Overlapping grid spectral collocation method for nonlinear PDEs: EMHD bioconvective flow in accelerated Carreau-Yasuda nanofluid

by Dr Musawenkhosi Patson Mkhatshwa

Department of Mathematics, University of KwaZulu Natal (South Africa)

Abstract : The focus of this work is on computational grid-manipulation (i.e using the overlapping grid idea) to enhance the accuracy, convergence and computational efficiency of spectral collocation approach for the solution of nonlinear partial differential equations with fluid mechanics applications. The improved method is used in the scrutiny of Darcy- Forchheimer bioconvection flow of Carreau-Yasuda nanofluid induced by an oscillatory moving surface. Sensitivity and error analysis are provided to confirm the accuracy and efficiency of using overlapping grids when discretizing the solution domain for Chebyshev spectral collocation method. The impact of different parameter values on fluid properties and transport phenomena is also discussed. Key findings, are inter alia, that the overlapping multi-domain spectral collocation approach is computationally efficient, produce stable and accurate results using a small number of grid points in each subinterval and in the entire computational domain. Using the overlapping grids yields less dense coefficient matrices that invert easily.

Brief Biography: Dr Musawenkhosi Patson Mkhatshwa is a mathematics researcher who specializes in numerical analysis and computational fluid dynamics. He recently graduated with a PhD in Applied Mathematics from the University of KwaZulu Natal (South Africa). He also holds MSc degree in Applied Mathematics and B.Ed Secondary (majoring in Mathematics and Education) from the University of Johannesburg (South Africa) and University of Swaziland (eSwatini), respectively. His main areas of research include the development of sufficiently accurate, convergent and computationally efficient numerical methods and application of these methods in solving differential equations modelling practical problems from fields such as fluid dynamics. He has participated in local and international conferences. His research work on the overlapping grid spectral collocation methods appears in more than 10 academic journals in applied mathematics, physics, and engineering so far.

30 September 2021
1600hrs – 1800hrs

Strong convergence method for monotone inclusion problem with alternating inertial steps

by Dr. Chibueze Christian Okeke

University of the Witwatersrand, Johannesburg South Africa.

Abstract : This article proposes a strong convergence of the forward-backward splitting method for monotone inclusion problem with alternated inertial extrapolation step in a real Hilbert space. The proposed method converges strongly under some suitable and easy to verify assumptions. The advantage of our iterative scheme is that the single-valued operator is Lipschitz continuous monotone rather than cocoercive and Lipschitz constant does not require to be known. Finally, we give some numerical experiments of the proposed algorithm to demonstrate the advantages of our algorithm over the existing related ones.

Brief Biography Chibueze Christian Okeke is a Lecturer at the University of the Witwatersrand, Johannesburg South Africa. He holds a Doctorate degree in Pure Mathematics in the area of fixed-point theory and its applications. He has published quite several good articles in this area.

23 September 2021
1600hrs – 1800hrs

Join via Zoom, Meeting ID: 870 5376 6741

Diversified Cryptoassets Portfolio Optimization

by Jules Mba

College of Business and Economics of the University of Johannesburg

Abstract: Nobel Prize winner Harry Max Markowitz in his modern portfolio theory (MPT) defined an approach to constructing portfolios in 1952 that has since become the model followed by most advisors and investors. This approach provides for the construction of investment portfolios that maximize expected returns based upon a targeted level of risk. Markowitz’s efficient frontier, which maximizes returns for a given level of risk, is reached by smartly combining assets in a portfolio. The financial crisis of 2008 caused many financial advisors and wealth managers to evaluate different approaches to diversified portfolio construction by including nontraditional assets that perform in a noncorrelated fashion to stocks and bonds. In the ashes of the 2008 financial crisis, Satoshi Nakamoto built the concept of Bitcoin. It ignited the cryptocurrency revolution, and its success has led to the birth of numerous other cryptocurrencies, making them significantly investable assets for innovative investors. The cryptocurrency market is known to be highly volatile due on one hand to its sensibility to new information, whether fundamental or speculative since it does not rely on the stabilizing policy of a central bank. On the other hand, the relative illiquidity of the market with no official market makers makes it fundamentally fragile to large trading volumes and to market imperfections, and thus more prone to large swings than traditional assets. In this talk, we present in terms of risk-reward performance, a diversified crypto portfolio able to achieve constant and positive return throughout the rebalancing periods. A brief survey on Automated Market Makers (AMMs) will also be presented.

