My recent focus has been the study of fractional Laplacian Equations with zero Dirichlet external condition. These equations are of the form,
For more information on the fractional Laplaican operator Click Here.
I interested in existence/uniqueness/multiplicity of positive solutions to fractional Laplacian equations. I am also interested in the computation of numerical solutions to fractional Laplacian equations and construction of numerical bifurcation diagrams. The following is a comparison of the of a numerical bifurcation diagram for the fractional Laplacian with s = 0.75 vs. the Laplacian where f(u) = u(1-u).
I am also interested in numerical experiments for the time dependent fractional Laplacian problem,
Consider the weighted logistic nonlinearity f(u(x,t))=u(x,t)[Q(x,t)-u(x,t)] where the initial distribution of resources with Q(x,0) and the initial population u(x,0) are defined as the following.
The following is an animation showing my numerical experiments for various time steps for different choices of the parameter s where the location of reources, defined by Q(x,t), randomly shifts location. Also given is the L1 norm of each numerical solution which measures the total size of the diffusing population. A question I am interested in studying is for a fixed time t>0 is there a choice of s which will maximize the L1 norm?
Sturm Liouville Two Paramemeter Differential Equation. This project was for my Masters Thesis at Wake Forest University completed under the supervision of Dr. Stephen Robinson
Survival Analysis of Olympic Records. This is joint work with Adam Zarn and Victoria Tevino, conducted as part of a Undergraduate research experience at UC Fresno Summer 2012