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There has recently been considerable interest in using a nonstandard piecewise approximation to formulate fractional order differential equations as difference equations that describe the same dynamical behaviour and are more amenable to a dynamical systems analysis. Unfortunately, due to mistakes in the fundamental papers, the difference equations formulated through this process do not capture the dynamics of the fractional order equations. We show that the correct application of this nonstandard piecewise approximation leads to a one parameter family of fractional order differential equations that converges to the original equation as the parameter tends to zero. A closed formed solution exists for each member of this family and leads to the formulation of a difference equation that is of increasing order as time steps are taken. Whilst this does not lead to a simplified dynamical analysis it does lead to a numerical method for solving the fractional order differential equation. The method is shown to be equivalent to a quadrature based method, despite the fact that it has not been derived from a quadrature. The method can be implemented with non-uniform time steps. An example is provided showing that the difference equation can correctly capture the dynamics of the underlying fractional differential equation.

Explicit numerical finite difference schemes for partial differential equations are well known to be easy to implement but they are particularly problematic for solving equations whose solutions admit shocks, blowups and discontinuities. Here we present an explicit numerical scheme for solving non-linear advection-diffusion equations admitting shock solutions that is both easy to implement and stable. The numerical scheme is obtained by considering the continuum limit of a discrete time and space stochastic process for non-linear advection-diffusion. The stochastic process is well posed and this guarantees the stability of the scheme. Several examples are provided to highlight the importance of the formulation of the stochastic process in obtaining a stable and accurate numerical scheme., Comment: 15 pages, 2 figures

We demonstrate an efficient high-precision optical spectroscopy technique for single trapped ions with non-closed transitions. In a double-shelving technique, the absorption of a single photon is first amplified to several phonons of a normal motional mode shared with a co-trapped cooling ion of a different species, before being further amplified to thousands of fluorescence photons emitted by the cooling ion using the standard electron shelving technique. We employ this extension of the photon recoil spectroscopy technique to perform the first high precision absolute frequency measurement of the $^{2}$D$_{3/2}$ $\rightarrow$ $^{2}$P$_{1/2}$ transition in $^{40}$Ca$^{+}$, resulting in a transition frequency of $f=346\, 000\, 234\, 867(96)$ kHz. Furthermore, we determine the isotope shift of this transition and the $^{2}$S$_{1/2}$ $\rightarrow$ $^{2}$P$_{1/2}$ transition for $^{42}$Ca$^{+}$, $^{44}$Ca$^{+}$ and $^{48}$Ca$^{+}$ ions relative to $^{40}$Ca$^{+}$ with an accuracy below 100 kHz. Improved field and mass shift constants of these transitions as well as changes in mean square nuclear charge radii are extracted from this high resolution data.

An introduction to periodic orbit techniques for deterministic dynamical systems is presented. The Farey map is considered as examples of intermittency in one-dimensional maps. The effect of intermittency on the Markov partition is considered. The Gauss map is shown to be related to the farey map by a simple transformation of trajectories. A method of calculating periodic orbits in the thermostated Lorentz gas is derived. This method relies on minimising the action from the Hamiltonian description of the Lorentz gas, as well as the construction of a generating partition of the phase space. This method is employed to examine a range of bifurcation processes in the Lorentz gas. A novel construction of the Sinai billiard is performed by using symmetry arguments to reduce two particles in a hard walled box to the square Sinai billiard. Infinite families of periodic orbits are found, even at the lowest order, due to the intermittency of the system. The contribution of these orbits is examined and found to be tractable at the lowest order. The number of orbits grows too quickly for consideration of any other terms in the periodic orbit expansion. A simple stochastic model for the diffusion in the Lorentz gas was constructed. The model produced a diffusion coefficient that was a remarkably good fit to more precise numerical calculations. This is a significant improvement to the Machta-Zwanzig approximation for the diffusion coefficient. We outline a general approach to constructing stochastic models of deterministic dynamical systems. This method should allow for calculations to be performed in more complicated systems.