Brief Biography: Jules Mba received a BSc degree in Mathematics in 1999 from University of Douala, an Advanced Diploma in Education in 2003 from Teachers Training College of the University of Yaoundé 1, a MSc in Differential Geometry in 2006 from the University of Yaoundé 1, Cameroon. In 2011, he received a PhD degree in Group Theory from the University of the Western Cape. In 2012, he joined the Department of Mathematics and Applied Mathematics of the University of Johannesburg as Lecturer. He completed a Masters of Commerce in Financial Economics in 2018 from the School of Economics of the University of Johannesburg. He has recently joined the College of Business and Economics of the University of Johannesburg as Senior Lecturer. His present research interests are mainly in financial modeling, risk and uncertainty, portfolio optimization, and group theory.

07 Oct 2021
1600hrs – 1800hrs

Join via Zoom, Meeting ID: 862 9053 7664

Stochastic differential game strategies in the presence of reinsurance and dividend payout.

by Dr. Farai Julius Mhlanga

Abstract: This talk presents and examines a problem in which two insurance companies apply non- proportional reinsurance to control risk. Additionally, each firm pays out dividends. The situation is modelled as a zero-sum stochastic differential game between the two companies. The goal of one company is to maintain business competitive advantage over the other by sustaining or increasing the difference between the respective liquid reserves of the two companies while the second company aims to minimise that difference. A verification theorem is formulated, proved and subsequently employed to derive the saddle point components. For the case of the payoff with a non-zero running cost function, we are able to solve explicitly the differential game. Numerical simulations are presented to illustrate the results as well as the economic interpretation.

Brief Biography: Farai Julius Mhlanga obtained a PhD in Applied Mathematics from the University of Cape Town, South Africa. He has worked as a Graduate Teaching Assistant at the University of Zimbabwe, Lecturer at the National University of Science and Technology (NUST), Zimbabwe, Part-time lecturer at the University of Cape Town, lecturer at the University of Zululand and is currently employed by the University of Limpopo where he is a senior lecturer. Farai Mhlanga is the coordinator of the SA-UK University Staff Capacity Doctoral Program: Building Capacity in Applied Mathematics. This program seeks to train permanent academic staff at South African institutions to acquire doctoral qualifications which will in turn increase the number of permanent academics with PhDs while also building supervisory capacity in postgraduate programs. Farai is also the University of Limpopo Node leader for the University Capacity Development Program. His research area includes Mathematics of Finance, Stochastic Analysis, Model-free Finance and Insurance Mathematics. He has supervised masters and doctoral students to completion. His recent PhD graduate graduated virtually on 28 April 2021. Farai is the current Vice President of Southern African Mathematical Sciences Association (SAMSA).

19 August 2021
1600hrs – 1800hrs

Join via Zoom, Meeting ID: 862 9053 7664

Adaptive arbitrary order block hybrid method

by Dr. S’yanda Mungwe

Abstract: Ordinary and partial differential equations widely model a vast of real-world problem such as in engineering, population dynamic, finance, etc. To gain insight of these problems, solutions to the aforementioned equation need to be computed. Solutions for some equations may be found analytically while others require numerical solvers in order to deduce their solutions. It is of this reason, in this talk, we present a numerical solution approach to these equations. Amongst the existing methods, the proposed approach offers flexibility to choose order of accuracy of the method depending on the complexity of the problem in hand. The method also chooses the step- size adaptively with respect to the variations that the solution function may possess. During the talk we shall demonstrate these advantages by solving some well know function such as Runge-function, Van der Pol’s equation, cosine problem, etc.

