<p> Much of the world's rapidly growing urban population relies upon water distribution systems to provide treated water through networks of pipes. Rather than continuously supplying water to users, many of these distribution systems operate intermittently, with parts of the network frequently losing pressure or emptying altogether. Such intermittent water supply deleteriously impacts water availability, infrastructure, and water quality for hundreds of millions of people around the world. In this work I introduce the problem of intermittent water supply through the lens of applied mathematics. I first introduce a simple descriptive mathematical model that captures some hydraulic features of intermittency not accounted for by existing water distribution system software packages. I then consider the potential uses of such a model in a variety of optimization examples motivated by real-world applications. In simple test networks, I show how to reduce pressure gradients while the network fills by changing either the inflow patterns or the elevation profile. I also show test examples of using measured data to potentially recover unknown information such as initial conditions or boundary outflows. I then use sensitivity analysis to better understand how various parameters control model output, with an eye to figuring out which parameters are worth measuring most carefully in field applications, and also which parameters may be useful in an optimization setting. I lastly demonstrate some progress in descriptively modeling a real network, both through the introduced mathematical model and through a fluid-mechanics-based method for identifying data where the model is most useful.</p>

<p> One of the reasons why predation is important is that no organism can live, grow, and reproduce without consuming resources. We have studied possible scenarios of pattern formation in three different predator-prey models; the Rozenweig-McArthur, and the Leslie-Gower model with alternative food for the predator with different functional responses; one uses a prey-dependent functional response while the other use ratio-dependent functional response. The Turing patterns observed for the Leslie-Gower model with prey dependence are of two types, hot spot patterns and cold spot patterns while the ratio dependent model exhibit only hot spot patterns. Also, the labyrinthine pattern is also observed for some choices of parameters values within the Turing-Hopf domain for both models.</p>

<p> In this dissertation we investigate the solution of boundary value problems on polygonal domains for elliptic partial differential equations. We observe that when the problems are formulated as the boundary integral equations of classical potential theory, the solutions are representable by series of elementary functions. In addition to being analytically perspicuous, the resulting expressions lend themselves to the construction of accurate and efficient numerical algorithms. The results are illustrated by a number of numerical examples.</p>

<p> Traditional finite difference methods for solving the partial differential equations (PDEs) associated with wave and heat transport often perform poorly when used in domains that feature jump discontinuities in model parameter values (interfaces). We present a radial basis function-derived finite difference (RBF-FD) approach that solves these types of problems to a high order of accuracy, even when curved interfaces and variable model parameters are present.</p><p> The method generalizes easily to a variety of different problem types, and requires only the inversion of small, well-conditioned matrices to determine stencil weights that are applied directly to data that crosses an interface. These weights contain all necessary information about the interface (its curvature; the contrast in model parameters from one side to the other; variability of model parameter value on either side), and no further consideration of the interface is necessary during time integration of the numerical solution. </p>

The Darboux transformation is one of the main techniques for finding solutions of integrable equations. The Darboux transformation is not only powerful in the construction of muilti-soliton solutions, recently, it is found that the Darboux transformation, after some modification, is also effective in generating the rogue wave solutions. In this thesis, we derive the rogue wave solutions for the Davey-Stewartson-II (DS-II) equation in terms of Darboux transformation. By taking the spectral function as the product of plane wave and rational function, we get the fundamental rogue wave solution and multi-rogue wave solutions via the normal Darboux transformation. Last but not least, we construct a generalized Darboux transformation for DS-II equation by using the limit process. As applications, we use the generalized Darboux transformation to derive the second-order rogue waves. In addition, an alternative way is applied to derive the N-fold Darboux transformation for the nonlinear Schrödinger (NLS) equation. One advantage of this method is that the proof for N-fold Darboux transformation is very straightforward.

In our work we analyze and implement numerical schemes for the infinite Prandtl number model for convection. This model describes the convection that is a potential driving force behind the flow and temperature of the Earth's mantle. There are many schemes available, but most are given with no mention of their ability to adequately capture the long time statistical properties of the model. We investigate schemes with the potential to actually capture these statistics. We further show numerically that our schemes align with current knowledge of the model's characteristics at low Rayleigh numbers. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Fall Semester 2015. / July 30, 2015. / infinite, long time, numerical, Prandtl, statistical / Includes bibliographical references. / Xiaoming Wang, Professor Directing Dissertation; Qing-Xiang Amy Sang, University Representative; Bettye Anne Case, Committee Member; Brian Ewald, Committee Member; Max Gunzburger, Committee Member.

