My research interests lie in the areas of numerical analysis, operator theory, differential equations, and nonlinear analysis. Within these fields, I also have interests in approximation methods, spectral theory, fractional and stochastic
differential operators, Lie group methods, and Hopf algebras. However, the majority of my contributions have been in the fields of operator theory and numerical analysis.

I am currently engaged in several areas of research. In order to help any interested readers, I have split this page into sections which contain my projects falling into the same general category. These categories are constantly expanding, so I make no claim that these descriptions are in any way complete. Readers may contact me via email if they would like more details regarding one (or all) of the listed areas. Each section may be expanded by clicking on the title of said section. For the sake
of clarity, within each section there is a tabbed “Results” addendum. This tabbed section briefly outlines my personal contributions to each subject.

2. Operator Splitting and Approximation

Operator theory, operator splitting, and approximation of operators have been my primary research areas, throughout my academic career. My endeavors have considered both deep theoretical issues and computational issues. Using techniques from each of these fields, I analyze various numerical methods via appropriate abstract techniques. Operator splitting methods enjoy numerous useful properties (such as stability, explicitness, efficiency, sub-problems may often be solved exactly, preserves geometric invariants, etc) and have been well-studied in numerous classical settings. However, there are still a wealth of open problems in regards to the abstract analysis and applications of these methods to non-classical settings (stochastic, fractional, optimization problems, and data analysis). Thus, many of my goals center around the exciting prospect of extending the notions of operator splitting to these novel settings.

2.1.

Let $X$ be a Banach space endowed with some norm given by $\|\cdot\|.$ I am interested in considering abstract initial
value problems of the form
\begin{equation}\label{eq1} u' = F(u), \quad 0\le t\le T, \quad u(0) = u_0 \in X, \end{equation}
for a Banach space-valued function $u\,:\,[0,T]\to X,$ where the structure of the (unbounded) nonlinear operator $F\,:\,D(F)\subseteq X\to X$ suggests a decomposition into two (or more) parts
\begin{equation}\label{eq2}
F(v) = A(v) + B(v), \quad v\in D(A)\cap D(B),
\end{equation}
and the (unbounded) nonlinear operators, $A\,:\,D(A)\subseteq X\to X$ and $B\,:\,D(B)\subseteq X\to X$ have suitable domains. Employing the formal calculus of Lie derivatives, one may write the solution to $(\ref{eq1})$
as
\begin{equation}\label{lie}
u(t) = e^{tD_F}u_0,\quad 0\le t\le T,
\end{equation}
where $D_F$ is the Lie derivative of the nonlinear operator $F.$ My primary interest in such problems may be
summarized as follows.

Question: What are the properties of the operator $e^{tD_F}?$ Moreover, how well do various approximations to this operator preserve these properties?

Thus, my goal is to study and develop operators which approximate $e^{tD_F}.$ Such approximations have been
well-studied, but there is still much work to be done when considering the qualitative aspects of these operators; such properties include underlying geometric features and long-time solution dynamics. Moreover, the application of such methods to operators involving stochastic components is fairly unexplored. To this end, I consider (unbounded) nonlinear operators of the form
\begin{equation}\label{approx1}
S(t) = \sum_{j=1}^K \gamma_j \prod_{k=1}^{N_j}H_A^{j,k}\left(\alpha_{j,k}tD_A\right)H_B^{j,k}\left(\beta_{j,k}tD_B\right)
\end{equation}
where $K,$ and $N_j$ are integers, and $\gamma_j,$ $\alpha_{j,k}$ and $\beta_{j,k}$ are complex constants. The $H_A^{j,k}$ and $H_B^{j,k}$ are appropriate approximations of the nonlinear evolution operators associated with the operators $A$ and $B,$ respectively—an example of such an approximation would be the Cayley operator associated with either $A$ or $B.$

With the formulations above, my research goals in this direction may be summarized as follows.

Goal: Develop novel operator splitting approximations, $S(t),$ which still respect the underlying qualitative properties of the original operator, $e^{tD_F}.$

Goal: Determine under what conditions and with what rate the operator $S(t)$ converges to the solution operator $e^{tD_F}.$ Moreover, how do these results depend on the underlying topology of the space?

The aforementioned goals are highly nontrivial. In the finite-dimensional case, one can easily verify the desired results via repeated applications of the Baker-Campbell-Hausdorff formula. However, the general setting is not so straightforward, and one must draw heavily upon operator theory, nonlinear functional analysis, and
differential equation theory. In particular, I often employ combinations of semigroup theory, Lie group methods, and Magnus-type integrators [5,11].