Brief Biography: S’yanda Mungwe is a lecturer at Stellenbosch University and a PhD candidate at the same institution. He specializes on Scientific Computing and Numerical Mathematics. On his spare time, he enjoys taking lessons on programming.

12 August 2021
1600hrs – 1800hrs

Join via Zoom, Meeting ID: 862 9053 7664

Department of Mathematics Colloquium

Brian Mudhara

An Expert in Business Analysis across various domains, Banking, Risk Management (Basel, IFRS9, ALM, Credit Risk, Market Risk and OP Risk), Information Technology, Insurance, Fintech and Digital Transformation, Brian is a result driven gentleman with an innovative mind. He is multi-talented, with great skill in understanding business, developing solutions for under-serviced markets, digital transformation and gifted with immaculate complex problem solving skills and analytical thinking power.

13 August 2021
1300hrs – 1500hrs

Join via Zoom, Meeting ID: 862 9053 7664

Mesenchymal stem cells used as carrier cells of oncolytic adenovirus results in enhanced oncolytic virotherapy

by Dr. Khaphetsi Joseph Mahasa

Abstract : Mesenchymal stem cells (MSCs) loaded with oncolytic viruses are presently being investigated as a new modality of advanced/metastatic tumors treatment and enhancement of virotherapy. MSCs can, however, either promote or suppress tumor growth. To address the critical question of how MSCs loaded with oncolytic viruses affect virotherapy outcomes and tumor growth patterns in a tumor microenvironment, we developed and analyzed an integrated mathematical-experimental model. We used the model to describe both the growth dynamics in our experiments of firefly luciferase- expressing Hep3B tumor xenografts and the effects of the immune response during the MSCs-based virotherapy. We further employed it to explore the conceptual clinical feasibility, particularly, in evaluating the relative significance of potential immune promotive/suppressive mechanisms induced by MSCs loaded with oncolytic viruses. One of the most impactful outcomes shown by this investigation, not inferred from the experiments alone, was the initially counter-intuitive fact that using tumor-promoting MSCs as carriers is not only helpful but necessary in achieving tumor control. Considering the fact that it is still currently a controversial debate whether MSCs exert a pro- or anti-tumor action, mathematical models such as this one help to quantitatively predict the consequences of using MSCs for delivering virotherapeutic agents in vivo. Taken together, our results show that MSC-mediated systemic delivery of oncolytic viruses is a promising strategy for achieving synergistic anti-tumor efficacy with improved safety profiles.

Biography Currently, Dr. Mahasa is a lecturer of Mathematics at the National University of Lesotho. His interdisciplinary research interests lie at the interface of Mathematics, Biological and Medical Sciences. Taking into account the complexity of immune system response to tumor growth and therapeutic application, he likes to develop models that are highly nonlinear and dynamic in nature. To solve such models, a wide variety of approaches are used ranging from mathematical to numerical analysis. In particular, “machinery” tools from diverse areas of mathematics such as ordinary, delayed and partial differential equations, stability and bifurcation theory, sensitivity analysis, and optimal control theory are used. Coupling experimental data with mathematical models is his major research expertise which warranties sound mathematical and medical predictions. Dr. Mahasa holds a PhD in Applied Mathematics from Stellenbosch University, MSc in Applied Mathematics from University of KwaZulu-Natal, MSc in Mathematical Sciences from AIMS and a BSc from National University of Lesotho.

25/03/2021
1600hrs – 1800hrs

Join via Zoom Meeting, Meeting ID: 862 9053 7664

The Kalman Filter

by Dr. Peter Mhone

Abstract : The Kalman filter is a practical finite dimensional solution to the real time optimal estimation for stochastic systems. The Kalman filter has been used to model many important applications involving noisy measurements for estimating the current conditions of dynamic systems. In this talk we develop the equations of the Kalmann filter problem, study the solutions and then consider a numerical example.

