Applications of partial differential equations and numerical simulations to biomedical sciences is a hot topic in today's scientific research. Being ``in the middle'' of the largest medical center in the world, the University of Houston is particularly attractive to scientists interested in biomedical engineering. Mathematics plays a crucial role in modeling, theoretical analysis of solutions, and verification of solutions obtained using numerical simulations. All of these aspects will be used and developed in the REU project related to conservation laws and hemodynamics.
The focus of the project will be on the study of blood flow through elastic blood vessels. Partial differential equations describing conservation of mass and momentum will be studied. The equations describe how the radius of a blood vessel and the axial velocity of blood flow change in space and time. Solutions of these equations may exhibit shock waves. A shock wave in this context means a sudden, abrupt change in the radius of a blood vessel and/or in the axial velocity. These are known to appear, for example, at the junction between a blood vessel and a prosthetic device used to treat abdominal aneurysm or atherosclerosis. Mathematical analysis of solutions of these partial differential equations and the implications of the theoretical result on the design of prosthetic devices, will be studied. Numerical simulations of solutions and virtual reality visualization will be utilized. A tour of the Texas Heart Institue and the University of Houston's Virtual Reality Laboratory will be organized.
The topics of the projects outlined here derive from the theory and applications of partial differential equations, with emphasis on conservation laws.
The first set of projects revolves around the Lighthill- Richards-Whitham model for one-way traffic flow. This is a well-known example of a scalar conservation law, for which a local solution can be found more or less explicitly by integration of a simple system of ordinary differential equations. At this level, the model has been discussed in undergraduate texts. Applications of the model are useful in traffic engineering. For the most part, the applications use rather little knowledge of conservation laws, and a number of appealing problems which appear amenable to solution have not appeared in the literature. These include questions such as staging traffic lights, the difference between linear and nonlinear models, and the effect of different assumptions for the nonlinearity. Also of interest are questions of comparing the model to real data and examining real-world solutions to problems of timing lights.
A second set of projects involves an analytical and numerical exploration of shock reflection. The problem of weak shock reflection at a wedge consists of studying what happens when an incident shock hits a corner of a wedge. Depending on the two parameters in the problem: the strength of the incident shock wave and the wedge angle, different reflection patterns occur. We have developed software to study a related initial-value problem.With the help of numerical simulations, and the use of analysis and modeling techniques we hope to be able to support our conjecture.
The research project we are proposing here has two goals. One goal is to first numerically explore whether solutions with shock reflection initial data can be recovered by using the existing software for all possible ranges of the two initial parameters, and not only for the one set of initial data mentioned above. The second goal is to study the relationship between the solutions in the two different initial-value problems and understand their relationship.
This work could certainly result in a student paper, and could provide an important insight into the theoretical study of shock reflection phenomena. The student will become familiar with certain theoretical aspects of solutions of two-dimensional systems of conservation laws, gain a considerable breath in modeling issues related to conservation laws and shock reflection, and be able to use an existing software to interpret numerical answers correctly and thus be able to make progress and conclude this project.