only the final location of the particle. To further improve the computational efficiency, we develop an adaptive approach to construct different subsamples of the whole particle trajectory based on a finite time interval. We will apply our method to two examples, including bounded and unbounded dynamic systems. Efficient Numerical Methods for Dynamic Interface Supervisor: LEUNG Shing Yu / MATH Student: LI Yuchen / SENG XU Minrui / SSCI Course: UROP1100, Summer This article uses MATLAB to implement the discrete QR method and visualize Lyapunov exponents of a regular dynamic system. We begin with a theoretical analysis of the FTLE, the Lagrangian method, and the discrete QR method, respectively, followed by their implementations. Finally, we present one numerical example and a comparison between these two algorithms.