Brief Biography : Dr. Peter Mhone has been teaching Mathematics and Statistics in Universities in East and Southern Africa since 2007. His areas of interest are in Mathematical Finance, Probability and Statistics and Mathematics of Insurance.

28/01/2021
1600hrs – 1800hrs

Application of Logical Analysis of Data (LAD) to credit risk ratings for banks: Case Study of Zimbabwean Banks.

by Dr. Isabel Moyo

Abstract : As data is now becoming available and accessible, credit risk management departments in financial institutions are now engaging machine learning techniques to produce more reliable internal credit risk rating systems. In this paper, data on 17 Zimbabwean banks are used to apply and test the Logical Analysis of Data (LAD) a supervised learning data mining technique, to generate an objective, transparent, consistent, accurate, self contained and generalizable credit risk rating system that has varying levels of granularity and is Basel compliant. This system gives an understanding of relationships between the use of credit ratings, the different options for rating system design and the effectiveness of internal credit rating systems. Such a system becomes useful in decision making pertaining to the determination of the amount allocated as regulatory capital in banks, which is a buffer in banks against distress and bank failure.

Brief Biography : Isabel has taught Mathematics, Statistics and Operations Research at institutions in Zimbabwe since 2004, currently lecturing Mathematics at UNESWA. She loves using latest techniques and statistical-OR tools to analyse data and solve community-and industry-based problems. Her research interests are in predictive analytics in Financial, Manufacturing and Health systems.

04/02/2021
1600hrs – 1800hrs

Modelling the spread of schistosomiasis in humans and non-human mammals incorporating concomitant immunity.

by Dr. Faraimunashe Chirove

Abstract : Although schistosomiasis containment campaigns have recorded substantial success in most developed countries, sub-Saharan Africa still suffers greatly under the burden of the disease. A basic mathematical model to assess the impact of concomitant immunity in humans and environmental transmission of schistosomiasis disease progression is formulated. Mathematical analysis is carried out to establish the existence of the equilibrium points providing necessary conditions for their local and global stability. Numerical simulations are done to analyse the effects of environmental transmission and processes associated with development of concomitant immunity. Our results suggest that schistosomiasis burden is increased by direct and indirect contribution of individuals with concomitant immunity to the schistosomiasis infection chain, increasing the shedding of miracidia up to the development of cercariae promoting the growth of cercariae, increase in environmental transmission due to cercariae, reducing the clearance rate of cercariae and reducing the development of humans and non-human mammals escape mechanisms from cercariae attack.

Brief Biography : Dr. Faraimunashe Chirove is a NRF C2 rated scientist and, holder of a Bachelor of Science honors in Mathematics (2003) and Master of Science in Mathematics (2005) from University of Zimbabwe and a PhD (2011) in Mathematics (Mathematical Biology) from University of Botswana. He is currently employed as a Senior lecturer at University of Johannesburg. His vision is to be one of the leading bio-mathematicians both nationally and internationally producing high quality and high impact research. His broad research objective is 'from omics to population dynamics', an objective that seeks to understand, interpret, explain, and account for the impact of infection across different hierarchical levels within and without the human body. Currently he is largely doing most of his research at cellular level and population dynamics predicting the impact of the cellular infection on the population dynamics and vice versa on human infectious diseases. His research interests are expanding into mathematical ecology, data- based modelling, agent-based modelling, applications into zoonotic diseases, stochastic modelling, multi-scale modelling, antimicrobial resistance in agricultural settings and, systems mathematical biology. He is also focusing on multi-and interdisciplinary research as to make his modelling skills manifest into realistic impact on public health and animal health policies.

15/04/2021
1600hrs – 1800hrs

A note on S-metrizable locales

by Dr. Cerene Rathilal

Abstract : Garcia-Maynez showed that S-metrizability of a topological space X is equivalent to the space having a perfect locally connected metrizable compacti cation. For the purpose of generalizing the above result, properties of dense metric sublocales shall be investigated. We show that every metric frame (M; d) is a dense metric sublocale of its metric frame completion. As a consequence, we obtain that if (M; d) is uniformly locally connected then its metric frame completion is also uniformly locally connected. Results on Property S will be presented before we obtain the analog of the result by Garcia-Maynez. We show that a connected locally connected metric frame (M; d) is S-metrizable i (M; d) has a perfect locally connected metrizable compacti cation.