2.2.

My contributions in this direction have primarily involved the analysis of various operator approximations in
generalized settings. In particular, I have extending existing results to handle unbounded operators, singular operators, stochastic operators, and coupled problems.

2.2.1.

Due to its wide range of applications in sciences and engineering, I have been interested in the following stochastic differential problem,
\begin{equation}\label{5}
du = [Au + f(u)]\,dt + g(u)\,dW,\quad 0\le t\le T,\quad u(0) = u_0\in H,
\end{equation}
where $H$ is a separable Hilbert space. In the above, $A\,:\,D(A)\subseteq H\to H$ is a linear operator whose domain is dense in $H$ and compactly embedded in $H.$ It is further assumed that $A$ generates an analytic semigroup $e^{tA},\ t\ge 0.$ The operators $f$ and $g$ are assumed to be Lipschitz continuous and possess continuous, uniformly bounded Fréchet derivatives up to order two. Moreover, $W$ is a standard Wiener process with respect to a normal filtration $\{\mathcal{F}_t\}_{t\in[0,T]}$ in a complete probability space $(\mathcal{B},\mathcal{F},\mathbb{P}).$

As noted earlier, I am interested in abstract operators $S\,:\,H\to H$ which approximate the nonlinear evolution operator solving $(\ref{5}).$ A particularly nice example is given by
\begin{equation}\label{7}
S(t) \mathrel{\mathop:}= e^{tA}e^{tD_f}e^{\Delta W(t)D_g},
\end{equation}
where the nonlinear operator $e^{tD_f}$ is associated with the differential equation $dv = f(v)\,dt,$ and $e^{\Delta W(t)D_g}$ is associated with the stochastic differential equation $dz = g(z)\,dW.$

When considering the operator given by $(\ref{7})$ we have the following result. For the sake of presentation, we represent the true solution operator for $(\ref{5})$ by $T(t).$

Theorem 1. Assume that $u_0 \in D(A)$ and that the operator $A$ and functions $f$ and $g$ satisfy certain (reasonable) regularity conditions. Then the mean square expected value of the approximation error is given by
$$\mathbb{E}\|T(t)u - S(t)u\|^2 \le Ct^{2+\beta},$$
where $\beta \in (0,1]$ is a parameter determined by the smoothness of the underlying Wiener process.

This result is the first such extension to the abstract setting and may also be employed to consider the global properties associated with such an approximation. This result is sharp as the corresponding finite-dimensional
case can demonstrate.

Results: To my knowledge, my work in applying operator splitting techniques to the general abstract stochastic setting is the first such consideration. My results have demonstrated the strong convergence properties of operator splitting approximations while also providing novel functional analytic insight into stochastic operators. In particular, I developed bounds on the logarithmic norms of such operators as they pertain to $(\ref{5}),(\ref{7})$ Such nonlinear functional bounds are extremely useful in practice as it provides insight into what spectral properties unbounded operators should possess for the operator splitting methods to be valid. Moreover, these bounds may be employed numerically to accurately and efficiently develop adaptive time-stepping methods for approximating differential equations.

2.2.2.

Much of my doctoral thesis work focused on the accurate approximation of the solution operators associated with problems exhibiting finite-time singularities in both the temporal and spatial regimes. To illustrate, let $(\mathcal{B},\mathcal{F},\mathbb{P})$ be a complete probability space and let $W(x,t)$ be a standard space-time Wiener process, in this space. I am then interested in problems of the form (disregarding initial and boundary conditions)
\begin{equation}\label{k1}
\sigma(x)du = [A(x,t)u + f(u)]\,dt + g(u)\,dW,
\end{equation}
where $A$ is some (elliptic) differential operator, $\sigma$ is a positive spatial function vanishing on portions of the domain boundary, and $f$ and $g$ are positive, monotonically increasing functions with the property that for some value $0 < c < \infty$ we have
\begin{equation}\label{k2}
\lim_{\|u\|\to c^-} f(u) = \infty\quad\mbox{and}\quad \lim_{\|u\|\to c^-}g(u) = \infty.
\end{equation}
Thus, any operator approximation must be able to handle the regularity issues induced by the function $\sigma,$ while also being robust enough to handle the solution dependent singularities arising due to $f$ and $g.$ Moreover, it is
the case that the condition $\|u\|\to c^-$ may only occur when the domain has a certain geometry or size, meaning that such approximations must be able to capture both of the qualitatively different solution
features.

