Date of Award

Fall 12-20-2017

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.

Rights

The University of New Orleans and its agents retain the non-exclusive license to archive and make accessible this dissertation or thesis in whole or in part in all forms of media, now or hereafter known. The author retains all other ownership rights to the copyright of the thesis or dissertation.

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