Brief Biography : Cerene Rathilal is a lecturer at the University of Johannesburg. She obtained her PhD in Pure Mathematics from the University of Kwa-Zulu Natal in 2019, and her more recent research focuses on Pointfree Topology. She has published in Topology and its Applications, and is the JFAC for Algebra and Topology at the CoE-MaSS.

30/07/2021
1600hrs – 1800hrs

On block hybrid methods for solving non-linear ordinary and partial differential equations

by Prof. Sandile Motsa

Abstract : A class of block hybrid method (BHM) for solving initial value problems for ordinary and partial differential equations is studied. The methods are developed using collocation in a block (sub-interval), In, of the integration domain and solved over M grid points, inside In. For the partial differential equations, spectral methods are used to discretize the space variables to form a hybrid method called spectral block hybrid method (SBHM). The proposed method is A-stable and has a flexible accuracy that can be progressively improved by adding additional nodes while keeping the step-size constant. The applicability of the BHM is tested on non-linear dynamical systems with applications in Mathematical biology, chaos theory, chemical kinetics and vibration theory. The SBHM is applied on non-linear evolution equations and hyperbolic equations. The numerical experiments considered in this study highlight the advantages of the proposed methods over existing methods that have appeared in the literature.

Brief Biography : Prof. Sandile Motsa is the Dean of the Faculty of Science and Engineering at the University of Eswatini. His research projects are mainly focused on developing new methods for solving mathematical models that emanate from all areas of Science and Engineering. Prof. Motsa’s research articles have appeared in over 200 leading academic journals in Applied Mathematics, Mathematical Physics and Engineering. He is ranked as one of the top 2% of the world’s most cited researchers in the world and his work has been cited over 3600 times. Prof. Motsa has successfully supervised (to completion) 12 MSc students, 13 PhD students and 5 Post-Doctoral fellows. As a service to the field of Mathematical Sciences, Prof. Motsa serves in the editorial board of three Scopus accredited journals, namely, the Journal of Applied Fluid Mechanics, Scientific African Journal, and Algorithms Journal. From 2014 to 2018, he served as the Vice President of the Southern Africa Mathematical Sciences Association (SAMSA), an organization that seeks to further the research and teaching of Mathematical Sciences in Southern African countries and beyond through holding of conferences, workshops, academic exchange visits and research schools. Prof. Motsa is now the current president of SAMSA after being elected into that position in November 2018. He is also the founding member of the newly established Kingdom of Eswatini Academy of Sciences.

25/02/2021
1600hrs – 1800hrs

Quadratic Hedging in Incomplete Markets

by Mr. Melusi Mavuso

Abstract : We begin by introducing some terminology and notation used in incomplete markets. We then present a solution to a problem of hedging a derivative security written on an illiquid asset but with unconstrained trading in a correlated asset. To this end, we apply the Kunita-Watanabe and Follmer-Schweizer decompositions and project the derivative onto the space of attainable claims. Finally, we present a solution to the mean- variance hedging problem found using Long-Short Term Memory Recurrent Neural Networks.

Brief Biography : Melusi Mavuso holds a Bachelor of Science in Actuarial Science, Mathematical Statistics, and Pure Mathematics, BSc(Hons) Pure Mathematics, MPhil Mathematical Finance and MSc Pure Mathematics from the University of Cape Town. He is currently a lecturer in the department of Statistical Sciences at the University of Cape Town (UCT), a position he has held since 2014. He also works for Liberty Financial Solutions as a Quantitative Analyst Consultant and have held similar consulting positions elsewhere before. He is also the director and head quant for NK Capital Management, a Cape Town-based hedge fund startup. His research interests are primarily in probability theory, mathematical finance, and machine learning. He has also worked in calculus of variations and PDE theory in the past. His research mainly involves applying machine learning techniques to various problems in mathematical finance, including calibration, hedging and pricing of derivatives, and the construction of trading strategies. He has also applied these techniques to solving PDEs and to problems in the calculus of variations.