Most of my research has focused on demonstrating that operator splitting was an appropriate choice for such problems. Such approximations allowed for the problems to be adaptively handled by employing monitoring functions that depend on the given nonlinearities. Moreover, such approximations were shown to preserve various features of the true solution relating to the singular nature of $f$ and $g,$ such as recovering the time, $t,$ for which $(\ref{k2})$ occured (almost surely)—a phenomenon known as quenching [7,9].

Results: While the initial goal of this work was to develop improved numerical techniques for solving singular problems, the results yielded a deeper understanding of operator splitting from a theoretical viewpoint. In particular, a deeper understanding of the spectral properties of the underlying operators and their approximations (in particular, abstract Jacobi operators) resulted from the analysis. Much of the work, and my resulting thesis, focused on developing a fully nonlinear convergence analysis that was valid up to the solution's singularity. This greatly improved upon existing analysis and yielded approximations that did not degrade as the singularity was approached. The operator splitting methods were also shown to behave equally well in conjunction with other quadrature or operator approximation methods. Most importantly, I showed that these operator splitting techniques preserved the important qualitative features of $(\ref{k1}),$ namely, solution positivity, monotonicity, and the quenching time.

The videos below demonstrate a quenching solution in both one- and two-dimensions. In order to emphasize the issues that arise, the maximum value of the temporal derivative is displayed for each time (you will need to press the "play" button to run the videos).

2.2.3.

Systems of reaction-diffusion equations are commonly used in biological models of food chains. The populations and their complicated interactions present numerous challenges in theory and in numerical approximation. Of particular interest, are models involving self- and cross-diffusion. Self-diffusion is a nonlinear term that models overcrowding of a particular species and cross-diffusion models the inclination of a prey to move towards or away from its prey [14,16]. These nonlinearities complicates attempts to construct approximations of the solution operator associated with such systems of equations. Thus, one of my goals has been to develop nonlinear operator splitting algorithms designed for such systems. To clarify the problem, an example of such a system is the following
\begin{eqnarray}
&& \partial_t u = \Delta\left(d_1u + s_1u^2 + c_{12}vu\right) + f(u,v)\label{E1}\\
&& \partial_t v = \Delta\left(d_2v + s_2v^2 + c_{21}uv\right) + g(u,v)\label{E2}
\end{eqnarray}
which would be coupled with appropriate boundary and initial conditions. Using this model my collaborators and I focused on the development and analysis of nonlinear splitting algorithms for approximating $(\ref{E1})-(\ref{E2}).$ This particular direction has numerous open problems as there is a need for a deeper understanding of standard existence and uniqueness results, as well as the potential application of operator splitting techniques.

Results: This problem has produced a considerable number of complications which are not present in the standard linear models. The nonlinearities and coupled features fit into the standard abstract framework, but this unfortunately is not so helpful when actually producing the approximation. Since these systems are involved in biological models, such as predator-prey and food chain models, any simulations must be quite large and be able to run for long periods with high accuracy. Thus, we introduced a novel operator splitting in conjunction with appropriate Padé approximations to the evolution operators. We then incorporated a novel factorization technique allowing us to decouple the problem and iterate the solution by composing solutions to multiple one-dimensional decoupled sub-problems. Such an approach results in greatly improved efficiency. My primary contribution in this direction was the associated abstract and applied analysis. The operator theoretical approach allowed for the consideration of a larger class of problems while also providing the necessary nonlinear convergence analysis. These operator methods were also demonstrated to accurately handle the potential finite-time blow-up of solutions for certain initial data (see, for instance, [22]).

The figure below demonstrates the efficiency of operator splitting for solving problems such as $(\ref{E1})-(\ref{E2}).$

2.3.

With respect to stochastic operator splitting, I am currently developing analysis for “higher-order” operator splitting methods. I am also working to analyze the ability to breach the so-called Sheng-Suzuki barrier imposed on stable operator splitting methods for parabolic-type problems. My intermediate and long-term research goals involve developing the weak convergence analysis of these splitting methods, which will involve a detailed analysis of the associated Kolmogorov equation associated with such operators (see [24] for details).

The analysis of operator splitting as applied to the Kawarada equations requires a much deeper nonlinear analysis. Currently I am working to provide an exact quantification of how operator splitting methods preserve quenching time (and other quenching characteristics). In particular, the goal is to demonstrate that the local order of the method is the rate at which the approximate quenching time converges to the true quenching time. A long-standing issue in these singular problems is the development of a fully nonlinear error analysis that may be used in numerical methods to develop efficient time-stepping methods. In the case of a quenching solution, any form of linearization immediately results in errors that restrict the numerical scheme's efficiency. As such, the development of such an analysis (or the improvement of existing analysis) is a long-term goal.

