Positive solutions of nonlinear elliptic boundary value problems

This dissertation focuses on the study of positive steady states to classes of nonlinear reaction diffusion (elliptic) systems on bounded domains as well as on exterior domains with Dirichlet boundary conditions. In particular, we study such systems in the challenging case when the reaction terms are negative at the origin, referred in the literature as semipositone problems. For the last 30 years, study of elliptic partial differential equations with semipositone structure has flourished not only for the semilinear case but also for quasilinear case. Here we establish several results that directly contribute to and enhance the literature of semipositone problems. In particular, we discuss existence, non-existence and multiplicity results for classes of superlinear as well as sublinear systems. We establish our results via the method of sub-super solutions, degree theory arguments, a priori bounds and energy analysis

Analysis of classes of singular steady state reaction diffusion equations

We study positive radial solutions to classes of steady state reaction diffusion problems on the exterior of a ball with both Dirichlet and nonlinear boundary conditions. We study both Laplacian as well as p-Laplacian problems with reaction terms that are p-sublinear at infinity. We consider both positone and semipositone reaction terms and establish existence, multiplicity and uniqueness results. Our existence and multiplicity results are achieved by a method of sub-supersolutions and uniqueness results via a combination of maximum principles, comparison principles, energy arguments and a-priori estimates. Our results significantly enhance the literature on p-sublinear positone and semipositone problems. Finally, we provide exact bifurcation curves for several one-dimensional problems. In the autonomous case, we extend and analyze a quadrature method, and in the nonautonomous case, we employ shooting methods. We use numerical solvers in Mathematica to generate the bifurcation curves

Analysis and computation for a class of semilinear elliptic boundary value problems

AbstractIn this paper, with the help of super-solutions and sub-solutions, we set up a general framework and get a positive threshold Λ for solution existence and non-existence of a class of semilinear elliptic Dirichlet boundary value problems. Moreover, a result on multiplicity is obtained when λ is large enough. We also give a numerical method to solve and visualize the positive solutions of the problem. Theoretical results are illustrated by numerical simulation

Non-trivial, non-negative periodic solutions of a system of singular-degenerate parabolic equations with nonlocal terms

We study the existence of non-trivial, non-negative periodic solutions for systems of singular-degenerate parabolic equations with nonlocal terms and satisfying Dirichlet boundary conditions. The method employed in this paper is based on the Leray-Schauder topological degree theory. However, verifying the conditions under which such a theory applies is more involved due to the presence of the singularity. The system can be regarded as a possible model of the interactions of two biological species sharing the same isolated territory, and our results give conditions that ensure the coexistence of the two species.Comment: 39 page

Mathematical and computational analysis of reaction diffusion models arising in ecology

The focus of this thesis is to study long term solutions for classes of steady state reaction diffusion equations. In particular, we study reaction diffusion models arising in mathematical ecology. We study how the patch size affects the existence, nonexistence, multiplicity, and uniqueness of the steady states. Our focus is also to study how various forms of density dependent emigrations at the boundary, and the effective matrix hostility, affect steady states. These considerations lead to the study of various forms of nonlinear boundary conditions. Further, they lead to the study of reaction diffusion models where a parameter (related to the patch size) gets involved in the differential equation as well as the boundary conditions. We establish analytical results in any dimension, namely, establish existence, nonexistence, multiplicity, and uniqueness results. Our existence and multiplicity results are achieved by a method of sub-supersolutions and uniqueness results via comparison principles and a-priori estimates. Via computational methods, we also obtain exact bifurcation diagrams describing the structure of the steady states. Namely, we obtain these bifurcation diagrams via a modified quadrature method and Mathematica computations in the one-dimensional case, and via the use of finite element methods and nonlinear solvers in Matlab in the two-dimensional case. This dissertation aims to significantly enrich the mathematical and computational analysis literature on reaction diffusion models arising in ecology

Solvability of nonlinear elliptic boundary value problems

This dissertation focuses on the study of steady states of reaction diffusion problems that are motivated by applications. In particular, we focus on elliptic boundary value problems where the nonlinear reaction may appear in the interior or on the boundary of a domain in the Euclidean space. First, we study linear elliptic problems with nonlinear reaction on the boundary. In this case, we establish the existence of maximal and minimal solutions for both monotone and non monotone cases. We then extend these results to the systems case. Next, we prove the existence, nonexistence, multiplicity and global bifurcation results of positive solutions of superlinear problems. To support our analytical results we numerically approximate solutions using finite difference methods including existence and stability analysis. Second, we study problems that are nonlinear inside the domain and linear on the boundary in the context of a model arising in mathematical ecology. To begin with we perform computational simulations for the problem in the one dimensional setting. Then, motivated by the bifurcation diagrams that are obtained, we prove several analytical results such as existence, uniqueness and nonexistence

Fractional Differential Equations, Inclusions and Inequalities with Applications

During the last decade, there has been an increased interest in fractional differential equations, inclusions, and inequalities, as they play a fundamental role in the modeling of numerous phenomena, in particular, in physics, biomathematics, blood flow phenomena, ecology, environmental issues, viscoelasticity, aerodynamics, electrodynamics of complex medium, electrical circuits, electron-analytical chemistry, control theory, etc. This book presents collective works published in the recent Special Issue (SI) entitled "Fractional Differential Equation, Inclusions and Inequalities with Applications" of the journal Mathematics. This Special Issue presents recent developments in the theory of fractional differential equations and inequalities. Topics include but are not limited to the existence and uniqueness results for boundary value problems for different types of fractional differential equations, a variety of fractional inequalities, impulsive fractional differential equations, and applications in sciences and engineering

Solutions to differential equations via fixed point approaches: new mathematical foundations and applications

The central aim of this thesis is to construct a fuller and firmer mathematical foundation for the solutions to various classes of nonlinear differential equations than is currently available in the literature. This includes boundary value problems (BVPs) that involve ordinary differential equations, and initial value problems (IVPs) for fractional differential equations. In particular, we establish new conditions that guarantee the existence, uniqueness and approximation of solutions to second-order BVPs, third-order BVPs, and fourth-order BVPs for ordinary differential equations. The results enable us, in turn, to shed new light on problems from applied mathematics, engineering and physics, such as: the Emden and Thomas-Fermi equations; the bending of elastic beams through an application of our general theories; and laminar flow in channels with porous walls. We also ensure the existence, uniqueness and approximation of solutions to some IVPs for fractional differential equations. An understanding of the existence, uniqueness and approximation of solutions to these problems is fundamental from both pure and applied points of view. Our methods involve an analysis of nonlinear operators through fixed-point theory in new and interesting ways. Part of the novelty involves generating new conditions under which these operators are contractive, invariant and/or establishing new a priori bounds on potential solutions. As such, we draw on: Banach fixed- point theorem, Schauder fixed-point theorem, Rus's contraction mapping theorem, and a continuation theorem due to A. Granas and its constructive version known as continuation method for contractive maps. The ideas in this thesis break new ground at the intersection of pure and applied mathematics. Thus, this work will be of interest to those who are researching the theoretical aspects of differential equations, and those who are interested in better understanding their applications