Chuanjie graduated with a MSc in statistics at the university of Warwick in 2022. His master’s thesis is about implementing operator splitting methods on Wright fisher stochastic differential equation, as well as studying the boundary condition and hitting probability. During his first-year study in SAMBa, he developed an interest in studying analysis tools in PDEs. Alongside math, chuanjie also enjoys basketball and swimming.

Project title:
Asymptotic analysis of operator norm convergence on periodic thin structure

Supervisor(s):
Kirill Cherednichenko

Project description:
The theme of the project is the asymptotic analysis of operators associated with PDEs on “thin” structures, i.e. those for which one or two of the linear dimensions is much smaller than the others (rods, plates, shells). More specifically, it will focus on the analysis of periodic thin structures with components that “resonate” in the presence of wave motion. Our method is based on “Ryzhov triples framework”, which reformulates the boundary value problem (BVP) in terms of operators defined in Hilbert space. This framework provides us benefits such as: 1) Extension of the original “boundary triples operators” to some wider classes, which builds a more convenient BVP model; 2) By applying Krein’s resolvent formula, the resolvent norm can be interpreted by the Weyl-Titchmarsh function, which is analytic and densely defined (therefore it is easier to study the asymptotic behaviours). To this end, we are aiming at deriving an order-sharp norm-resolvent convergence estimates for the solution of PDEs. Above methodology is effective in studying the asymptotic resolvent operator in Maxwell system (defined on periodic “thin” structure domain embedded in R^3). Moe precisely, by regarding the tubular material (domain) as boundaryless manifold, we are aiming at defining the “Ryzhov triples” operators on graph-like manifold for the convenience to derive an order-sharp estimate of asymptotic resolvent in the uniform norm topology.