The nonlinear coupled problem has numerous open avenues of research, many of which are amenable to beginning graduate and advanced undergraduate students. The three-species model is completely unexplored with respect to operator splitting, and as such, an immediate goal is the development and analysis of such a method. Moreover, including stochastic influences into the model will provide novel hurdles for any operator theoretical approach. With such a generalization, the problem introduces the additional long-term goal of developing the necessary abstract analysis of the partial differential equation.

3. Fractional Differential Equations

Fractional differential equations are a newly blossoming field of interest that have applications in numerous scientific fields. My primary interest in such problems revolves around the study of the highly non-local solution operators which arise when solving equations involving fractional derivatives. This direction yields numerous interesting mathematical problems which have important open questions that deserve both theoretical and computational attention.

I have considered two distinct problems involving fractional operators. I briefly outline the projects below.

3.1.

This problem was first introduced to me by Akif Ibraguimov, who was kind enough to work with me during my first year as a postdoctoral scholar. His invaluable insight improved the project greatly.

Let $X$ be a real Banach space with norm denoted $\|\cdot\|.$ I am interested in the following singular Bessel-type problem
\begin{eqnarray}
&& u''(t) + \frac{\alpha}{t}u'(t) = -Au(t),\quad t\in (0,\infty),\label{b1}\\
&& u(0) = u_0\in X, \label{b2}
\end{eqnarray}
where $\alpha\mathrel{\mathop:}=1-2s,\ s\in (0,1),$ and $A\,:\,D(A)\subseteq X\to X$ is linear. Moreover, I am interested in the case when the operator $A$ is closed, densely defined, and strictly $m-$dissipative in $X$&emdash;such an operator would include the standard Laplacian with Dirichlet boundary data. Under these assumptions, I showed the following result.

Theorem 2. Let $u_0\in D((-A)^s).$ Then a solution to $(\ref{b1})-(\ref{b2})$ is given by
\begin{equation}\label{usol1}
u(t) = \frac{1}{\Gamma(s)}\int_0^\infty z^{s-1}e^{-t^2/4z}T(z)(-A)^su_0\,dz
\end{equation}
and also satisfies
\begin{equation}\label{uder}
\lim_{t\to 0^+} t^{1-2s}u'(t) = c_s (-A)^su_0,
\end{equation}
where $T$ is the standard heat semigroup generated by $A$ and $c_s\mathrel{\mathop:}= 2^{1-2s}\Gamma(1-s)/\Gamma(s).$

Thus, the problem $(\ref{b1})-(\ref{b2})$ may be seen as being related to the fractional powers of the operator $-A.$ In particular, $(\ref{uder})$ demonstrates that taking the trace of the co-normal derivative of $(\ref{usol1})$ yields an expression for the fractional power of the operator $A.$ This connection is made via a special form of the Poincaré-Steklov operator and is in fact a generalization of the celebrated results of Caffarelli and Silvestre [4]. While the numerous applications and studies in the direction of this problem are all worthy of mention, for brevity I will omit them and be happy to pursue $(\ref{b1})-(\ref{b2})$ simply as an interesting mathematical problem with even more interesting difficulties.

As before, the goal is to develop and analyze an approximation to the above solution operator. To aid in my goal, I showed that the solution given by $(\ref{usol1})$ decays exponentially with respect to $t,$ and thus one may truncate the improper integral with an error term depending upon the spectral properties of the underlying operator, $A.$ To that end, fix $M\in \mathbb{N}$ (reasonably large) and $M < N\in\mathbb{N}.$ Then for $u_0\in D((-A)^s),$ one may consider
\begin{equation}\label{approx}
v(t)\mathrel{\mathop:}= \sum_{k=0}^N \gamma_k(t) S^k(h)S(h/2)(-A)^su_0,
\end{equation}
where $h\mathrel{\mathop:}= M/(N+1),$
$$
\gamma_k(t)\mathrel{\mathop:}= \frac{h^s[(k+1)^s-k^s]}{\Gamma(1+s)}e^{-t^2/4z_{k+1/2}},\quad\mbox{and}\quad
S(h)\mathrel{\mathop:}= \left(I-\frac{h}{2}A\right)^{-1}\left(I+\frac{h}{2}A\right),
$$
with $z_{k+1/2}\mathrel{\mathop:}= z_k + h/2.$ Under the assumption that the Cayley operator $S(w)$ is nonexpansive in $X,$ which is always true if $X$ is in fact a Hilbert space, I showed the following result.

