In the mathematical modeling of the classical option pricing models it is assumed that the underlying stock price process follows a geometric Brownian motion, but through statistical analysis persistency was found in the log-returns of some South African stocks and Brownian motion does not have persistency. We suggest the replacement of Brownian motion with fractional Brownian motion which is a Gaussian process that depends on the Hurst parameter that allows for the modeling of autocorrelation in price returns. Three fractional Black-Scholes (Black) models were investigated where the underlying is assumed to follow a fractional Brownian motion. Using South African options on futures and warrant prices these models were compared to the classical models. / Dissertation (MSc)--University of Pretoria, 2012. / Mathematics and Applied Mathematics / unrestricted

In this work, we construct numerical methods to solve a wide range of problems in finance. This includes the valuation under affine jump diffusion processes, chaotic and hyperchaotic systems, and pricing fractional cryptocurrency models. These problems are of extreme importance in the area of finance. With today’s rapid economic growth one has to get a reliable method to solve chaotic problems which are found in economic systems while allowing synchronization. Moreover, the internet of things is changing the appearance of money. In the last decade, a new form of financial assets known as cryptocurrencies or cryptoassets have emerged. These assets rely on a decentralized distributed ledger called the blockchain where transactions are settled in real time. Their transparency and simplicity have attracted the main stream economy players, i.e, banks, financial institutions and governments to name these only. Therefore it is very important to propose new mathematical models that help to understand their dynamics. In this thesis we propose a model based on fractional differential equations. Modeling these problems in most cases leads to solving systems of nonlinear ordinary or fractional differential equations. These equations are known for their stiffness, i.e., very sensitive to initial conditions generating chaos and of multiple fractional order. For these reason we design numerical methods involving Chebyshev polynomials. The work is done from the frequency space rather than the physical space as most spectral methods do. The method is tested for valuing assets under jump diffusion processes, chaotic and hyperchaotic finance systems, and also adapted for asset price valuation under fraction Cryptocurrency. In all cases the methods prove to be very accurate, reliable and practically easy for the financial manager. / Thesis (PhD)--University of Pretoria, 2021. / Mathematics and Applied Mathematics / PhD / Unrestricted

This study seeks to develop a method that generalises the use of Adams-Bashforth to solve or treat partial differential equations with local and non-local differentiation by deriving a two-step Adams-Bashforth numerical scheme in Laplace space. The resulting solution is then transformed back into the real space by using the inverse Laplace transform. This is a powerful numerical algorithm for fractional order derivative. The error analysis for the method is studied and presented. The numerical simulations of the method as applied to the four-dimensional model, Caputo-Lu-Chen model and the wave equation are presented. In the analysis, the bifurcation dynamics are discussed and the periodic doubling processes that eventually caused chaotic behaviour (butterfly attractor) are shown. The related graphical simulations that show the existence of fractal structure that is characterised by chaos and usually called strange attractors are provided. For the Caputo-Lu-Chen model, graphical simulations have been realised in both integer and fractional derivative orders. / Mathematical Sciences / M. Sc. (Applied Mathematics)

Fractional calculus has been introduced as an e cient tool for modeling physical phenomena, thanks to its memory and hereditary properties. For example, fractional models have been successfully used to describe anomalous di↵usion processes such as contaminant transport in soil, oil flow in porous media, and groundwater flow. These models capture important features of particle transport such as particles with velocity variations and long-rest periods. Mathematical modeling of physical phenomena requires the identification of pa- rameters and variables from available measurements. This is referred to as an inverse problem. In this work, we are interested in studying theoretically and numerically inverse problems for space Fractional Advection-Dispersion Equation (FADE), which is used to model solute transport in porous media. Identifying parameters for such an equa- tion is important to understand how chemical or biological contaminants are trans- ported throughout surface aquifer systems. For instance, an estimate of the di↵eren- tiation order in groundwater contaminant transport model can provide information about soil properties, such as the heterogeneity of the medium. Our main contribution is to propose a novel e cient algorithm based on modulat-ing functions to estimate the coe cients and the di↵erentiation order for space FADE, which can be extended to general fractional Partial Di↵erential Equation (PDE). We also show how the method can be applied to the source inverse problem. This work is divided into two parts: In part I, the proposed method is described and studied through an extensive numerical analysis. The local convergence of the proposed two-stage algorithm is proven for 1D space FADE. The properties of this method are studied along with its limitations. Then, the algorithm is generalized to the 2D FADE. In part II, we analyze direct and inverse source problems for a space FADE. The problem consists of recovering the source term using final observations. An analytic solution for the non-homogeneous case is derived and existence and uniqueness of the solution are established. In addition, the uniqueness and stability of the inverse problem is studied. Moreover, the modulating functions-based method is used to solve the problem and it is compared to a standard Tikhono-based optimization technique.

