abstract: Mathematical models are important tools for addressing problems that exceed experimental capabilities. In this work, I present ordinary and partial differential equation (ODE, PDE) models for two problems: Vicodin abuse and impact cratering. The prescription opioid Vicodin is the nation's most widely prescribed pain reliever. The majority of Vicodin abusers are first introduced via prescription, distinguishing it from other drugs in which the most common path to abuse begins with experimentation. I develop and analyze two mathematical models of Vicodin use and abuse, considering only those patients with an initial Vicodin prescription. Through adjoint sensitivity analysis, I show that focusing efforts on prevention rather than treatment has greater success at reducing the total population of abusers. I prove that solutions to each model exist, are unique, and are non-negative. I also derive conditions for which these solutions are asymptotically stable. Verification and Validation (V&V) are necessary processes to ensure accuracy of computational methods. Simulations are essential for addressing impact cratering problems, because these problems often exceed experimental capabilities. I show that the Free Lagrange (FLAG) hydrocode, developed and maintained by Los Alamos National Laboratory, can be used for impact cratering simulations by verifying FLAG against two analytical models of aluminum-on-aluminum impacts at different impact velocities and validating FLAG against a glass-into-water laboratory impact experiment. My verification results show good agreement with the theoretical maximum pressures, and my mesh resolution study shows that FLAG converges at resolutions low enough to reduce the required computation time from about 28 hours to about 25 minutes. Asteroid 16 Psyche is the largest M-type (metallic) asteroid in the Main Asteroid Belt. Radar albedo data indicate Psyche's surface is rich in metallic content, but estimates for Psyche's composition vary widely. Psyche has two large impact structures in its Southern hemisphere, with estimated diameters from 50 km to 70 km and estimated depths up to 6.4 km. I use the FLAG hydrocode to model the formation of the largest of these impact structures. My results indicate an oblique angle of impact rather than a vertical impact. These results also support previous claims that Psyche is metallic and porous. / Dissertation/Thesis / Psyche asteroid impact simulation initialization / Psyche asteroid impact simulation video / Doctoral Dissertation Applied Mathematics 2019

Numerical methods to solve initial value problems of differential equations progressed quite a bit in the last century. We give a brief summary of how useful numerical methods are for ordinary differential equations of first and higher order. In this thesis both computational and theoretical discussion of the application of numerical methods on differential equations takes place. The thesis consists of an investigation of various categories of numerical methods for the solution of ordinary differential equations including the numerical solution of ordinary differential equations from a number of practical fields such as equations arising in population dynamics and astrophysics. It includes discussion what are the advantages and disadvantages of implicit methods over explicit methods, the accuracy and stability of methods and how the order of various methods can be approximated numerically. Also, semidiscretization of some partial differential equations and stiff systems which may arise from these semidiscretizations are examined.

The aim of the thesis is to find a method, in conjunction with the ordinary differential equation (ODE) based method of lines (MOL) solution of Richards’ equation, to model the steep wetting front infiltration in very dry soils, accurately and efficiently. Due to the steep pressure head or steep water volumetric content gradients, highly nonlinear soil hydraulic properties and the rapid movement of the wetting front, accurate solutions for infiltration into a dry soil are usually difficult to obtain. Additionally, such problems often require very small time steps and large computation times. As an enhancement to the used ODE/MOL approach, Higher Order Finite Differencing, Varying Order Finite Differencing, Vertical Scaling, Adaptive Schemes and Non-uniform Stretching Techniques have been implemented and tested in this thesis. Success has been found in the ability of Vertical Scaling to simulate very steep moving front solution for the Burgers’ equation. Unfortunately, the results also show that Vertical Scaling needs significant research and improvement before their full potential in routine applications for difficult nonlinear problems, such as Richard’s equation with very steep moving front solution, can be realized. However, we have also shown that the use of the composed form of RE and a 2nd order finite differencing for the first order derivative approximation is conducive for modelling steep moving front problem in a very dry soil. Additionally, with the combination of an optimal influx value at the edges of the inlet, the ODE/MOL approach is able to model a 2-D infiltration in very dry soils, effectively and accurately. Furthermore, one of the strengths of this thesis is the use of a MATLAB PDE template. Implementing the ODE/MOL approach via a MATLAB PDE template has shown to be most suitable for modelling of partial differential equations. The plug and play mode of modifying the PDE template for solving time-dependent partial differential equations is user-friendly and easy, as compared to more conventional approaches using Pascal, Fortran, C or C++. The template offers greater modularity, flexibility, versatility, and efficiency for solving PDE problems in both 1-D and 2-D spatial dimensions. Moreover, the 2-D PDE template has been extended for irregular shaped domains.

