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A new method with regularization for solving split variational inequality problem in real Hilbert spaces.
(2022) Akutsah, Francis.; Narain, Ojen Kumar
The concept of the optimization problem, fixed point theory and its application constitute the nucleus of nonlinear analysis, which is a major branch of mathematics. Optimization theory, fixed point theory and its applications have a wide range of application in practically every field of science, particularly mathematical sciences. The theory of optimization and fixed point have received great attention from authors around the world and these areas will continue to receive such great attention. The theory has been well developed by well-known researchers in these areas. However, there are still a lot of work to be done. The goal of this thesis is to advance the theory of optimization and fixed point in the framework of Hilbert and Banach spaces. The substance of this thesis is separated into two parts. The research efforts of the first part of this thesis (Chapter 3 to Chapter 6) has to do with introducing some new iterative methods for approximating the solution of a variational inequality problems, split variational inequality problems, equilibrium problems, split monotone variational inclusion problem, split generalized mixed equilibrium problem and fixed point problems in the framework of a Hilbert and Banach spaces. In addition, we introduce a new class of bilevel problem in the framework of real Hilbert spaces and a new regularization technique, and inertial terms for approximating solutions of split bilevel variational inequality problems. Furthermore, we establish that the proposed iterative methods converges strongly to the solution of the aforementioned problems as the case may be. Then, we present some numerical experiments to show the efficiency and applicability of our proposed methods in comparison with other state-of-the-art iterative methods in the literature. The second part (Chapter 7) of this thesis deals with developing iterative algorithms and introducing some nonlinear mappings in the framework of the Hilbert and Banach spaces. First, we present a modified (improved) generalized Miteration with the inertial technique for three quasi-nonexpansive multivalued mappings in a real Hilbert space. In addition, we present some fixed point results for a general class of nonexpansive mappings in the framework of the Banach space and also proposed a new iterative scheme for approximating the fixed point of this class of mappings in the framework of uniformly convex Banach spaces. Finally, we apply our convergence results to certain optimization problems, integral equations, and we present some numerical experiments to show the efficiency and applicability of the proposed method in comparison with other existing methods in the literature.
School managers’ perceptions on the use of technological resources in curriculum supervision in the context of COVID-19.
(2023) Biyela, Thokozani Andreas.; Mpungose, Cedric Bheki.
This dissertation presents a qualitative case study of sixteen school managers specifically school principals. This study was conducted with the main purpose of exploring the school
managers’ perceptions on the use of technological resources in curriculum supervision in the context of COVID-19 in the Zululand District. Furthermore, the study employed an interpretive paradigm. In addition, this paradigm has been utilised because the study aims at exploring the three missing levels of perception (knowledge, skill, and attitude) during curriculum supervision. The study sought to understand why school managers resist to use EdTech (educational technological) resources to supervise curriculum. In addition, an interpretive paradigm and case study were used on sixteen participants to gain the meaning in real situation. Subsequently, reflective activity, one-on-one semi-structured interviews, and focus groups were used to generate data. Moreover, the study used non-probability sampling methods, comprising purposive and convenient sampling. This study was framed by the by the curricular spider web (Van den Akker et al., 2010). Further to this, the theory that shapes this study is technological pedagogical content knowledge (TPACK). Data were analysed using guided analysis, which employed both deductive and inductive methods. Finally, ethical considerations related to qualitative research were explored such as trustworthiness, confirmability, dependability, transferability, and credibility. The findings of the study reveal that school managers were driven by knowledge perception to supervise curriculum when using EdTech supervision aids. The school managers could indicate the lack of technological resources and knowledge in schools. Even though there is a policy in place, some school managers are non-compliant with that policy. The study recommends the Department of Basic Education to install Wi-Fi in all schools to enforce the policy implementation. Moreover, the study further recommends that curriculum developers work alongside information and communication technologies (ICT) team specialists to advocate paperless curriculum supervision, train, and enhance school managers on software and ideological-ware EdTech supervision aids.
Numerical analysis and long-term dynamics of Fourier-type pseudo-spectral schemes applied to the Klein-Gordon equation.
(2024) Lukumon, Gafari Abiodun.; Shindin, Sergey Konstantinovich.; Parumasur, Nabendra.