08/04/2021
1600hrs – 1800hrs

A Liu-Storey-type conjugate gradient algorithm for unconstrained optimization with application in portfolio selection

by Dr. Auwal Bala Abubakar

Abstract : Conjugate gradient methods have played a vital role in finding the minimizers of large- scale unconstrained optimization problems due to the simplicity of their iteration, convergence properties and their low memory requirements. Based on the Liu-Storey conjugate gradient method, in this paper, we present a Liu-Storey type method for finding the minimizers of large-scale unconstrained optimization problems. The direction of the proposed method is constructed in such a way that the sufficient descent condition is satisfied. Also, an adaptive conjugacy condition and an intrinsic self-restarting mechanism are revealed, a dynamical adjustment can be regarded as the inheritance and development of properties of standard Liu-Storey method. Further, we establish the global convergence result for general functions using both the Wolfe and Armijo-type line search. Numerical findings indicate that our presented approach is efficient and robust, thus effective in solving large-scale test problems. In addition, practical application of the algorithm in portfolio selection is also explored.

Brief Biography : Auwal Bala Abubakar received the Master's degree in Mathematics in 2015 and the Ph.D. degree in applied mathematics from the King Mongkut's University of Technology Thonburi, Thailand in 2020. He is currently a Lecturer II with the Department of Mathematical Sciences, Faculty of Physical Sciences, Bayero University, Kano, Nigeria, and also a Postdoctoral fellow at The King Mongkut's University of Technology Thonburi. He has published more than 40 research articles in highly reputable journals. His main research interest includes iterative methods for solving unconstrained optimization problems, nonlinear monotone operator equations with application in motion control, portfolio selection, signal, and image recovery.

22/04/2021
1600hrs – 1800hrs

Everywhere Arbitrage: The Special Case of the Food Markets.

by Dr. Sani Sulaiman

Abstract : Stock and derivative pricing have proved valuable in hedging financial risk of several businesses especially those businesses with high level free lunch tendencies. The existence of pseudo assets within certain category of contingencies makes market trading below capacity due to hyper level pricing arbitrage and free lunch. The food industry is one industry with this bug and in need of Mathematical analysis for business continuity and stability. This paper analyses pseudo assets lying within the contingencies of food assets in trade and provides a stochastic pricing strategy under no arbitrage condition. Demonstration of the application of derived formulas is provided as examples and remarks over imposed effects sequel to sub pseudo contingencies. The results are aids of assistance for business owners, investors and governments relative to financing pseudo asset ventures for the purpose of valuation for sustainability and completeness.

Brief Biography : Sulaiman Sani is a Senior Lecturer-Mathematics, University of Eswatini, Kwaluseni- Eswatini. He holds a Doctorate Degree in Mathematics of the University of Botswana, Gaborone since 2015. Sulaiman Sani is actively engaged in Mathematics of Finance, Operational Research and Optimization of service systems of Economics, Engineering and Environment to date. He has published several research papers in these areas and was invited by many academic institutions worldwide for presentation of research papers at conferences of International repute including the prestigious African Academy of Sciences Conference of 2016 in Abuja-Nigeria. Sulaiman Sani received the Annual Best Lecturer award for Physical Sciences of the School of Engineering and Applied Sciences, Kampala International University Uganda in December 2018. He is also a recipient of the Jalingo local Government, Nigeria merit medal in 2020. He has taught and tutored Mathematics in several African Universities including the University of Botswana, Gaborone. Sulaiman Sani has supervised to completion five (5) M.Sc. theses to which three (3) led to the award of M.Sc. Mathematics and two (2) led to the award of M.SEE. Electrical Engineering.

22/04/2021
1600hrs – 1800hrs