Advisor: Muhammad Usman, Ph.D
Stander Symposium 2013
We demonstrate numerically the eventual time-periodicity of the solutions of the Korteweg-de Vries equation with periodic forcing at the boundary using the sinc-collocation method. This method approximates the space dimension of the solution with a cardinal expansion of sinc functions, thus allowing the avoidance of a costly finite difference grid for a third-order boundary value problem. The first-order time derivative is approximated with a weighted finite difference method. The sinc-collocation method was found to be more robust and more efficient than other numerical schemes when applied to this problem
2. Simulation of Nonlinear Waves Using Sinc Collocation-Interpolation
Eric A Gerwin, Jessica E Steve
In this project we explore the Sinc collocation method to solve an initial and boundary value problem of nonlinear wave equation. The Sinc collocation method is based upon interpolation technique, by discretizing the function and its spatial derivatives using linear combination of translated Sinc functions. Our project will focus on multiple boundary conditions such as the well known Dirichlet and Neumann conditions. Our project will also focus on two established nonlinear partial differential equations: the Sine-Gordon equation and the Kortweg-de Vries equation.
3. Exploring the Sinc-Collocation Method for Solving the Integro-Differential Equation
In this project we study the Sinc approximation method to solve a family of integral differential equations. First we will apply the Sinc-collocation method to solve the second order Fredholm integro-differential equation. Numerical results and examples demonstrate the reliability and efficiency of this method. Secondly, various types of integro-differential equations are solved by Sinc-collocation technique and the numerical results are compared, to explore the stability of this method.
Stander Symposium 2012
Comparison of Numerical Methods for Analysis of the Diffusion of Soluble Proteins Through Sensory Cilia
A recent paper in the Journal of General Physiology disproved the hypothesis that the ciliary axoneme and the basal bodies of cilia impose selective barriers to the movement of proteins into and out of the the cilium using a combination of numerical modeling and observation with confocal and multiphoton microscopy. We compare the accuracy and computational efficiency of the numerical method used in the paper, known as the method of lines, to another method, known as sinc collocation, and discuss the possible use of other methods for improving the algorithm.
2. Stability Analysis of a Model for In Vitro Inhibition of Cancer Cell Mutation
Human homeostasis is the body's ability to physiologically regulate its inner environment to ensure its stability in response to changes in the outside environment. An inability to maintain homeostasis may lead to death or disease, which is caused by a condition known as homeostatic imbalance. Normal cells follow the homeostasis when they proliferate and cancer cells do not. This work describes a model consisting of three reaction-diffusion equations representing in vitro interaction between two drugs. One inhibits proliferation of cancerous cells, and the other destroys these cells. A stability analysis of the model is performed with and without diffusion applied to the model. MATLAB is used to perform the stability analysis of the model.
3. Numerical Investigation into a Computational Approximation of Bifurcation Curves
Joshua R Craven
In this project, I use computational tools to study the bifurcations in nonlinear oscillators. Matlab is first used to determine the slow flow phase portrait of each region and the characteristics of each critical point. Next, the parameters are discretized and for each set of values we find the locations of the real critical points and the eigenvalues of the Jacobian matrix. With this knowledge, we can approximate the bifurcation diagram. These results are compared with results from preexisting software.
4. Numerical Study of a Mathematical Model of IL-2 Adoptive Immunotherapy on Patients with Metastatic Melanoma
Alyssa C Lesko
IL-2 treatments have recently been identified to significantly reduce metastatic melanoma tumors and in some cases eliminate them. The problem with these treatments is that a set plan of administration varies from patient to patient and methods for determining treatment steps are still in the process of being developed. Previous research by Asad Usman and colleagues has used a numerical technique using MATLAB to decide treatment protocols. This research used the MATLABâ??s built in ode15 function to addresses treatment procedures including the starting and stopping of each treatment and the period in between each treatment. Building on this data and existing model, my project will explore several other numerical techniques such as ode23 and ode45 solvers, Eulerâ??s method, and the predictor corrector method to study IL-2 treatments in metastatic melanoma patients. A comparison will be made using error plots and tables, and a stability analysis using pplane7 will be investigated.
