Dissertation Defense for PhD in Mathematics

These are the slides I presented during the defense of my doctoral research and dissertation. My research involved extending a method that my advisor and his collaborators had introduces for numerically solving the spectral fractional Laplacian. There were two major accomplishments that I contributed. The first was to avoid the problem of numerically computing the eigenvalues of an ODE needed by the method with analytically solving the ODE. This overcomes the stability issue in finding the eigenvalues and eigenfunctions. The second was showing that the method with analytic eigenvalues and eigenfunctions was essentially a quadrature for a Balakrishnan integral formulation of the spectral fractional Laplacian. This discovery was made while demonstrating the convergence rate of the algorithm. I implemented the method in C++ using the deal.ii FEM library and was able to match the theoretical convergence order as well as demonstrate the parallel efficiency of the algorithm.