Theorem 3. Let $u_0\in D((-A)^{\ell + s}),\ \ell = 1,2.$ Then, we have
$$\left\|u(t) - v(t)\right\| \le C\left(M^{s-1}e^{\mu(A)M} + h^{\ell+s-1}\right),\quad t\in [0,\infty),$$
where $\mu(A)$ is the logarithmic norm of $A$ and $C$ is a constant independent of $h$ and $k=0,\ldots,N.$

Moreover, it is the case that this operator approximation method also preserves the co-normal derivative property demonstrated in $(\ref{uder}).$

While this result may not seem terribly exciting, it demonstrates that fractional powers of an operator may be realized via the composition of simple operator approximations associated with the original operator. This allows one to bypass the considerable difficulty that arises with the nonlocal nature of fractional operators and apply analysis techniques which have been well established for the classical problems.

Results: The results presented herein are, to my knowledge, the first attempts to extend the notion of operator splitting and Padé approximations to the fractional Laplace equation (and its related problems). The results are an improvement upon many existing approximations as one does not need explicit knowledge of the eigenvalues of the Laplacian, which is the standard approach of many methods. Moreover, the proposed approximation works well on all relevant domains (and can readily be extended to arbitrary manifolds with their associated fractional Laplace-Beltrami operator). My results are more general than simply considering fractional powers of the Laplace operator, allowing one to consider any appropriate elliptic operator (and thus having a wider range of applications). Moreover, the analysis still holds valid for appropriate approximations of the operator $A,$ and the logarithmic norm incorporation provides an explicit method to control the truncation error of $(\ref{approx}).$

3.2.

The second problem of interest is a generalization of the Kawarada problem from Section 2. It is worth noting that this generalization is not one without reason. The Kawarada equations are known to model various solid-fuel combustion processes [6]. Such processes are known to be highly nonlocal and the associated evolution process exhibits the property of “long-term memory” and anomalous diffusion [21]. Such processes require a more generalized model, such as one involving fractional differential operators.

Let $\Omega$ be an open bounded domain in $\mathbb{R}^d$ with smooth boundary $\partial\Omega.$ We then define $Q_T \mathrel{\mathop:}= \Omega\times (0,T)$ and the parabolic boundary $\Gamma_T = \partial\Omega\times (0,T).$ I am interested in the following nonlocal Kawarada problem
\begin{eqnarray}
&&\partial_t^\alpha u = -(-\Delta)^s u + f(u), \quad (x,t) \in Q_T,\label{kk1}\\
&&u = 0, \qquad \qquad\qquad\qquad\quad(x,t) \in \Gamma_T,\label{kk2}\\
&&u(x,0) = u_0(x), \qquad\qquad\qquad~\, x \in \Omega,\label{kk3}
\end{eqnarray}
where $\partial_t^\alpha$ denotes the Caputo time-fractional derivative of order $\alpha\in(0,1),$ $(-\Delta)^s$ is the fractional Laplacian with $s\in(0,1),$ and the continuous initial data $u_0\,:\,\Omega\to \mathbb{R}_+$ is such that $0 \le u_0 \ll c.$ The nonlinear reaction term $f\,:\,B_\rho\to \mathbb{R}_+,$ where $0<\rho< c$ and $B_\rho\mathrel{\mathop:}=\{u\in L^\infty(\Omega)\,:\, \|u\|_\infty < \rho\},$ is a given continuous, convex function satisfying a local Lipschitz condition on $B_\rho.$ We further assume that $f$ is a monotonically increasing function on $B_\rho$ and
\begin{equation}\label{lim}
\lim_{u\to c^-}f(u) = \infty.
\end{equation}
The formulation of this problem requires a bit of terminology and I refer the interested reader to [23]. for more information. To avoid confusion, it suffices to say that the fractional power of the negative Laplacian may be defined as in Section 3.1 or through standard functional calculus, while the Caputo derivative (for smooth enough functions) may be defined as
\begin{equation}\label{caputo}
\partial_t^\alpha v(t) \mathrel{\mathop:}= \frac{1}{\Gamma(1-\alpha)}\int_0^t (t-s)^{-\alpha}v'(s)\,ds,\quad \alpha\in (0,1].
\end{equation}

My particular interest in this problem lies in the study of the associated nonlinear solution operator. The solution operator is given in terms of the Mittag-Leffler function, as opposed to the heat semigroup in the case of integer order derivatives.

