Nicholas Hale

As an example in their complex variables textbook, Mark Ablowitz and Athanassios Fokas apply the method of steepest descent to derive an asymptotic estimate for the solution of the linear Korteweg–de Vries (KdV) equation for large times. Here the analysis is extended to an equation with a fractional dispersion relation that contains this KdV equation (which models dispersion), the heat equation (diffusion), and the one-way wave equation (advection) as special cases.

A framework for Chebyshev spectral collocation methods for the numerical solution of functional and delay differential equations (FDEs and DDEs) is described. The framework combines interpolation via the barycentric resampling matrix with a multidomain approach used to resolve isolated discontinuities propagated by nonsmooth initial data. Geometric convergence in the number of degrees of freedom is demonstrated for several examples of linear and nonlinear FDEs and DDEs with various delay types, including discrete, proportional, continuous and state-dependent delay. The framework is a natural extension of standard spectral collocation methods and can be readily incorporated into existing spectral discretizations, such as in Chebfun/Chebop, allowing the automated and efficient solution of a wide class of nonlinear FDEs and DDEs.

Despite its limitations, Prandtl’s mixing length model is widely applied in modelling turbulent free shear flows. Prandtl’s extended model addresses many of the shortfalls of the original model, but is not so widely used, in part due to additional mathematical complexities that arise in its derivation and implementation. Furthermore, in both models, Prandtl neglects the kinematic viscosity on the basis that it is much smaller in magnitude than the turbulent viscosity. Recent work has shown that including the kinematic viscosity in the original model has both mathematical and physical advantages. In the present work, a novel derivation of the extended model is provided, and it is demonstrated that similar advantages are again obtained when the kinematic viscosity is included. Additionally, through the use of scaling techniques, similarity mean velocity profiles of the extended model are derived, resulting in a single nonlinear ordinary differential equation that is solved numerically with a Hermite spectral method. The computed profiles for the normalized similarity mean velocity and shear stress are compared with experimental observations and shown to be in excellent agreement.

This paper applies an algorithm for the convolution of compactly supported Legendre series (the CONLeg method) (cf. Hale and Townsend, An algorithm for the convolution of Legendre series. SIAM J. Sci. Comput., 2014, 36, A1207–A1220), to pricing European-type, early-exercise and discrete-monitored barrier options under a Lévy process. The paper employs Chebfun (cf. Trefethen et al., Chebfun Guide, 2014 (Pafnuty Publications: Oxford), Available online at: http://www.chebfun.org/) in computational finance and provides a quadrature-free approach by applying the Chebyshev series in financial modelling. A significant advantage of using the CONLeg method is to formulate option pricing and option Greek curves rather than individual prices/values. Moreover, the CONLeg method can yield high accuracy in option pricing when the risk-free smooth probability density function (PDF) is smooth/non-smooth. Finally, we show that our method can accurately price options deep in/out of the money and with very long/short maturities. Compared with existing techniques, the CONLeg method performs either favourably or comparably in numerical experiments.

This paper applies an algorithm for the convolution of compactly supported Legendre series (the CONLeg method) (cf. Hale and Townsend 2014a), to pricing/hedging European-type, early-exercise and discrete-monitored barrier options under a Levy process. The paper employs Chebfun (cf. Trefethen et al. 2014) in computational finance and provides a quadrature-free approach by applying the Chebyshev series in financial modelling. A significant advantage of using the CONLeg method is to formulate option pricing and option Greek curves rather than individual prices/values. Moreover, the CONLeg method can yield high accuracy in option pricing and hedging when the risk-free smooth probability density function (PDF) is smooth/non-smooth. Finally, we show that our method can accurately price/hedge options deep in/out of the money and with very long/short maturities. Compared with existing techniques, the CONLeg method performs either favourably or comparably in numerical experiments.

The Legendre-based ultraspherical spectral method for ordinary differential equations (Olver, S. & Townsend, A. (2013) A fast and well-conditioned spectral method. SIAM Rev., 55, 462–489.) is combined with a formula for the convolution of two Legendre series (Hale, N. & Townsend, A. (2014a) An algorithm for the convolution of Legendre series. SIAM J. Sci. Comput., 36, A1207–A1220.) to produce a new technique for solving linear Fredholm and Volterra integro-differential equations with convolution-type kernels. When the kernel and coefficient functions are sufficiently smooth, then the method is spectrally accurate and the resulting almost-banded linear systems can be solved with linear complexity.

A fast algorithm (linear in the degrees of freedom) for the solution of linear variable-coefficient rational-order fractional integral and differential equations is described. The approach is related to the ultraspherical method for ODEs [S. Olver and A. Townsend, SIAM Rev., 55 (2013), pp. 462--489] and involves constructing two different bases, one for the domain of the operator and one for the range of the operator. The bases are constructed from direct sums of suitably weighted ultraspherical or Jacobi polynomial expansions, for which explicit representations of fractional integrals and derivatives are known, and are carefully chosen so that the resulting operators are banded or almost banded. Geometric convergence is demonstrated for numerous model problems when the variable coefficients and right-hand side are sufficiently smooth.