Browsing by Subject "partial differential equations"
Now showing 1 - 4 of 4
Results Per Page
Sort Options
Item 2D Peskin Problems of an Immersed Elastic Filament in Stokes Flow(2018-05) Rodenberg, AnaliseIn the work that follows we investigate a class of problems where a one dimensional closed elastic structure is immersed in a plane of steady Stokes flow. The dynamics are governed by a boundary integral equation describing the configuration of the immersed structure. Depending on the choice of elasticity law, we break our class into either a semilinear or fully nonlinear system of equations. In the nonlinear setting, we prove that the linearization of the system generates an analytic semigroup and use it to prove local existence and uniqueness in low regularity H\"{o}lder spaces. In the semilinear setting, we remove the principle operator via small scale decomposition and use it to build similar local existence results. Further, we establish spatial smoothness of solutions by careful estimates on a class of commutators. Using these regularity results, we are able to establish that the only equilibria of the system are uniformly parameterized circles which we then prove nonlinear stability about. Finally, we identify a quantity which classifies global-in-time behavior.Item Fundamental Solutions and Green Functions for Nonhomogeneous Second Order Elliptic Operators with Discontinuous Coefficients(2017-06) Hill, JonathanThis thesis is devoted to the properties of fundamental solutions, Green functions, and Neumann-Green functions for general non-homogeneous second order elliptic systems with discontinuous coefficients. We establish existence, uniqueness, and scale-invariant estimates for fundamental solutions of non-homogeneous second order elliptic systems. We impose certain non-homogeneous versions of de Giorgi-Nash-Moser bounds on the weak solutions and investigate in detail the assumptions on the lower order terms sufficient to guarantee such conditions. Our results, in particular, establish the existence and fundamental estimates for the Green functions associated to the Schrödinger, magnetic Schrödinger, and generalized Schrödinger operators with bounded measurable real and complex coefficients on arbitrary domains. Most of the results above rely on the construction of the averaged fundamental solutions and Green functions with sharp uniform estimates. We also showcase a different approach to Green and Neumann-Green functions via layer potentials which yields, in addition, certain new mapping properties for the Green operators. A substantial portion of the results of this thesis gave rise [DHM17], submitted for publication.Item Numerical analysis of prediction with expert advice(2022-05) Mosaphir, DrisanaThis work investigates the online machine learning problem of prediction with expert advice through numerical analysis of a related PDE. The problem is a repeated two-person game involving decision-making at each step informed by n experts with geometric stopping condition; the continuum limit of the consequences of this game over a large number of steps leads to an elliptic PDE. This work presents a numerical scheme that allows us to solve this PDE for general number of experts n, and gives numerical results for n < 9.Item Studying Synergies between SDEs and PDEs; Analysis of Kolmogorov and Feynman-Kac Results(2021) Dey, ArnabIn this report, the synergy between stochastic differential equations (SDEs) and partial differential equations (PDEs) is studied. There are important results by Kolmogorov, Feynman and Kac that can be utilized to solve PDEs using SDEs and vice-versa. In this report, the results in the articles by Black-Scholes, and Harrison are analyzed and their importance critiqued. These synergies between PDEs and SDEs are poised to be utilized in the energy markets, demand-response programs and reserve planning for the grid. The report is presented by instantiating the analysis to European options; the general theory is applicable widely.