5. Mathematical Study of the Foot and Mouth Outbreak Model
Student - Jungmi Johnson
The foot and mouth outbreak in the UK in 2001 was a disastrous event for the country and the economic. The disease did not only cost UK government so much money to stop the disease, but it also affected the tourism industry. Mathematical epidemic models can provide clear strategy for minimizing the effect of such a disease, determining the expected manner of its progression in the event of a future outbreak based upon the latest available data on the epidemic. This project is to explore how to minimize the cost, how to contain the disease in minimal time, and how realistic these models will be considering the limitation of the model. Numerical and qualitative tools such as MATLAB's built in ode solver will be used.
6. Applying Mathematical Epidemic Modeling to Discover Commercially Beneficial Outbreak Control Methods
This project is a mathematical analysis and computational study of the 2001 foot and mouth epidemic in the UK. This model includes an application of the SIR model, developed by W. O. Kermack and A. G. McKendrick, with two additional factors: vaccination and incubation period infectives. The incubation period infectives represent the population of individuals infected with the disease that do not show symptoms, but still have the possibility of infecting other individuals.
7. Qualitative Study of an SIR epidemic model with an asymptotically homogeneous transmission function
Karoline E. Hoffman
I will be exploring and analyzing an SIR epidemic model. This particular model has an asymptotically homogeneous transmission function which means the transmission rate is proportional to the fraction of the number of infective individuals to the total population. I will also look at a qualitative analysis of the model and then discuss the implications of the results of the model.
Stander Symposium 2011A Numerical Study of In Vitro Inhibition of Mutation of Cancer Cells Using Two Different
The growth of in-vitro cancer cells has been studied using two numerical methods: the Predictor-Corrector and the Operator Splitting method. The mathematical model developed by Dey (2000) is used, which consists of three reaction-diffusion equations representing in vitro interaction between two drugs, one which inhibits the proliferation of the cancer cells and the other which destroy these cells. The solutions resulting from the application of the two methods are in excellent agreement. In addition stability analyses of model and diffusion free case have been performed.
Stander Symposium 2010
1. A Graphical User Interface for Solving the Falkner-Skan Equation (Download all MATLAB codes here)
A Graphical User Interface (GUI) has been developed to solve the Falkner-Skan equation. This famous nonlinear third order Falkner-Skan equation on infinite interval describes several fluid dynamic problems under varying the value of two constant coefficients. The developed GUI enables the user to input these two coefficients, which will characterize the behavior of the corresponding solution, and the parameters necessary for the iterative methods used to solve the equation. At this regard, a shooting method developed by Zhang J. and Chen B. has been adopted. The required parameters for the numerical solution are represented by the initial shooting angle, the initial free boundary and two tolerance criteria.
2. A Computational Study of Adaptive Residual Subsampling Method for Radial Basis Functions Interpolation
In this work we revisit some relatively new techniques based on radial basis functions (RBFs) to interpolate, boundary-value and initial-boundary-value problems with high degrees of localization in space and/or time. First, we generate an initial discretization using equally spaced points and find the RBF approximation of the function. Next, we compute the interpolation error at points halfway between the nodes. Points at which the error exceeds a threshold become centers, and centers that lie between two points with error less than a smaller threshold are removed. The two end points are always left intact. We also adapt the shape parameters of RBFs based on the node spacing to prevent the growth of the conditioning of the interpolation matrix. The shape parameter of each center is chosen based on the spacing with nearest neighbors, and the RBF approximation is recomputed using the new center set. Recent work in the literature on radial basis functions method has shown some promising results in terms of accuracy and efficiency to solve higher order nonlinear partial differential equations. Since radial basis functions methods are completely meshfree, requiring only interpolation nodes and a set of points called centers defining the radial basis functions. Adaptive radial basis function approach is based on refining and coarsening nodes based on shape parameter, interpolation error and condition number of the interpolation matrix.
