Fractional calculus, which is a generalization of classical calculus, has been the subject of numerous applications in physics and engineering during the last decade. In this thesis, fractional calculus has been implemented for chemical engineering applications, namely in process control and in the modeling mass transfer in adsorption. With respect to process control, some researchers have proposed fractional PIλDμ controllers based on fractional calculus to replace classical PI and PID controllers. The closed-loop control of different benchmark dynamic systems using optimally-tuned fractional PIλDμ controllers were investigated to determine for which dynamic systems this more computationally-intensive controller would be beneficial. Four benchmark systems were used: first order plus dead time system, high order system, nonlinear system, and first order plus integrator system. The optimal tuning of the fractional PIλDμ controller for each system was performed using multi-objective optimization minimizing three performance criteria, namely the ITAE, OZ, and ISDU. Conspicuous advantages of using PIλDμ controllers were confirmed and compared with other types of controllers for these systems. In some cases, a PIλ controller was also a good alternative to the PIλDμ controller with the advantage of being less computationally intensive. For the optimal tuning of fractional controllers for each benchmark dynamic system, a new version of the non-dominated sorting genetic algorithm (NSGA-III) was used to circumscribe the Pareto domain. However, it was found that for the tuning of PIλDμ controllers, it was difficult to circumscribe the complete Pareto domain using NSGA-III. Indeed, the Pareto domain obtained was sometimes fragmentary, unstable and/or susceptible to user-defined parameters and operators of NSGA-III. To properly use NSGA-III and determine a reliable Pareto domain, an investigation on the effect of these user-defined operators and parameters of this algorithm was performed. It was determined that a reliable Pareto domain was obtained with a crossover operator with a significant extrapolation component, a Gaussian mutation operator, and a large population. The findings on the proper use of NSGA-III can also be used for the optimization of other systems. Fractional calculus was also implemented in the modeling of breakthrough curves in packed adsorption columns using finite differences. In this investigation, five models based on different assumptions were proposed for the adsorption of butanol on activated carbon. The first four models are based on integer order partial differential equations accounting for the convective mass transfer through the packed bed and the diffusion and adsorption of an adsorbate within adsorbent particles. The fifth model assumes that the diffusion inside adsorbent particles is potentially anomalous diffusion and expressed by a fractional partial differential equation. For all these models, the best model parameters were determined by nonlinear regression for different sets of experimental data for the adsorption of butanol on activated carbon. The recommended model to represent the breakthrough curves for the two different adsorbents is the model that includes diffusion within the adsorbent particles. For the breakthrough experiments for the adsorption of butanol on activated carbon F-400, it is recommended using a model which accounts for the inner diffusion within the adsorbent particles. It was found that instantaneous or non-instantaneous adsorption models can be used. Best predictions were obtained with fractional order diffusion with instantaneous adsorption. For the adsorption of butanol on activated carbon Norit ROW 0.8, it is recommended using an integer diffusion model with instantaneous adsorption. The gain of using fractional order diffusion equation, given the intensity in computation, was not sufficient to recommend its use.

An analysis of the pertinent literature showed that there were two commonly used instructional strategies for teaching calculus to engineering technology students. Theory-problem instructional strategy is being used when an instructor first develops-the new calculus theory and then "illustrates" this theory with an applied problem. Problem-theory instructional strategy is being used when an instructor first "initiates" the new calculus theory with an applied problem and then develops the new calculus theory. The purpose of this study is to investigate which of the above instructional strategies is best for teaching calculus to engineering technology students. The evaluation was done by comparing the achievement of the theory-problem group and the problem-theory group on two tests. For the length of the study the two groups were taught identical content by the same instructor. The only difference was the order in which the applied problems were presented. There was no significant difference in achievement of the two groups on the test designed to measure understanding of techniques, principles and concepts of calculus. There was no significant difference in the achievement of the two groups on the test designed to measure success at solving applied problems. The results of this study indicate that students will do as well if they are taught by an instructional strategy which uses problems to illustrate calculus theory as they will if they are taught by an instructional strategy which uses problems to initiate the study of calculus theory. / Education, Faculty of / Graduate

