Analysis of positive solutions for classes of nonlinear reaction diffusion equations and systems

UNCG Author/Contributor (non-UNCG co-authors, if there are any, appear on document)
Amila K. Muthunayake (Creator)
The University of North Carolina at Greensboro (UNCG )
Web Site:
Ratnasingham Shivaji

Abstract: The focus of this thesis is to study positive solutions for classes of nonlinear reaction diffusion equations and systems. In particular, we consider three focuses. In Focus 1, we establish existence and uniqueness of positive solutions for a class of infinite semipositone problems with nonlinear boundary conditions. In Focus 2, we explore the consequences of fragmentation and trait-mediated dispersal on the coexistence of a system of two mutualists by employing a model built upon the reaction diffusion framework. We establish several coexistence and nonexistence results. Finally, in Focus 3, we develop and analyze a radial finite difference method that directly approximate solutions for classes of semipositone problems with Dirichlet boundary conditions. Our existence results in Focus 1 and Focus 2 are achieved by methods of sub and supersolutions. In Focus 3, via computational methods, we obtain bifurcation diagrams describing the structure of positive solutions. Namely, we obtain these bifurcation diagrams via a modified finite difference method and MATLAB computations in the case when the domain is the unit disc in R2. This dissertation aims to significantly enrich the mathematical and computational analysis literature on reaction diffusion equations and systems. [This abstract may have been edited to remove characters that will not display in this system. Please see the PDF for the full abstract.]

Additional Information

Language: English
Date: 2021
Reaction-diffusion systems, Nonlinear systems, Positive solutions
Differential equations, Nonlinear $x Numerical solutions
Reaction-diffusion equations $x Numerical solutions
Nonlinear systems $x Mathematical models
Diffusion $x Mathematical models

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