To better understand what I mean, I will briefly introduce these operators (which have primarily been of interest in complex analysis and special function theory). Let $\alpha,\beta\in\mathbb{C}$ with $\mbox{Re}(\alpha),\mbox{Re}(\beta)>0.$ Then we may define the two parameter Mittag-Leffler function to be
\begin{equation}\label{ml1}
E_{\alpha,\beta}(z) \mathrel{\mathop:}= \sum_{n=0}^\infty \frac{z^n}{\Gamma(\beta+\alpha n)} = \frac{1}{2\pi i}\int_{\gamma}\frac{\lambda^{\alpha-\beta}e^\lambda}{\lambda^\alpha - z}d\lambda,
\end{equation}
where $\gamma$ is a contour which starts at $-\infty$ and encircles the disc $|\lambda|\le |z|^{1/\alpha}$ counterclockwise. Moreover, let $\sigma(A)$ and $\rho(A)\mathrel{\mathop:}= \mathbb{C}-\sigma(A)$ be the spectrum and resolvent set of the operator $A \mathrel{\mathop:}= (-\Delta)^s,$ respectively.
We then define the family of operators $\{S_\alpha(t)\}_{t\in[0,T]}$ and $\{P_\alpha(t)\}_{t\in[0,T]}$ to be
\begin{equation}\label{ml2}
S_\alpha(t)\mathrel{\mathop:}= E_{\alpha,1}(-zt^\alpha)(A) = \frac{1}{2\pi i}\int_{\Gamma_\theta}E_{\alpha,1}(-zt^\alpha)R(z; A)\,dz,
\end{equation}
\begin{equation}\label{ml3}
P_\alpha(t)\mathrel{\mathop:}= E_{\alpha,\alpha}(-zt^\alpha)(A) = \frac{1}{2\pi i}\int_{\Gamma_\theta} E_{\alpha,\alpha}(-zt^\alpha)R(z;A)\,dz,
\end{equation}
where $\Gamma_\theta$ is any contour containing $\sigma(A)$ and $R(z,A)$ is the resolvent operator defined as $R(z;A) \mathrel{\mathop:}= (zI-A)^{-1},\ z\in\rho(A).$

With these fractional operators in place, one can show that any mild solution (weak solutions are of less
interest in practical combustion applications) of $(\ref{kk1})-(\ref{kk3})$ is of the form
\begin{equation}\label{ksol1}
u(x,t) = S_\alpha(t)u_0(x) + \int_{0}^t (t-s)^{\alpha-1}P_\alpha(t-s)f(u(x,s))\,ds,\quad x\in\Omega.
\end{equation}
After obtaining a solution, it becomes important to explore the properties of this solution and see how they compare with the known properties of the integer-order differential problem (which is a limiting case of $(\ref{kk1})-(\ref{kk3})$). The results of my work may be summarized as follows.

Results: This work, thus far, has resulted in one publication (see [14]) whose focus was on the properties of the solution operator given in $(\ref{ksol1})$ and the recovery of expected solution properties. It was shown under reasonable (and expected) assumptions on the initial data that the solution may only exist locally in time. The local solution was then shown to be positive and monotonically increasing on this interval of existence. The same held in the case of a global solution, as well. Moreover, I demonstrated that the existence of a global solution depends explicitly upon the given spatial domain. That is, for a fixed domain shape, the volume of the domain will uniquely determine the existence of global solutions. While not a focus of the article, the results demonstrated that for a fixed domain shape, monotonically increasing the volume of the domain will result in a monotonically decreasing interval of existence for the mild solution.

3.3.

Research involving fractional powers of abstract operators is still in its infancy, thus, there are numerous open problems as well as points of entry for interested graduate students. There is a need to explore improved operator approximations and provide more detailed analysis. I am also interested in extending this work to include parameters that correspond to larger fractional powers of the operator (which are qualitatively different operators). My long-term goal, with respect to this problem, is to develop a method for the nonlinear fractional problem. Such problems have been studied from a differential equation viewpoint, but have received
little operator theoretical attention. In particular, my goal is to develop a method similar to the approach of Magnus in Lie group theory, which would allow me to contruct a locally viable solution to the nonlinear problem which resembles $(\ref{usol1})$ (where one would replace $T(t)$ by the appropriate nonlinear
semigroup.

