Integro-differential equations: theory and applications
The traditional partial differential equations (PDEs) are relations of an unknown function of several variables and its partial derivatives. To verify such an equation at a point, we only need the values of this function in an arbitrarily small neighborhood. Integro-differential equations in general involve integrals and derivatives of the unknown function. Therefore, in order to verify an integro-differential equation at a point, information of the values of this function far from the point is also needed. This is why they are also called nonlocal equations.
Integro-differential equations arise naturally in many contexts such as probability, geometry, physics and ecology. The class of integro-differential equations is very rich. For example,second order PDEs can actually be obtained as limits of integro-differential equations. It would be ideal if one can find explicit solutions of given PDEs or integro-differential equations. But this is only possible for very few special equations.
This project aims at introducing the basics (such as existence, uniqueness, regularity, applications in other fields) of solutions of integro-differential equations (and PDEs) to motivated undergraduate students, through reading some important journal papers in this area. Students who have excellent performance will be invited to continue on the project for more original research.
The students will read some important journal papers on integro-differential equations under the guidance of the supervisor. Students who take part in this project should already have a solid foundation on Multivariable Calculus, Real Analysis and Linear Algebra. Prior knowledge in Differential Equations will be a plus, but is not required. At the end of the project, students are expected to write a survey paper on either some special integro-differential equation or a specific property of solutions to general integro-differential equations.
Applicant's Learning Objectives:
1. Strengthen their background knowledge on Calculus, Real Analysis and linear algebra;
2. Read research papers in pure math;
3. Learn how to search references during the research;
4. Build solid background on differential equations for future research in pure and applied math (such as geometry, numerical analysis, etc)