(1) Research Interests
Our understanding of the fundamental processes of the natural world is based to a large extent on partial differential equations. These are equations that describe changes. Examples include the vibrations of solids, the flow of liquids, the diffusion of chemicals, the spread of heat, the structure of molecules, and the radiation of electromagnetic waves. As efforts are continually being made to gain a better understanding of the world around us, increasing efforts are being made to better understand the area of partial differential equations and betters ways to solve them.
My present research efforts can be basically described as: (1) making fundamental contributions to a well-established field, that of linear partial differential equations, and (2) the construction of exact solutions to nonlinear partial differential equations.
Linear Partial Differential Equations
Linear partial
differential equations are fundamental in the description of the flow of heat.
When the flow of heat in a one-dimensional rod is modeled, the partial
differential equation,
(1)
is normally used. In this equation, u is the temperature in the rod which is a function of time, t, and position, x. Often the underlying medium is assumed to be homogeneous and thus can be characterized by a diffusion coefficient, D, and is assumed constant. However, in most realistic situations, the medium is not homogeneous and the above equation is usually replaced with an equation with a non-constant diffusion coefficient,
(2)
The first equation is one that is easily solved whereas the second is much more difficult and cannot be solved in general.
Darboux transformations
My research on the heat equation with variable diffusion focused on obtaining new exactly solvable equations. Our work involved a technique called Darboux transformations. These are transformations that link the solutions of different differential equations, and typically that link the solutions of simple equations to the solutions of complex equations. For example, in our work, we made the link between equations with D constant and D not constant. This gave rise to a large number of exactly solvable equations.
Our work has also been extended to equations that model both sound and wave propagation in a varying medium.
Nonlinear Partial Differential Equations
Since most real world phenomena are modeled by nonlinear partial differential equations, the search for exact solutions continues to be an important part of applied mathematics. Nobel Laureate Werner Heisenburg (1932) once said:
"... the progress of physics will to a large extent depend on the progress of nonlinear mathematics and of methods to solve nonlinear equations."
My research in
this area is to use symmetry analysis to construct exact solutions to nonlinear
partial differential equations. In essence, symmetry analysis seeks to exploit
the symmetry of the equation. Once obtained, this process reduces the
complexity of the original equation. My work has focused mostly on the reaction
diffusion equation,
Studies of these types of equations are very important since they have been used in a wide variety of applications including modeling heat by microwave radiation and in the growing field of mathematical biology.
Recently, we demonstrated that the results from symmetry analysis can be recovered simply and easily by requiring compatibility of the original equation with a second, simpler equation. We believe that this result indicates that more complex equations might be found that are compatible with the original equation and that it might be possible to find new exact solutions. Our present focus is trying to determine compatible relations with higher dimensional diffusion equations. In particular, we are considering the compatibility between
in two spatial dimensions. Preliminary investigations have shown that in two dimensions, if Q is of the form
then F is of the form
which are the results from nonclassical symmetry method. In three space dimension, we have only the symmetry method regardless of the nature of Q. Continued investigations are currently underway.
Undergraduate Research
My involvement with undergraduates both individually and in collaboration with colleagues in my department and college has led to an undergraduate research presence that is currently very strong in the Department of Mathematics at UCA. My work with students has resulted in a long list of oral presentations; 15 different students over the last 5 years have made 64 student presentations at local, regional and national meetings.
A question that is often asked, presumably prompted by the relative inexperience of the undergraduates involved in research, is: “Are undergraduate students necessary to complete your research work?” My answer is definitely no. I’ve acquired many years of training so that I may be called a research mathematician. I could do the research work myself, given the time. So why should I involve undergraduates? It is because I am also a teacher. I thrive when introducing a student to a new world - the world of applied mathematics. I love to watch them in wonderment when they discover that not all is known about mathematics and there is really much that we don’t know. I enjoy engaging them and pushing them towards discovery. I enjoy sharing the ups and downs that are so often associated with research. I enjoy introducing them to new people and places involved in mathematics. What I am really doing is providing the means for our students to embark on this exciting journey.
In addition to the research mentioned above and my active involvement with students, I am also in the process of writing a book on the “Symmetry analysis of differential equations.” I have taught a special topics course at UCA in the symmetry analysis of differential equations for the last 5 years and I have at present a set of 22 typeset lecture notes that are presently available on the web for my students. We hope to convert the lecture notes into a draft of the book which we anticipate by early 2006.