3. Kinetic Modeling of A Spherical Catalytic Particle
It is critical for chemical engineers to understand the kinetics behavior associated with catalytic particles. This is crucial in the design and fabrication of catalytic reactors. In this work, a mathematical model for the interplay between the rates of molecular transport (diffusion) and the intrinsic activity (chemical kinetics) is studied. The concept of effectiveness factor in catalytic first order chemical reactions, exothermic and endothermic, is addressed in detail. Also the behavior of chemical reaction rates and the temperature gradient in the catalytic particle with respect to stability is discussed. The Numerical solution of the model has been computed for various values of the parameters. For all of our computations and numerical simulations we have used MATLAB.
4. Measles Epidemic: Studying the Spread Using Numerical Techniques
Jaye S. Flavin
Studying past epidemics is a necessary step in understanding and preventing the spread of future contagions. The measles epidemic in New York in the mid-1960s is an ideal case study for mathematical epidemiology because of the detailed records kept on those infected and the unique properties of measles as a disease. Using a Computer Algebra System (CAS), we will revisit the qualitative properties of the measles epidemic model and compare solutions using different numerical techniques.
Undergraduate Research in Mathematical Biology
Advisors: Muhammad Usman, Ph.D,
Amit Singh, Ph.D (Department of Biology)
Stander Symposium 2010
5. A Computational Study of the Fitzhugh-Nagumo Action Potential System
Joseph R Salomone, Anna M. Stcyr, Angela Q. Umstead
The most celebrated example of mathematical modeling is the Hodgkin-Huxley model of nerve physiology. Their experiments were carried out on a giant axon of a squid, which was large enough for the implantation of electrodes. The Hodgkin-Huxley mathematical model for nerve cell action potential is a system of four coupled ordinary differential equations. The Fitzhugh-Nagumo two-variable action potential system behaves qualitatively like the Hodgkin-Huxley space-clamped system. Being simpler by two variables, action potentials and other properties of the Hodgkin-Huxley model may be visualized as phase-plane plots. We use MATLAB to study the numerical solutions as well as the qualitative properties of the model.
6. Mathematical Modeling of H1N1 Flu
William E. Balbach Nathan B. Frantz Brett R. Mershman William T. Weger
Mathematical models have been used to understand the dynamics of infectious diseases and to predict the future epidemic or pandemics. In 2009, a new strain of the influenza A (H1N1) virus spread rapidly throughout the world. This “swine flu” as it is commonly known, increased to what is considered an epidemic in a matter of months. In order to understand the spread of this virus, and similar patterns in future outbreaks, we study a simplified SIR mathematical model to answer some epidemiological questions. We solve the model numerically and also study the qualitative properties of the model. It is important
to mention that a solution of a mathematical model is not necessarily a solution to the real problem, but a solution to a simplified idealization of the real world problem.
7. Mathematical Modelling of Infectious Diseases
Kevin M. George, Branden J. King
Study of infectious diseases has become more important with increased global connectivity and personal contact. The discovery of the microscope in the 17th century caused a revolution in biology by revealing otherwise invisible. Mathematics broadly interpreted is a more general non optical microscope. Mathematical model helps to understand dynamics of how they spread, how many people are infected, resist the infection, or recover. In this work we study infectious diseases models qualitatively. These mathematical models are solved numerically using MATLAB.
Student Research 2009
Miriam Poteet (Currently a Ph.D. Student at AFIT)
Stander Symposium 2009 Poster Presentation
David Aaby (Currently a graduate student in Biostatistics at University of Michigan)
Undergraduate Research in Mathematical Biology: A comparative study of numerical methods for the Hodgkin-Huxley model of nerve cell action potential, (Citation)
Advisors: Muhammad Usman, (Mathematics), Amit Singh, (Biology)
Stander Symposium 2009 Poster Presentation