This thesis investigates the Klein-Gordon equation (KGE) using a combined theoretical and numerical approach. We develop a robust numerical approximation scheme for the KGE that demonstrates good convergence properties for various data types and explore the long-time behavior of semi-discrete KGE solutions near finite- and infinite-dimensional invariant subsets of an appropriate space. In the first part of the thesis, we establish convergence results for the semidiscrete, Fourier pseudo-spectral spatial approximation of the KGE with smooth potentials. We present an extensive stability and convergence analysis for finite Sobolev regularity data in T and R, as well as for smooth data from Gevrey classes in T. We demonstrate that the convergence rate is algebraic in the first case and (sub-)geometric in the second case. The second part of the thesis deals with the numerical studies of the long-time dynamics of semi-discrete numerical solutions in periodic settings. Through an extensive set of simulations, we show that the pseudospectral semi-discretization is capable of preserving finite- and infinite-dimensional invariant structures over very long time intervals.
Experiences of adolescents transitioning into blended families from a selected high school in the Umlazi District.
(2024) Harisingh, Keiasha.; Mkhize-Mthiyane, Ncamisile Parscaline.
Research indicates that adolescents across the globe experience stressful life events such as their parents’ divorce, separation and other forms of family breakdown, which
often results in them having to transition into blended families. For some adolescents, the transition process is often confusing and stressful and can affect their psychological, physical, mental, and socio-emotional health and educational achievements. The South African government and the Department of Basic Education have neglected to acknowledge or include provisions and support for blended families. This lack of inclusion is also experienced by teachers who are often ill-equipped and inexperienced to assist their learners who are transitioning into their new blended families. This study explored the experiences of adolescents transitioning into blended families and examined how they navigated their adjustment to their blended families. This study was underpinned by an interpretivist paradigm and theoretically framed by the McMaster Model of Family Functioning (MMFF). A qualitative case study design was applied, and semi-structured interviews and diaries were utilised to gather data from twelve participants at a selected high school in the Umlazi District in KwaZulu-Natal. Ethical principles, including voluntary participation, non-maleficence, beneficence and confidentiality, were ensured, and the data was analysed using thematic analysis. The findings revealed that how adolescents received their transition announcements influenced their receptiveness to their new blended families. It was also discovered that the blended families engaged in varied methods of conflict resolution and levels of family involvement that ranged from effective to ineffective in terms of their family functioning. The findings further revealed the types of support that adolescents received throughout their transition as well as their evaluation of their emotional stability and the effectiveness of their transition into their respective blended families. The study concludes that all adolescents who transition into blended families have unique experiences based on varied circumstances, and this indicates that support initiatives for these adolescents must be personalised and cater to their individual needs.
Geography teachers’ reflections on teaching topography.
(2024) Kumalo, Lucky Nhlanhla.; Shoba, Makhosazana.
In view of the scant existing research on topography teaching, this study focused on how topography is taught in South African schools and aimed to fill this gap by exploring geography teachers’ reflections on teaching topography in four schools in Chatsworth. This dissertation adopted an exploratory case study rooted within the interpretive paradigm. It explored geography teachers’ reflections on teaching topography to understand their reflections, what informs them, and what we can learn from them. This exploration therefore necessitated the use of the interpretive paradigm. The study used reflective activity and semi structured interviews to draw out teachers’ reflections on teaching topography. The convenience and purposive sampling techniques were used to select the geography teachers from whom the data to fulfil the aims of this study was generated. The study employed the guided data analysis approach to analyse the elicited data. The reflective activity and interview questions used to obtain teachers’ reflections on teaching topography were underpinned by the guiding questions of the curricular spider web concepts. The findings of this study indicate that geography teachers mainly engage in practical reflection when evaluating their teaching of topography. It appears that teachers are largely impelled by their professional rationale in their reflections. Thus, their teaching of topography is mainly guided by the prescriptions of the curriculum policy rather than their personal perspectives or societal needs. As such, teachers adhered to the guidelines of the Curriculum Assessment Policy Statements (CAPS) curriculum when deciding what to teach (content), when to teach (time) and how to teach it (teaching goals). However, the findings of this study further indicate that teachers’ compliance to the predetermined standards of the intended curriculum (professional rationale) made teachers unable to transform topography curriculum to suit learners’ needs. Therefore, it is suggested in this study that teachers draw from all three teaching rationales (personal, professional, and societal) for a balanced implementation of topography where teachers’ personal perspectives, professional knowledge, and societal needs are considered in topography teaching.