The fractional Laplacian is a promising mathematical tool due to its ability to capture the anomalous diffusion and model the complex physical phenomenon with long-range interaction, such as fractional quantum mechanics, image processing, jump process, etc. One of the important applications of fractional Laplacian is a turbulence intermittency model of fractional Navier-Stokes equation which is derived from Boltzmann's theory. However, the efficient computation of this model on bounded domains is challenging as highly accurate and efficient numerical methods are not yet available. The bottleneck for efficient computation lies in the low accuracy and high computational cost of discretizing the fractional Laplacian operator. Although many state-of-the-art numerical methods have been proposed and some progress has been made for the existing numerical methods to achieve quasi-optimal complexity, some issues are still fully unresolved: i) Due to nonlocal nature of the fractional Laplacian, the implementation of the algorithm is still complicated and the computational cost for preparation of algorithms is still high, e.g., as pointed out by Acosta et al \cite{AcostaBB17} 'Over 99\% of the CPU time is devoted to assembly routine' for finite element method; ii) Due to the intrinsic singularity of the fractional Laplacian, the convergence orders in the literature are still unsatisfactory for many applications including turbulence intermittency simulations. To reduce the complexity and computational cost, we consider two numerical methods, finite difference and spectral method with quasi-linear complexity, which are summarized as follows. We develop spectral Galerkin methods to accurately solve the fractional advection-diffusion-reaction equations and apply the method to fractional Navier-Stokes equations. In spectral methods on a ball, the evaluation of fractional Laplacian operator can be straightforward thanks to the pseudo-eigen relation. For general smooth computational domains, we propose the use of spectral methods enriched by singular functions which characterize the inherent boundary singularity of the fractional Laplacian. We develop a simple and easy-to-implement fractional centered difference approximation to the fractional Laplacian on a uniform mesh using generating functions. The weights or coefficients of the fractional centered formula can be readily computed using the fast Fourier transform. Together with singularity subtraction, we propose high-order finite difference methods without any graded mesh. With the use of the presented results, it may be possible to solve fractional Navier-Stokes equations, fractional quantum Schrodinger equations, and stochastic fractional equations with high accuracy. All numerical simulations will be accompanied by stability and convergence analysis.

