Physical Sciences and Mathematics Commons^™

Well-Posedness And Symmetry Properties Of Free Boundary Problems For Some Non-Linear Degenerate Elliptic Second Order Partial Differential Equations, Alaa Haj Ali

Wayne State University Dissertations

In the first part of this thesis, a bifurcation about the uniqueness of a solution of a singularly perturbed free boundary problem of phase transition associated with the $p$-Laplacian, subject to given boundary condition is proved in the first chapter. We show this phenomenon by proving the existence of a third solution through the Mountain Pass Lemma when the boundary data decreases below a threshold. In the second chapter and third chapter, we prove the convergence of an evolution to stable solutions, and show the Mountain Pass solution is unstable in this sense.

In the second part of this thesis, …

Teachers' Reflection On Their Beliefs And Question-Asking Practices During Mathematics Instruction, Kaili Takiyah Hardamon

Wayne State University Dissertations

Teachers’ daily instructional practices are a critical component in creating a rich and meaningful educational experience for students. Thus, factors that inform instructional practices are of particular importance and interest to education researchers and other stakeholders. Beliefs about teaching and learning are a known factor influencing teachers’ instructional practices (Ernest, 1989). This study focused on a specific instructional practice, question-asking, which has a profound impact on students’ experience with mathematics (Weiland, Hudson, and Amador (2014). Understanding the relationship between teachers’ beliefs and practice helps to make sense of teachers’ decision-making processes, particularly as they choose questions to ask students during …

Second-Order Generalized Differentiation Of Piecewise Linear-Quadratic Functions And Its Applications, Hong Do

Wayne State University Dissertations

The area of second-order variational analysis has been rapidly developing during the recent years with many important applications in optimization. This dissertation is devoted to the study and applications of the second-order generalized differentiation of a remarkable

class of convex extended-real-valued functions that is highly important in many aspects of nonlinear and variational analysis, specifically those related to optimization and stability.

The first goal of this dissertation is to compute the second-order subdifferential of the functions described above, which will be applied in the study of the stability of composite optimization problems associated with piecewise linear-quadratic functions, known as extended …

Numerical Approaches To A Thermoelastic Kirchhoff-Love Plate System, Zeyu Zhou

Wayne State University Dissertations

In this work, theory background of the sobolev spaces and finite element spaces are

reviewed first. Then the details of how the thermoelastic Kirchhoff-Love(KL) plates numerically established are presented. Later we approaches to the thermoelastic KL system numerically with mixed element method, H^1−Galerkin method and interior penalty discontinuous galerkin method(IP-DG).

What is more, the SIP-DG also applied to solve this KL system numerically. The well-posedness, existence, uniqueness and convergence properties are theoretical analyzed. The gain of the convergence rate is also O(h^k), that is 1 less than the observed convergence rate.

When discussing the H1-Galerkin method, the main advantages over …

Spectral Methods For Hamiltonian Systems And Their Applications, Lewei Zhao

Wayne State University Dissertations

Hamiltonian systems typically arise as models of conservative physical systems and have many applications. Our main emphasis is using spectral methods to preserve both symplectic structure and energy up to machine error in long time. An engery error estimation is given for a type of Hamiltonian systems with polynomial nonlinear part, which is numerical verified by solving a Henon-Heiles systems. Three interesting applications are presented : the first one is the N-body problems. The second one is approximation for Weyl's Law and the third one is simulating quantum cooling in an optomechanical system to study the dissipative dynamics. Moreover, nonsmooth …