In qualitative theory of differential equations, an important role is played by special classes of solutions, like periodic solutions or solutions to some boundary value problems. When a system of ordinary differential equations has equilibria, i.e. constant solutions, whose stability properties are known, it is significant to search for connections between them by trajectories of solutions of the given system. These are called homoclinic or heteroclinic, according to whether they describe a loop based at one single equilibrium or they "start" and "end" at two distinct equilibria. This thesis is devoted to the study of heteroclinic solutions for a specific class of ordinary differential equations related to the Extended Fisher-Kolmogorov equation and the Swift-Hohenberg equation. These are semilinear fourth order bi-stable evolution equations which appear as mathematical models for problems arising in Mechanics, Chemistry and Biology. For such equations, the set of bounded stationary solutions is of great interest. These solve an autonomous fourth order equation. In this thesis, we focus on such equations having a variational structure. In that case, the solutions are critical points of an associated action functional defined in convenient functional spaces. We then look for heteroclinic solutions as minimizers of the action functional. Our main contributions concern existence and multiplicity results of such global and local minimizers in the case where the functional is defined from sign changing Lagrangians. The underlying idea is to impose conditions which imply a lower bound on the action over all admissible functions. We then combine classical arguments of the Calculus of Variations with careful estimates on minimizing sequences to prove the existence of a minimum.

Integrodifference equations are discrete in time and continuous in space, and are used to model the spread of populations that are growing in discrete generations, or at discrete times, and dispersing spatially. We investigate optimal harvesting strategies, in order to maximize the profit and minimize the cost of harvesting. Theoretical results on the existence, uniqueness and characterization, as well as numerical results of optimized harvesting rates are obtained. The order of how the three events, growth, dispersal and harvesting, are arranged also affects the harvesting behavior. Cholera remains a public health threat in many parts of the world and improved intervention strategies are needed. We investigate a key intervention strategy, vaccination, with optimal control applied to a cholera model. This system of differential equations has human compartments with susceptibles with different levels of immunity, symptomatic and asymptomatic infecteds, and two cholera vibrio compartments, hyperinfectious and non-hyperinfectious. The spread of the infection in the model is shown to be most sensitive to certain parameters, and the effect of varying these parameters on the optimal vaccination strategy is shown in numerical simulations. Our simulations also show the importance of the infection rate under various parameter cases.

Species augmentation is a method of reducing species loss via augmenting declining or threatened populations with individuals from captive-bred or stable, wild populations. In this dissertation, species augmentation is analyzed in an optimal control setting to determine the optimal augmentation strategies given various constraints and settings. In each setting, we consider the effects on both the target/endangered population and a reserve population from which the individuals translocated in the augmentation are harvested. Four different optimal control formulations are explored. The first two optimal control formulations model the underlying population dynamics with a system of ordinary differential equations. Each of these two formulations utilizes a different function to model the cost of augmentation. For each optimal control formulation we find a characterization for the optimal control and show numerical results for scenarios of different illustrative parameter sets. The second two optimal control formulations model the underlying population dynamics with systems of discrete difference equations. The difference between these two optimal control formulations is the order in which events occur within each time step in the population models. In the first formulation the population is augmented before the natural growing season in each time step (augment then grow model), whereas in the second formulation the population is augmented after the natural growing season in each time step (grow then augment model). These two discrete time models, which differ only in their order of events, lead to structurally different models. The formulation with the augment then grow model cannot utilize discrete time optimal control theory and a brute force method of finding the optimal augmentation strategy is used. The formulation with the grow then augment model does utilize optimal control theory and we find the characterization of the optimal control. For both formulations, we explore several scenarios of different illustrative parameter sets. In each of the four optimal control formulations, the numerical results provide considerably more detail about the exact dynamics of optimal augmentation than can be readily intuited. The work presented here are the first steps toward building a general theory of population augmentation, which accounts for the complexities inherent in many conservation biology applications.

This thesis consists of two main chapters along with an introduction andconclusion. In the introduction, we address the inspiration for the thesis, whichoriginates in a common calculus problem wherein travel time is minimized across two media separated by a single, straight boundary line. We then discuss the correlation of this problem with physics via Snells Law. The first core chapter takes this idea and develops it to include the concept of two media with a circular border. To make the problem easier to discuss, we talk about it in terms of running and swimming speeds. We first address the case where the starting and ending points for the passage are both on the boundary. We find the possible optimal paths, and also determine the conditions under which we travel along each path. Next we move the starting point to a location outside the boundary. While we are not able to determine the exact optimal path, we do arrive at some conclusions about what does not constitute the optimal path. In the second chapter, we alter this problem to address a rectangular enclosed boundary, which we refer to as a swimming pool. The variations in this scenario prove complex enough that we focus on the case where both starting and ending points are on the boundary. We start by considering starting and ending points on adjacent sides of the rectangle. We identify three possibilities for the fastest path, and are able to identify the conditions that will make each path optimal. We then address the case where the points are on opposite sides of the pool. We identify the possible paths for a minimum time and once again ascertain the conditions that make each path optimal. We conclude by briefly designating some other scenarios that we began to investigate, but were not able to explore in depth. They promise insightful results, and we hope to be able to address them in the future.

