UROP Proceedings 2020-21

School of Science Department of Mathematics 57 Integro-Differential Equations: Theory and Applications Supervisor: JIN Tianling / MATH Student: SZETO Chun / SSCI Course: UROP1000, Summer The eigenvalues and eigenfunctions of the fractional Laplacian have been of great interest, for both mathematical and physical reasons. In the following, two fundamental properties of the fractional Laplacian will be proved: 1. the first eigenvalue is simple and positive; 2. the first eigenfunction does not vanish on its domain U, which is a bounded, Lipchitz open set. As the first eigenfunction is continuous, without loss of generality one usually assumes that the first eigenfunction is positive. The proofs mainly build on a wellknown compactness result of the fractional Sobolev space. Then, one can mimic the corresponding proofs for the Laplacian (−∆) to prove similar results in the fractional case. Efficient Algorithms for Visualizing Dynamical Systems Supervisor: LEUNG Shing Yu / MATH Student: ZHOU Siyuan / DSCT Course: UROP1100, Fall Using Lagrangian approaches to calculate a point’s (denoted by P(x0,y0)) finite-time Lyapunov exponent (FTLE), we need to know its terminal condition after time T and the behavior of points near P. The approach requires a velocity field of any point to construct an ODE to figure out the terminal condition. However, in reality, we can only collect discrete random points at different time levels, which means we do not know the explicit expression of the velocity field. Now we want to use the Gaussian basis function to approximate the FTLE. The approach is as follows: 1. Collect a set of points at different time levels. Use (x(T)-x(T0))/dt to approximate each point’ s velocity at different time level. 2. Getting a set of points with velocity in step 2, use the Gaussian basis function to reconstruct a velocity field. 3. Use the reconstructed velocity field to develop the ODE function to get the terminal condition of different initial points. Calculate FTLE of different points. This report will use a known velocity field to get a set of random points at different time levels, then follow the above steps to get an approximated FTLE. Efficient Algorithms for Visualizing Dynamical Systems Supervisor: LEUNG Shing Yu / MATH Student: QIN Hanmo / MATH-CS Course: UROP2100, Spring The backward phase flow method is an Eulerian approach to approximate the Lyapunov Exponent of a dynamic system. The method uses interpolation to evaluate the midway short time flow maps of a given long time flow map. A suitable interpolation method means the map will be closer to reality and is more accurate. In this report, we will discuss and compare several different interpolation skills and conclude which one is more suitable for the moderation of the backward phase flow map.

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