There has been a rapid growth in the data rate carried by cellular services, and this increase along with the emergence of new multimedia applications have motivated the 3rd Generation Partnership (3GPP) Project to launch Long-Term Evolution (LTE) [1]. LTE is the latest standard in the mobile network technology and is designed to meet the ubiquitous demands of next-generation mobile networks. LTE assures significant spectral and energy efficiency gains in both the uplink and down- link with low latency. Multiple access schemes such as Orthogonal Frequency Division Aultiple Access (OFDMA) and Single Carrier Frequency Division Multiple Access (SC-FDMA) which is a modified version of OFDMA have been recently adopted in 3GPP LTE downlink and uplink, respectively [1]. A typical feature of OFDMA is the decomposition of available bandwidth into multiple narrow orthogonal subcarriers. The orthogonality among subcarriers causes minimal intra-cell interference, however, the inter-cell interference (ICI) incurred on a given subcarrier is relatively impulsive and poses a fundamental challenge for the network designers. Moreover, as the number of interferers on a given subcarrier can be relatively limited it may not be accurate to model ICI as a Gaussian random variable by invoking the central limit theorem. The nature of ICI relies on a variety of indeterministic parameters which include frequency reuse factor, channel conditions, scheduling decisions, transmit power, and location of the interferers. This thesis presents a combination of algorithmic and theoretical studies for efficient modeling and management of ICI via radio resource management. In the preliminary phase, we focus on developing and analyzing the performance of several centralized and distributed interference mitigation and rate maximization algorithms. These algorithms relies on optimizing the spectrum allocation and user’s transmission powers to maximize the system capacity. Even though, the developed algorithms possesses low complexity, the simulation run-time may become challenging in the practical scenarios with very large number of users and subcarriers. Motivated by this fact, we then develop several statistical models that can accurately capture the dynamics of interference with distinct applications in the performance analysis of single carrier and multicarrier future wireless networks. The developed models can be customized for (i) various state-of-the-art coordinated and uncoordinated scheduling algorithms; (ii) slow and fast power control mechanisms; (iii) partial and fractional frequency reuse systems; and (iv) various composite fading distributions. The developed framework is useful in evaluating important system performance metrics such as outage probability, ergodic capacity, and average fairness numerically without the need of time consuming Monte-Carlo simulations. The theoretical framework is expected to enhance the planning tools for OFDMA based wireless networks by providing fast estimates of the typical performance metrics. Finally, we investigate and quantify the spectral and energy efficiency of two tier heterogeneous networks (HetNets) by employing power-control based interference mitigation technique. In particular, we analyze the performance of two tier HetNets deployment by deriving the theoretical bounds on the area spectral efficiency and exact analytical expressions for the energy efficiency by considering slow and fast power control mechanisms. The derived expressions are expected to be useful in providing insights for the design of efficient HetNet deployments.

The fractional–order capacitors add an additional degree of freedom over conventional capacitors in circuit design and facilitate circuit configurations that would be impractical or impossible to implement with conventional capacitors. We propose a generic strategy for fractional-order capacitor fabrication that integrates layers of conductive, semiconductor and ferroelectric polymer materials to create a composite with significantly improved constant phase angle, constant phase zone, and phase angle variation performance. Our approach involves a combination of dissolving the polymer powders, mixing distinct phases and making a film and capacitor of it. The resulting stack consisting of ferroelectric polymer-based composites shows constant phase angle over a broad range of frequencies. To prove the viability of this method, we have successfully fabricated fractional-order capacitors with the following: nanoparticles such as multiwall carbon nanotube (MWCNT), Molybdenum sulfide (MoS2) inserted ferroelectric polymers and PVDF based ferroelectric polymer blends. They show better performance in terms of fabrication cost and dynamic range of constant phase angle compared to fractional order capacitor from graphene percolated polymer composites. These results can be explained by a universal percolation model, where the combination of electron transport in fillers and the dielectric relaxation time distribution of the permanent dipoles of ferroelectric polymers increase the constant phase angle level and constant phase zone of fractional-order capacitors. This approach opens up a new avenue in fabricating fractional capacitors involving a variety of heterostructures combining the different fillers and different matrixes.

This work presents a new control strategy using fractional order operators in threephase grid-connected photovoltaic generation systems with unity power factor for any situation of solar radiation. The modeling of the space vector pulse width modulation inverter and fractional order control strategy using Park’s transformation are proposed. The system is able to compensate harmonic components and reactive power generated by the loads connected to the system. A fractional order extremum seeking control and “Bode’s ideal cut-off extremum seeking control” are proposed to control the power between the grid and photovoltaic system, to achieve the maximum power point operation. Simulation results are presented to validate the proposed methodology for grid-connected photovoltaic generation systems. The simulation results and theoretical analysis indicate that the proposed control strategy improves the efficiency of the system by reducing the total harmonic distortion of the injected current to the grid and increases the robustness of the system against uncertainties. Additionally, the proposed maximum power point tracking algorithms provide more robustness and faster convergence under environmental variations than other maximum power point trackers.

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