The fractional Kawarada problem provides two avenues of research. First, I intend to continue studying the properties of the differential equation. I hope to obtain explicit representations of the relationship between existence of solutions and domain geometries. In particular, I am looking into the effects of nonsmooth
boundaries on this phenomenon. Moreover, I hope to better understand how such singularities affect the weak solutions to $(\ref{kk1})-(\ref{kk3}),$ and what properties are still retained by the weak solution. The second avenue of research
is a long-term goal and one which I find extremely interesting. I intend to develop a notion of operator splitting for the Mittag-Leffler operators considered in Section 3.2. These operators are highly nonlocal, which is problematic for splitting (as it is a local process). However, I have developed (and will publish) a method to make the operators more local and am currently working on appropriate splitting methods for this formulation. The analysis involving fractional operators is intriguing and provides many avenues for interested students to join the work, as
well. This approach has also offered me the opportunity to begin working with Angie Peace (Texas Tech University), as well, which will provide numerous new avenues to apply these results to problems in mathematical epidemiology.

4. Spectral Theory for Anderson-type Hamiltonians

Some of, what I believe to be, my most interesting work has been done in conjunction with the phenomenal analyst Constanze Liaw, from the University of Deleware, and physicist Eva Kostadinova, Lorin Matthews, and Truell Hyde, from Baylor University. Conni pioneered a novel approach for studying spectral properties of operators, which allows one to mathematically study the Anderson localization phenomenon independent of artificially imposed boundary conditions or contruction of eigenfunctions. Continuing in this direction, we have now extended this approach to consider fractional operator models, as well. Most importantly, this work has
demonstrated results that better match observed experimental data due to the chosen approach and our improved model.

4.1.

In 1958 P. W. Anderson suggested that sufficiently large impurities in a semiconductor could lead to spatial localization of electrons. The fact that some incoming electrons will remain trapped within the medium can be reformulated in mathematical terms as a nontrivial singular component of an operator's spectral measure. A general mathematical model associated with this physical phenomenon is called the Anderson-type Hamiltonian. This model consists of a self-adjoint operator plus a self-adjoint perturbation, which has a probabilistic component to model the random impurities. Examples of Anderson-type Hamiltonians include the discrete
random Schrödinger and Jacobi operators [12].

To demonstrate the problem, let $n\in\mathbb{N}$ and consider the probability space $\Omega_n = (\mathbb{R},\mathcal{B},\mu_n),$ where $\mathcal{B}$ is the Borel sigma-algebra on $\mathbb{R}$ and $\mu_n$ is a probability measure. Let $\Omega = \textstyle\prod_{n=0}^\infty \Omega_n$ be a product space with the probability measure $\mathbb{P}$ on $\Omega$ introduced as the product measure of the corresponding measures on $\Omega_n$ in the product sigma-algebra $\mathcal{A}.$ The elements of $\Omega$ are points in $\mathbb{R}^\infty,$ $\omega = (\omega_1,\omega_2,...)$ for $\omega_n\in\Omega_n.$
Let $H$ be a separable Hilbert space and let $\{\varphi_n\}_{n\in\mathbb{N}}$ be a countable collection of unit vectors in $H.$ For each $\omega\in\Omega$ we define an Anderson-type Hamiltonian on $H$ as a self-adjoint operator of the form
\begin{equation}\label{and1}
A_\omega = A + V_\omega,\quad\mbox{where}\ V_\omega = \sum_n\omega_n\langle\cdot,\varphi_n\rangle\varphi_n.
\end{equation}
Since in most cases the perturbation $V_\omega$ is almost surely a non-compact operator, one cannot apply results from classical perturbation theory to study the spectrum of $A_\omega.$ Thus, it became imperative to develop a novel approach.

While there are numerous definitions of delocalization (or equivalently, localization) in the literature, our approach defines the term delocalization to mean the existence of absolutely continuous spectrum almost surely. One can note that such a delocalization implies the standard dynamical delocalization. It is well-known that if an Anderson-type Hamiltonian has purely singular spectrum almost surely then the operator is cyclic almost surely. The equivalent statement—if an Anderson-type Hamiltonian is not cyclic with positive probability then there are energies which are diffusive with nonzero probability—is the basis of the proposed method for studying localization properties of such operators. The idea is summarized as follows (see [10]).

