Date of Award
Fall 12-2017
Degree Type
Dissertation-Restricted
Degree Name
Ph.D.
Degree Program
Engineering and Applied Science
Department
Physics
Major Professor
Ashok Puri, Ph. D.
Second Advisor
Salvadore J. Guccione, Ph. D.
Third Advisor
Juliette W. Ioup, Ph. D.
Fourth Advisor
Vesselin P. Jilkov, Ph. D.
Fifth Advisor
Nikolas Xiros, Ph. D.
Abstract
In this work, we design a linear, two step implicit finite difference method to approximate the solutions of a biological system that describes the interaction between a microbial colony and a surrounding substrate. Three separate models are analyzed, all of which can be described as systems of partial differential equations (PDE)s with nonlinear diffusion and reaction, where the biological colony grows and decays based on the substrate bioavailability. The systems under investigation are all complex models describing the dynamics of biological films. In view of the difficulties to calculate analytical solutions of the models, we design here a numerical technique to consistently approximate the system evolution dynamics, guaranteeing that nonnegative initial conditions will evolve uniquely into new, nonnegative approximations. This property of our technique is established using the theory of M-matrices, which are nonsingular matrices where all the entries of their inverses are positive numbers. We provide numerical simulations to evince the preservation of the nonnegative character of solutions under homogeneous Dirichlet and Neumann boundary conditions. The computational results suggest that the method proposed in this work is stable, and that it also preserves the bounded character of the discrete solutions.
Recommended Citation
Landry, Richard S. Jr., "An Application of M-matrices to Preserve Bounded Positive Solutions to the Evolution Equations of Biofilm Models" (2017). University of New Orleans Theses and Dissertations. 2418.
https://scholarworks.uno.edu/td/2418
Included in
Biological and Chemical Physics Commons, Biological Engineering Commons, Catalysis and Reaction Engineering Commons, Computational Engineering Commons, Engineering Physics Commons, Numerical Analysis and Computation Commons, Partial Differential Equations Commons, Water Resource Management Commons
Rights
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