Malaria is a preventable and treatable blood-borne disease whose complications can be fatal. Although many interventions exist in order to reduce the impacts of malaria, the optimal method of distributing these interventions in a geographical area with limited resources must be determined. This thesis refines a model that uses an integer linear program and a compartmental model of epidemiology called an SIR model of ordinary differential equations. The objective of the model is to find an intervention strategy over multiple time steps and multiple geographic regions that minimizes the number of days people spend infected with malaria. In this paper, we refine the resolution of the model and conduct sensitivity analysis on its parameter values.

Delay differential equations (DDEs) are a class of differential equations that have received considerable recent attention and been shown to model many real life problems, traditionally formulated as systems of ordinary differential equations (ODEs), more naturally and more accurately. Ideally a DDE modeling package should provide facilities for approximating the solution, performing a sensitivity analysis and estimating unknown parameters. In this thesis we propose new techniques for efficient simulation, accurate sensitivity analysis and reliable parameter estimation of DDEs. We propose a new framework for designing a delay differential equation (DDE) solver which works with any supplied initial value problem (IVP) solver that is based on a general linear method (GLM) and can provide dense output. This is done by treating a general DDE as a special example of a discontinuous IVP. We identify a precise process for the numerical techniques used when solving the implicit equations that arise on a time step, such as when the underlying IVP solver is implicit or the delay vanishes. We introduce an equation governing the dynamics of sensitivities for the most general system of parametric DDEs. Then, having a similar view as the simulation (DDEs as discontinuous ODEs), we introduce a formula for finding the size of jumps that appear at discontinuity points when the sensitivity equations are integrated. This leads to an algorithm which can compute sensitivities for various kind of parameters very accurately. We also develop an algorithm for reliable parameter identification of DDEs. We propose a method for adding extra constraints to the optimization problem, changing a possibly non-smooth optimization to a smooth problem. These constraints are effectively handled using information from the simulator and the sensitivity analyzer. Finally, we discuss the structure of our evolving modeling package DDEM. We present a process that has been used for incorporating existing codes to reduce the implementation time. We discuss the object-oriented paradigm as a way of having a manageable design with reusable and customizable components. The package is programmed in C++ and provides a user-friendly calling sequences. The numerical results are very encouraging and show the effectiveness of the techniques.

Delay differential equations (DDEs) are a class of differential equations that have received considerable recent attention and been shown to model many real life problems, traditionally formulated as systems of ordinary differential equations (ODEs), more naturally and more accurately. Ideally a DDE modeling package should provide facilities for approximating the solution, performing a sensitivity analysis and estimating unknown parameters. In this thesis we propose new techniques for efficient simulation, accurate sensitivity analysis and reliable parameter estimation of DDEs. We propose a new framework for designing a delay differential equation (DDE) solver which works with any supplied initial value problem (IVP) solver that is based on a general linear method (GLM) and can provide dense output. This is done by treating a general DDE as a special example of a discontinuous IVP. We identify a precise process for the numerical techniques used when solving the implicit equations that arise on a time step, such as when the underlying IVP solver is implicit or the delay vanishes. We introduce an equation governing the dynamics of sensitivities for the most general system of parametric DDEs. Then, having a similar view as the simulation (DDEs as discontinuous ODEs), we introduce a formula for finding the size of jumps that appear at discontinuity points when the sensitivity equations are integrated. This leads to an algorithm which can compute sensitivities for various kind of parameters very accurately. We also develop an algorithm for reliable parameter identification of DDEs. We propose a method for adding extra constraints to the optimization problem, changing a possibly non-smooth optimization to a smooth problem. These constraints are effectively handled using information from the simulator and the sensitivity analyzer. Finally, we discuss the structure of our evolving modeling package DDEM. We present a process that has been used for incorporating existing codes to reduce the implementation time. We discuss the object-oriented paradigm as a way of having a manageable design with reusable and customizable components. The package is programmed in C++ and provides a user-friendly calling sequences. The numerical results are very encouraging and show the effectiveness of the techniques.