The variation of the eigenvalues and eigenfunctions of an ordinary linear self-adjoint differential operator L is considered under perturbations of the domain of L. The basic problem is defined as a suitable singular eigenvalue problem for L on the open interval ω_ < a < ω+ and is assumed to have at least one real eigenvalue λ of multiplicity k. The perturbed problem is a regular self-adjoint problem defined for L on a closed subinterval [a,b] of (ω_,ω+). It is proved under suitable conditions on the boundary operators of the perturbed problem that exactly k perturbed eigenvalues [ Formula omitted ] as a,b ⇢ ω_,ω+. Further, asymptotic estimates are obtained for [ Formula omitted ] as a,b ⇢ ω_,ω+. The other results are refinements which lead to asymptotic estimates for the eigenfunctions and variational formulae for the eigenvalues. The conditions on the limiting behaviour of the boundary operators depend strongly on the nature of the singularities ω_,ω+. If for some numberℓ⃘, ℓ⃘not an eigenvalue, linearly independent solutions of Lx = ℓ⃘x exist which are asymptotically ordered at ω_, then ω_ is palled a class 1 singularity. In the case that both ω_,ω+ are class 1 singularities, very general boundary operators permit the convergence of [ Formula omitted ], Class 2 singularities are defined as follows: If all solutions of Lx = ℓ⃘x are square-integrable on (ω_, c] for any c satisfying ω_ < c < ω+, then ω_ is called a class 2 singularity. An asymptotic ordering of the solutions is not assumed in this case. Since the behaviour of the solutions of Lx = ℓ⃘x is essentially arbitrary when both ω_,ω+ are class 2 singularities, the generality of the boundary operators has to be sacrificed to ensure that [ Formula omitted ]. Certain one end perturbation problems and examples also are considered. / Science, Faculty of / Mathematics, Department of / Graduate

Let U⊗V be a tensor product space over an algebraically closed field F ; let dim U = m and dim V = n ; let T be a linear transformation on U⊗V such that T preserves rank two tensors. We show that T preserves rank one tensors and this enables us to characterize T for all values of m and n. / Science, Faculty of / Mathematics, Department of / Graduate

In this thesis we explore the concept of q-calculus and its generalisation. We begin by defining q-combinatorics which uses a real number to define a new set of numbers and then use these numbers to get classic combinatoric elements. These results have use when we work on our algebra that are related with this specific real number. We then work out some results involving one of the operators in the algebra. This operator together with a similar operator produces some special differential equations that we explore. Then we go on to define integrals as the inverse operator to the one used for our differential equations. In the last chapter we try to generalise everything we have explored until then.

Discusses types of mathematical errors that may occur in scientific work. / Typescript. / "Submitted to the Graduate Council of Florida State University in partial fulfillment of the requirements for the degree of Master of Arts." / Advisor: H. C. Griffith, Professor Directing Paper. / "August, 1958." / Includes bibliographical references (leaf 26).

"There is no more fundamental theorem in calculus than the mean-value theorem. Much of the theory of calculus depends, either directly or indirectly, on this theorem. As a consequence of its importance, the theorem has been investigated by a number of mathematicians with the result that various modifications and extensions of the basic theorem have been made"--Introduction. / "May 1956." / Typescript. / "Submitted to the Graduate Council of Florida State University in partial fulfillment of the requirements for the degree of Master of Arts." / Includes bibliographical references (leaf 17).

A Dissertation submitted to the Department of Mathematics, Science and Technology Education In fulfilment of the requirement for the Degree of Master of Education (Mathematics Education) in the Faculty of Education at the University of Zululand, 2019. / Studies from interdisciplinary have noted positive correlation between spatial-visualization skills and mathematical problem solving. However, majority of these studies that interrogated this shared link between spatial-visualization and problem solving were carried in the urban settings only few interrogated rural settings. Also, studies have identified family social economic status (SES) which mainly described one’s geographical settlement to be one of the major effects on cognitive development. Thus, research finding from cognitive discipline revealed that students from poor SES background are less advantagous to cognitive activities (e.g., problem solving) compare to their counterpart. However, one of research achievements is providing evidence-based that cognitive skills can be enhanced through computer technology and spatial activities hence, the integration of several graphical tools such as: MATLAB, GeoGebra, and many other computer environments in mathematics education. These graphical tools are believed to enhance students’ conceptual and procedural knowledge in problem solving in mathematics areas such as: Euclidean geometry, multivariate calculus, and trigonometry which require more spatial skills in their problem solving. However, little has been researched on vector calculus even though vector calculus by its definition is accompanied by spatial reasoning. Students find it easy to evaluate a given vector integral using analytical techniques for integrations but struggle to visualize and transform it from one coordinate system to another. Objectives Based on the background, the current research employed the theoretical frameworks of Duval semiotic representation and the visual-analyser (VA) proposed by Zazkis et al., to analyse MATLAB instruction on rural-based pre-service teachers' spatial-visualisation skills and problem solving in vector calculus. The examination was guided by the analysis of the dynamic software MATLAB instruction on Spatial-Visualization, problem solving, and achievement in Vector Calculus. The three objectives were to 1) Analyse how rural-based v pre-service teachers apply their spatial-visualisation skills in problem solving in vector calculus. 2) To investigate the degree to which rural-based pre-service teachers’ spatial-visualisation skills correlate with their vector calculus achievement and 3) To assess how a dynamic software environment such as MATLAB influences rural-based pre-service teachers’ spatial-visualisation skills.

Yes / This technical report describes one technique for implementing π-calculus style actions in a programming language. It first attempts to clarify the nature of actions, then gives requirements a primitive in a programming language must satisfy if it is to be used as the basis for the implementation of actions. Finally an example is given of how actions may be implemented in Ada.

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