Eben Maré

Solving the Generalized Regularized Long Wave Equation Using a Distributed Approximating Functional Method

mite local spectral kernel. Test problems including propagation of single solitons, interaction of two and three solitons, and conservation properties of mass, energy, and momentum of the GRLW equation are discussed to test the efficiency and accuracy of the method. Furthermore, using the Maxwellian initial condition, we show that the number of solitons which are generated can be approximately determined. Comparisons are made between the results of the proposed method, analytical solutions, and numerical methods. It is found that the method under consideration is a viable alternative to existing numerical methods.

Discrete singular convolution for the generalized variable-coefficient Korteweg-de Vries equation

Abstract Numerical solutions of the generalized variable-coefficient Korteweg-de Vries equation are obtained using a discrete singular convolution and a fourth order singly diagonally implicit Runge-Kutta method for space and time discretisation, respectively. The theoretical convergence of the proposed method is rigorously investigated. Test problems including propagation of single solitons and interaction of solitary waves are performed to verify the efficiency and accuracy of the method. The numerical results are checked against available analytical solutions and compared with the Sinc numerical method. We find that our approach is a very accurate, efficient and reliable method for solving nonlinear partial differential equations.

A lagrange regularized kernel method for solving multi-dimensional time-fractional heat equations

Abstract: Evolution equations containing fractional derivatives can provide suitable mathematical models for describing important physical phenomena. In this paper, we propose an accurate method for numerical solutions of multi-dimensional time-fractional heat equations. The proposed method is based on a fractional exponential integrator scheme in time and the Lagrange regularized kernel method in space. Numerical experiments show the effectiveness of the proposed approach.

Fractional Black-Scholes Option Pricing, Volatility Calibration and Implied Hurst Exponents

This paper addresses several theoretical and practical issues in option pricing and implied volatility calibration in a fractional Black-Scholes market. In particular, we discuss how the fractional Black-Scholes model admits a non-constant implied volatility term structure when the Hurst exponent is not 0.5, and also that one-year implied volatility is independent of Hurst exponent and equivalent to fractional volatility. Building on these observations, we introduce a novel 8-parameter fractional Black-Scholes inspired, or FBSI, model. This deterministic volatility surface model is based on the fractional Black-Scholes framework and uses Gatheral’s (2004) SVI pamaterisation for the fractional volatility skew and a quadratic parameterisation for the Hurst exponent skew. The issue of arbitrage-free calibration for the FBSI model is addressed in depth and it is proven in general that any FBSI volatility surface will be free from calendar-spread arbitrage. The FBSI model is empirically tested on implied volatility data on a South African equity index as well as the USDZAR exchange rate. Results show that the FBSI model fits the equity index implied volatility data very well and that a more flexible Hurst exponent parameterisation is needed to accurately fit the USDZAR implied volatility surface data.

Market Risk Management in the Context of Engineering Asset Management

Engineering asset management has a broad scope and covers a wide variety of areas. These would typically include general management, operations and production areas as well as financial aspects. It is essential to consider risk management aspects arising from asset management activities, in particular if we view financial assets of the firm as financial derivatives of our engineering assets. A coordinated strategic framework is required to ascertain Enterprise Risk Management. In this paper we focus on market risk aspects in the context of engineering asset management. We demonstrate the market risk process and note implementation requirements.

Sinc collocation method for solving the Benjamin-Ono equation

We propose a simple, though powerful, technique for numerical solutions of the Benjamin-Ono equation. This approach is based on a global collocation method using Sinc basis functions. Some properties of the Sinc collocation method required for our subsequent development are given and utilized to reduce the computation of the Benjamin-Ono equation to a system of ordinary differential equations. The propagation of one soliton and the interaction of two solitons are used to validate our numerical method. The method is easy to implement and yields accurate results.