Symmetric positive definite (SPD) matrices have become fundamental computational objects in many areas. It is often of interest to average a collection of symmetric positive definite matrices. This dissertation investigates different averaging techniques for symmetric positive definite matrices. We use recent developments in Riemannian optimization to develop efficient and robust algorithms to handle this computational task. We provide methods to produce efficient numerical representations of geometric objects that are required for Riemannian optimization methods on the manifold of symmetric positive definite matrices. In addition, we offer theoretical and empirical suggestions on how to choose between various methods and parameters. In the end, we evaluate the performance of different averaging techniques in applications. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Summer Semester 2018. / July 20, 2018. / Includes bibliographical references. / Kyle A. Gallivan, Professor Co-Directing Dissertation; Pierre-Antoine Absil, Professor Co-Directing Dissertation; Gordon Erlebacher, University Representative; Giray Okten, Committee Member; Martin Bauer, Committee Member.

The definition of partial differential equation (PDE) models usually involves a set of parameters whose values may vary over a wide range. The solution of even a single set of parameter values may be quite expensive. In many cases, e.g., optimization, control, uncertainty quantification, and other settings, solutions are needed for many sets of parameter values. We consider the case of the time-dependent Navier-Stokes equations for which a recently developed ensemble-based method allows for the efficient determination of the multiple solutions corresponding to many parameter sets. The method uses the average of the multiple solutions at any time step to define a linear set of equations that determines the solutions at the next time step. In this work we incorporate a proper orthogonal decomposition (POD) reduced-order model into the ensemble-based method to further reduce the computational cost; in total, three algorithms are developed. Initially a first order accurate in time scheme for low Reynolds number flows is considered. Next a second order algorithm useful for applications that require long-term time integration, such as climate and ocean forecasting is developed. Lastly, in order to extend this approach to convection dominated flows a model incorporating a POD spatial filter is presented. For all these schemes stability and convergence results for the ensemble-based methods are extended to the ensemble-POD schemes. Numerical results are provided to illustrate the theoretical stability and convergence results which have been proven. / A Dissertation submitted to the Department of Scientific Computing in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Summer Semester 2018. / June 27, 2018. / ensemble computation, Navier-Stokes equations, Proper Orthogonal Decomposition / Includes bibliographical references. / Max Gunzburger, Professor Directing Thesis; Mark Sussman, University Representative; Janet Peterson, Committee Member; Gordon Erlebacher, Committee Member; Chen Huang, Committee Member.

Diseases such as tuberculosis, chronic pneumonia, and inner ear infections are caused by bacterial biofilms. Biofilms can form on any surface such as teeth, floors, or drains. Many studies show that it is much more difficult to kill the bacteria in a biofilm than planktonic bacteria because the structure of biofilms offers additional layered protection against diffusible antimicrobials. Among the bacteria in planktonic-biofilm populations, persisters is a subpopulation that is tolerant to antibiotics and that appears to play a crucial role in survival dynamics. Understanding the dynamics of persister cells is of fundamental importance for developing effective treatments. In this research, we developed a method to better describe the behavior of persistent bacteria through specific experiments and mathematical modeling. We derived an accurate mathematical model by tightly coupling experimental data and theoretical model development. By focusing on dynamic changes in antibiotic tolerance owing to phenotypic differences between bacteria, our experiments explored specific conditions that are relevant to specifying parameters in our model. We deliver deeper intuitions to experiments that address several current hypotheses regarding phenotypic expression. By comparing our theoretical model to experimental data, we determined a parameter regime where we obtain quantitative agreement with our model. This validation supports our modeling approach and our theoretical predictions. This model can be used to enhance the development of new antibiotic treatment protocols. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Spring Semester 2018. / March 1, 2018. / Antibiotic resistance, Biofilm, Mathematical Modeling, Persisters / Includes bibliographical references. / Nick Cogan, Professor Directing Dissertation; Peter Beerli, University Representative; Richard Bertram, Committee Member; Giray Okten, Committee Member; Theodore Vo, Committee Member.

We consider a classical risk process with arrival of claims following a non-stationary Hawkes process. We study the asymptotic regime when the premium rate and the baseline intensity of the claims arrival process are large, and claim size is small. The main goal of the article is to establish a diffusion approximation by verifying a functional central limit theorem and to compute the ruin probability in finite-time horizon. Numerical results will also be given. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Fall Semester 2018. / November 12, 2018. / diffusion approximation, Hawkes process, risk model / Includes bibliographical references. / Lingjiong Zhu, Professor Directing Dissertation; Xufeng Niu, University Representative; Arash Fahim, Committee Member; Sanghyun Lee, Committee Member.