- Spectral theory of nonlocal operator
- Numerical methods for Nonlocal PDEs
- Stochastic differential equations in physical and chemical systems
- Population model in biology and ecology
Publications
- X. Xie, J. Duan, X. Li, and G. Lv. A regularity result for the fractional Fokker-Planck equation with Ornstein-Uhlenbeck drift,preprint arXiv:1504.04631
- W.Shen and X.Xie. Spectral theory for nonlocal dispersal operators with time periodic indefinite weight functions and applications (submitted)
- W.Shen and X.Xie. Approximation of Random Dispersal Operators by Nonlocal Dispersal Operators Journal of Differential Equations, 259 (2015), no. 12, 7375-7405.
- W.Shen and X.Xie. On principal spectrum points/principal eigenvalues of nonlocal dispersal operators and applications Discrete and Continuous Dynamical Systems-Series A, 35 (2015), no. 4, 1665-1696.
- L.Zhao, P.Zhao and X.Xie. Existence and multiplicity of solutions for divergence type elliptic equations. Electronic Journal of Differential Equations, Vol.2011(2011), No.43, pp.1-9
Work in process
- X.Xie. Classical, Nonlocal and Fractional Dispersal Operators and Connections to the Winner, Compound Poisson and L'evy Processes.
- X.Xie and X.Li. Numerical Method for the nonlocal operators with singular kernel and applications in Fokker-Planck equations on bounded domain
- W.Shen and X.Xie. Minimization of the generalized principal eigenvalue for nonlocal boundary value problem with indefinite weight, and applications to population dynamics
- J.Coville and X.Xie. Persistence criteria in time periodic environment for populations with non-local dispersion
- X. Xie, J. Duan, and X. Li. A regularity result for the fractional Fokker-Planck equation with Ornstein-Uhlenbeck drift with Dirichlet boundary condition
Conference Presentations
- Invited Talk, the 11th AIMS Conference on Dynamical Systems,Differential Equations and Applications Orlando, FL, Jul.2016
- Invited Talk, at the Special Session \textit{Random and Complex Dynamics of Reaction-Diffusion Systems, I at the 2016 Joint Mathematics Meeting Seattle, WA. Jan.2016
- Invited Talk, at the East Lake International Forum for Outstanding Oversees Young Scholars Huazhong University of Science and Technology Wuhan, Hubei,China Dec.2015
- Poster Presentation, the Nonlocal Models in Mathematics, Computation, Science, and Engineering Oak Ridge National Laboratory, Oak Ridge, TN. Oct.2015
- Invited Talk, the 35th Southeastern Atlantic Regional Conference on Differential Equations University of North Carolina at Greensboro, Greensboro, NC. Oct.2015
- Co-Organizer and Invited Talk, at the Special Session Nonlocal Diffusion the AMS Central Fall Sectional Meeting Loyola University Chicago, Chicago, IL. Oct.2015
- Invited Talk, the Workshop on Mathematical Biology and Nonlinear Analysis University of Miami, Coral Gables, FL. Dec.2014
- Invited Talk, at the Special Session Reaction Diffusion Equations and Applications at the 2014 Joint Mathematics Meeting, Baltimore,MD, Jan.2014
- Invited Talk, at the 9th Annual UNCG Regional Mathematics and Statistics Conference, University of North Carolina at Greensboro,Greensboro,NC, Nov.2013
- Seminar Talk, at the Applied and Computational Mathematics Seminar, Department of Mathematics and Statistics, Auburn University, Auburn, AL, Sep.2013
- Seminar Talk, at the Graduate Student Seminar, Department of Mathematics and Statistics, Auburn University, Auburn, AL, Sep.2013
- Poster Presentation, in the IMA Annual Program Year Workshop: Lattice and Nonlocal Dynamical Systems and Applications, IMA (Institute of Mathematics and its Applications), Minneapolis, MN, Dec.2012
- Invited Talk, at Special Session Dissipative Systems and Applications at t the 9th American Institute of Mathematical Sciences Conference on Dynamical Systems, DifferentialEquations and Applications, Orlando, FL, Jul.2012
- Poster Presentation, in the SAMSI summer program: Nonlocal Continuum Models for Diffusion, Mechanics, and Other Applications, SAMSI(Statistical and Applied mathematical Sciences Institute), Research Triangle Park, NC Jun.2012
- Poster Presentation, at the 36th Annual SIAM (Society for Industrial and Applied Mathematics) Southeastern Atlantic Section Conference, Huntsville,AL Mar.2012
- Invited Presentation, at the 60th Annual Meeting of the Alabama Association of College Teachers of Mathematics Tuskegee,AL Feb.2010