A dissertation submitted to the Faculty of Engineering and the Built Environment, University of the Witwatersrand, in ful llment of the requirements for the degree of Master of Science in Engineering, Johannesburg, 2016 / The aim of this research project is evaluating the application of the Crum-based transformation in solving engineering systems modelled as two-point boundary value problems. The boundary value problems were subjected to the various combinations of Dirichlet, Non-Dirichlet and Affine boundary conditions. The engineering systems that were modelled were in the elds of electrostatics, heat conduction and longitudinal vibrations. Other methods such as the Z-transforms and iterative methods have been discussed. An attractive property of the Crum-based transformation is that it can be applied to cases where the eigenparameters (function of eigenvalues) generated in the discrete case are negative and was therefore chosen to be explored further in this dissertation. An alternative matrix method was proposed and used instead of the algebraic method in the Crum- based transformation. The matrix method was tested against the algebraic method using three unit intervals. The analysis revealed, that as the number of unit intervals increase, there is a general increase in the accuracy of the approximated continuous-case eigenvalues generated for the discrete case. The other observed general trend was that the accuracy of the approximated continuous- case eigenvalues decrease as one ascends the continuous-case eigenvalue spectrum. Three cases: (Affine, Dirichlet), (Affine, Non-Dirichlet) and (Affine, Affine) generated negative eigenparameters. The approximated continuous-case eigenvalues, derived from the negative eigenparameters, were shown not to represent true physical natural frequencies since the discrete eigenvalues, derived from negative eigenparameters, do not satisfy the condition for purely oscillatory behaviour. The research has also shown that the Crum-based transformation method was useful in approximating the shifted eigenvalues of the continuous case, in cases where the generated eigenparameters were negative: since, as the number of unit intervals increase, the post-transformed approximated eigenvalues improved in accuracy. The accuracy was also found to be better in the post-transformed case than in the pre-transformed case. Furthermore, the approximated non-shifted and shifted continuous- case eigenvalues (except the approximated continuous-case eigenvalues generated from negative eigenparameters) satis ed the condition for purely oscillatory behaviour. / MT2017

The investigation of positive steady states to reaction diffusion models in bounded domains with Dirichlet boundary conditions has been of great interest since the 1960’s. We study reaction diffusion models where the reaction term is negative at the origin. In the literature, such problems are referred to as semipositone problems and have been studied for the last 30 years. In this dissertation, we extend the theory of semipositone problems to classes of singular semipositone problems where the reaction term has singularities at certain locations in the domain. In particular, we consider problems where the reaction term approaches negative infinity at these locations. We establish several existence results when the domain is a smooth bounded region or an exterior domain. Some uniqueness results are also obtained. Our existence results are achieved by the method of sub and super solutions, while our uniqueness results are proved by establishing a priori estimates and analyzing structural properties of the solution. We also extend many of our results to systems.

In this dissertation, we establish new existence, multiplicity, and uniqueness results on positive radial solutions for classes of steady state reaction diffusion equations on the exterior of a ball. In particular, for the first time in the literature, this thesis focuses on the study of solutions that satisfy a general class of nonlinear boundary conditions on the interior boundary while they approach zero at infinity (far away from the interior boundary). Such nonlinear boundary conditions occur naturally in various applications including models in the study of combustion theory. We restrict our analysis to reactions terms that grow slower than a linear function for large arguments. However, we allow all types of behavior of the reaction terms at the origin (cases when it is positive, zero, as well as negative). New results are also added to ecological systems with Dirichlet boundary conditions on the interior boundary (this is the case when the boundary is cold). We establish our existence and multiplicity results by the method of sub and super solutions and our uniqueness results via deriving a priori estimates for solutions.

In this dissertation we study positive solutions to a singular p-Laplacian elliptic boundary value problem on a bounded domain with smooth boundary when a positive parameter varies. Our main focus is the analysis of a challenging class of singular p-Laplacian problems. We establish the existence of a positive solution for all positive values of the parameter and the existence of at least two positive solutions for a certain explicit range of the parameter. In the Laplacian case, we also prove the uniqueness of the positive solution for large values of the parameter. We extend our existence and multiplicity results to classes of singular systems and to the case when a domain is an exterior domain. We prove our existence and multiplicity results by the method of sub and supersolutions and our uniqueness result by establishing apriori and boundary estimates. Such results are well known in the literature for the nonsingular case. In this study, we extend these results to the more difficult singular case.

In this thesis we study two reaction-diffusion models that have been used to analyze the existence of alternate stable states in ecosystems. The first model describes the steady states of a logistic growth model with grazing in a spatially homogeneous ecosystem. It also describes the dynamics of the fish population with natural predation. The second model describes phosphorus cycling in stratified lakes. The same equation has also been used to describe the colonization of barren soils in drylands by vegetation. In this study we discuss the existence of multiple positive solutions, leading to the occurrence of S-shaped bifurcation curves. We were able to show that both the models have alternate stable states for certain ranges of parameter values. We also introduce a constant yield harvesting term to the first model and discuss the existence of positive solutions including the occurrence of a Sigma-shaped bifurcation curve in the case of a one-dimensional model. Again we were able to establish that for certain ranges of parameter values the model has alternate stable states. Thus we establish analytically that the above models are capable of describing the phenomena of alternate stable states in ecological systems. We prove our results by the method of sub-super solutions and quadrature method.