Theorem 4. Under appropriate technical assumptions, if localization occurs, then the orbit of any nonzero vector under the operator $A_\omega$ is almost surely dense in $H.$ More specifically, fix $0\neq f\in H.$ If $A_\omega$ has purely singular spectrum almost surely, then for all $v\in H$ with norm $\|v\|=1$ one has
$$\mbox{dist}\left(v,\mbox{span}\left\{A_\omega^kf\,:\,k\in\{0,1,2,...,n\}\right\}\right) \to 0,$$
as $n\to\infty,$ $\mathbb{P}-$almost surely.

4.2.

With Theorem 4 (and its various implications) in hand, we have studied the localization properties of various discrete Anderson-type Hamiltonians on connected (infinite) graphs. These studies have focused mostly on the standard integer lattice graphs, but have also included more exotic graphs with fractional Hausdorff dimensions. Moreover, the studies have demonstrated that the critical threshold of the magnitude of the perturbation is much lower than had been previously expected [8]. Interestingly, the computation proposed by Theorem 4 can be significantly simplified by taking advantage of the self-adjoint structure of the operator $A_{\omega}$ during a Gram-Schmidt procedure. It can be shown that the computation only requires the use of the current vector in the orbit and the two prior ones
[8,10,15].

My main contributions have stemmed from the introduction of the fractional Laplacian as a choice for $A.$ It is known that delocalization always occurs for the standard Laplacian in dimension one for any nontrivial perturbation $V_\omega.$ However, it has recently been experimentally observed that delocalization in one-dimentional chains can occur \cite{billy2008direct}. This motivated the consideration of the more general discrete fractional Anderson-type Hamiltonian. From this, I have been able to develop the following conjecture.

Conjecture 1. Let $H = \mathbb{Z}$ and $A = -(-\Delta)^s,\ s\in(0,2).$ Then, if $s\in [1,2),$ the operator $A_\omega$ exhibits purely singular spectrum almost surely, and if $s\in (0,1),$ the operator $A_\omega$ exhibits purely absolutely continuous spectrum almost surely.

This conjecture provides a critical parameter on the fractional Laplacian for which we have a “spectral state change” for Anderson-type Hamiltonians. With this change there is a monotonic change of the necessary critical magnitude of the perturbation which can once again shift the spectrum. This conjecture is supported by
numerous numerical experiments and is the focus of ongoing research and a collaborative NSF grant proposal with the Plasma Physics group at Baylor University. I will close this section with a final conjecture which is topologically of great interest.

Conjecture 2. Let $H$ be a (infinite) graph and $A$ be the standard Laplacian operator defined on this graph. Then there exists a critical Hausdorff dimension for which the spectrum will “switch” to being purely singular, regardless of the magnitude of a non-degenerate perturbation.

This conjecture is loosely worded due to the obvious issue of the interplay between the actual geometry of the graph and its associated dimension. However, this conjecture follows from the intimate relationship between fractional derivatives and fractal domains. While this second conjecture is very premature, it opens a new area of research which will have far-reaching implications in both experimental physics and spectral theory of self-adjoint operators.

Results: This work has currently resulted in one publication, with two publications submitted, and an NSF grant proposal (pending). Initially, the work was a continuation of the study of the spectral properties of the discrete random Schrödinger operator, with the goal being to determine critical values for the impurities which could induce localization. The newer studies now focus on better understanding the spectral properties of fractional operators in the presence of non-compact perturbations, as well as generalizing the models to better fit experimental data resulting from plasma crystal experiments. This project has yielded exciting results in the realm of spectral theory and functional analysis, while allowing for exact numerical simulations. Moreover, the project has allowed me to develop novel closed-form representations of higher powers of the fractional Laplacian (with the associated discrete harmonic analysis to come).

4.3.

Currently, my focus in this direction has been the development of a collaborative NSF grant employing spectral methods in the study of certain physical phenomena. By studying the operator in this abstract fashion, we intend to provide novel insights which may typically be obfuscated by the complicated standard algorithms. Moreover, we are relating the onset of turbulence to the spectral properties of the Navier-Stokes operator.

With respect to the aforementioned work. I am still actively working to better understand the spectrum of the discrete fractional operator. Classical analysis involves the consideration of Jacobi operators, but the fractional operator does not possess this “banded” feature that allowed the classical analysis to prove useful. Thus, I am currently working to use the Poincaré-Steklov operator to map the operator to a non-fractional operator and apply spectral theory to the newly resulting local (but singular) operator. Long-term research includes formulating closed-form fractional operators on general graphs and also generalizing the spectral approach to include normal operators or Banach space-valued
operators.

5. Algebraic Theory of Numerical Integration

This section is still under construction...Feel free to contact me with questions...

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