A recent method for solving singular perturbation problems is examined. It is designed for the applied mathematician or engineer who needs a convenient, useful tool that requires little preparation and can be readily implemented using little more than an industry-standard software package for spreadsheets. In this paper, we shall examine singularly perturbed two point boundary value problems with the boundary layer at one end point. An initial-value technique is used for its solution by replacing the problem with an asymptotically equivalent first order problem, which is, in turn, solved as an initial value problem by using cubic splines. Numerical examples are provided to show that the method presented provides a fine approximation of the exact solution. The first chapter provides some background material to the cubic spline and boundary value problems. The works of several authors and a comparison of different solution methods are also discussed. Finally, some background into the specific singularly perturbed boundary value problems is introduced. The second chapter contains calculations and derivations necessary for the cubic spline and the initial value technique which are used in the solutions to the boundary value problems. The third chapter contains some worked numerical examples and the numerical data obtained along with most of the tables and figures that describe the solutions. The thesis concludes with some reflections on the results obtained and some discussion of the error bounds on the calculated approximations to the exact solutions for the numeric examples discussed

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A theoretical study of convective heat transfer is presented for a laminar flow subjected to an axial variation in the external heat transfer coefficient (or dimensionless Biot number). Since conventional techniques fail for a variable boundary condition parameter, a variable eigenfunction approach is developed. An analysis is carried out for a periodic heat transfer coefficient, which serves as a model for heat transfer from a duct fitted with an array of evenly spaced fins. Three solution methods for the variable eigenfunction technique are examined: an Nth order approximation method, an iterative method and a stepwise periodic method. The stepwise periodic method provides the most convenient and accurate solution for a stepwise periodic Biot number. Graphical results match exactly to ones obtained by Charmchi and Sparrow from a finite-difference scheme. A connected region technique is also developed to provide limited exact results to test the validity of the three solution methods. The study of a finned duct by a stepwise periodic Biot number is carried out via a parametric study, an average (constant) Biot number approximation and an assumed velocity profile analysis. Results for the parametric study show that external finning yields substantial heat transfer enhancement over an unfinned duct, especially when the Biot number of the unfinned regions is low. A decrease in the interfin spacing causes increased enhancement. Variations of the period of the Biot number causes relatively small changes in enhancement as long as the ratio of finned to unfinned surface remains unchanged. An average (constant) Biot number approximation for a specified finned tube is compared to the stepwise periodic Biot number solution. The results show that the constant Biot number approximation provides accurate results. Finally, the results for the influence of the assumed velocity profile demonstrate that a constant velocity flow provides increased heat transfer and more effective enhancement by external finning than a laminar fully developed flow, especially at high Biot numbers. This study provides insight into heat transfer enhancement due to finning and also develops a solution methodology for problems involving variable boundary condition parameters. / M.S.

Numerical schemes for the solution of two-point boundary value problems arising from the application of optimal control theory to mathematical models of dynamic systems, are discussed. Optimal control problems are formulated for rotational maneuvers of multiple rigid body, asymmetric spacecraft configurations with both external torques and/or internal torques. Necessary conditions for optimality are derived through Pontryagin’s principle; solutions to the problems are obtained numerically. Comparison studies using competing numerical methods and various choices of performance indices are reported. / Ph. D.

Ordinary differential equations (ODEs) play an important role in mathematics. Although intrinsically, the setting for describing ODEs is the continuous framework, where differential operators are considered as maps from one function space to another, common numerical algorithms for ODEs discretise problems early on in the solution process. This thesis is about continuous analogues of such discrete algorithms for the numerical solution of ODEs. This thesis shows how Newton's method for finite dimensional system can be generalised to function spaces, where it is known as Newton-Kantorovich iteration. It presents affine invariant damping strategies for increasing the chance of convergence for the Newton-Kantorovich iteration. The derivatives required in this continuous setting are Fréchet derivatives, the continuous analogue of Jacobian matrices. In this work, we present how automatic differentiation techniques can be applied to compute Fréchet derivatives. We introduce chebop, a Matlab solver for nonlinear boundary-value problems, which combines damped Newton iteration in function space and automatic Fréchet differentiation. By proving that affine operators have constant Fréchet derivatives, it is demonstrated how automatic linearity detection of computed quantities can be implemented. This is valuable for black-box solvers, which can use the information to determine whether an iteration scheme has to be employed for solving a problem. Like nonlinear systems of equations, nonlinear boundary-value problems can have multiple solutions. This thesis present two techniques for obtaining multiple solutions of operator equations: deflation and path-following. An algorithm combining the two techniques is proposed.