Applications of symbolic computing for symmetry analysis of differential equations

 This thesis presents a number of applications of symbolic computing to the study of differential equations. In particular, three packages have been produced for the computer algebra system MAPLE and used to find a variety of symmetries (and corresponding invariant solutions) for a range of differential systems.

Assessing the performance of two lung age equations on the Australian population: using data from the cross-sectional BOLD-Australia study

Lung age, a simple concept for patients to grasp, is frequently used as an aid in smoking cessation programs. Lung age equations should be continuously updated and made relevant for target populations. We observed how new lung age equations developed for Australian populations performed when utilizing the Burden of Obstructive Lung Disease (BOLD)-Australia dataset compared to more commonly used equations.

Evaluation of Actical equations and thresholds to predict physical activity intensity in young children

This study examined the validity of current Actical activity energy expenditure (AEE) equations and intensity cut-points in preschoolers using AEE and direct observation as criterion measures. Forty 4–6-year-olds (5.3 ± 1.0 years) completed a ~150-min room calorimeter protocol involving age-appropriate sedentary behaviours (SBs), light intensity physical activities (LPAs) and moderate-to-vigorous intensity physical activities (MVPAs). AEE and/or physical activity intensity were calculated using Actical equations and cut-points by Adolph, Evenson, Pfeiffer and Puyau. Predictive validity was examined using paired sample t-tests. Classification accuracy was evaluated using weighted kappas, sensitivity, specificity and area under the receiver operating characteristic curve. The Pfeiffer equation significantly overestimated AEE during SB and underestimated AEE during LPA (P < 0.0125 for both). There was no significant difference between measured and predicted AEEs during MVPA. The Adolph cut-point showed significantly higher accuracy for classifying SB, LPA and MVPA than all others. The available Actical equation does not provide accurate estimates of AEE across all intensities in preschoolers. However, the Pfeiffer equation performed reasonably well for MVPA. Using cut-points of ≤6 counts · 15 s−1, 7–286 counts · 15 s−1 and ≥ 287 counts · 15 s−1 when classifying SB, LPA and MVPA, respectively, is recommended.

FracSym: automated symbolic computation of Lie symmetries of fractional differential equations

In this paper, we present an algorithm for the systematic calculation of Lie point symmetries for fractional order differential equations (FDEs) using the method as described by Buckwar & Luchko (1998) and Gazizov, Kasatkin & Lukashchuk (2007, 2009, 2011). The method has been generalised here to allow for the determination of symmetries for FDEs with n independent variables and for systems of partial FDEs. The algorithm has been implemented in the new MAPLE package FracSym (Jefferson and Carminati 2013) which uses routines from the MAPLE symmetry packages DESOLVII (Vu, Jefferson and Carminati, 2012) and ASP (Jefferson and Carminati, 2013). We introduce FracSym by investigating the symmetries of a number of FDEs; specific forms of any arbitrary functions, which may extend the symmetry algebras, are also determined. For each of the FDEs discussed, selected invariant solutions are then presented. © 2013 Elsevier B.V. All rights reserved.

Isotherm, kinetic, and thermodynamic equations for cefalexin removal from liquids using activated carbon synthesized from loofah sponge

Activated carbon (AC) developed from loofah sponge with phosphoric acid activation was applied to absorb cefalexin (CEX) in aqueous solution. AC was characterized by N2 adsorption–desorption isotherms and Fourier transform infrared spectroscopy (FTIR). Factors influencing the adsorption process were investigated. The equilibrium adsorption isotherms and kinetics of CEX were also studied. The results showed that AC prepared from loofah sponge had rough surface and abundant pores. The determination results of specific surface area (810.12 m²/g) and average pore size (5.28 nm) suggested the high adsorption capability. At low concentration, the AC could adsorb about 95% of CEX. The adsorption effect was independent of the temperature and pH. The maximum adsorption amount of CEX was about 55.11 mg/g at 308 K. The equilibrium data agreed well with Freundlich isotherm equation (R² = 0.9957) at 308 K, which indicated multilayer adsorption. FTIR analysis suggested the existence of phosphorus-containing functional groups, C–O bond, and C=C bond on the surface of AC of which the peak intensity of AC after adsorption was slightly lower after adsorption, indicating that the AC surface groups interacted with or were covered by the CEX species.

Robust optimal motion cueing algorithm based on the linear quadratic regulator method and a genetic algorithm

The aim of this paper is to design and develop an optimal motion cueing algorithm (MCA) based on the genetic algorithm (GA) that can generate high-fidelity motions within the motion simulator's physical limitations. Both, angular velocity and linear acceleration are adopted as the inputs to the MCA for producing the higher order optimal washout filter. The linear quadratic regulator (LQR) method is used to constrain the human perception error between the real and simulated driving tasks. To develop the optimal MCA, the latest mathematical models of the vestibular system and simulator motion are taken into account. A reference frame with the center of rotation at the driver's head to eliminate false motion cues caused by rotation of the simulator to the translational motion of the driver's head as well as to reduce the workspace displacement is employed. To improve the developed LQR-based optimal MCA, a new strategy based on optimal control theory and the GA is devised. The objective is to reproduce a signal that can follow closely the reference signal and avoid false motion cues by adjusting the parameters from the obtained LQR-based optimal washout filter. This is achieved by taking a series of factors into account, which include the vestibular sensation error between the real and simulated cases, the main dynamic limitations, the human threshold limiter in tilt coordination, the cross correlation coefficient, and the human sensation error fluctuation. It is worth pointing out that other related investigations in the literature normally do not consider the effects of these factors. The proposed optimized MCA based on the GA is implemented using the MATLAB/Simulink software. The results show the effectiveness of the proposed GA-based method in enhancing human sensation, maximizing the reference shape tracking, and reducing the workspace usage.

Primary voltage control of a single-phase inverter using linear quadratic regulator with integrator

This paper proposes a linear quadratic regulator with integral action, ensuring fast dynamic response and resisting capability of voltage deviation from instantaneous reference grid voltage, to control the inverter voltage that can also be used in a microgrid. The proposed control strategy is based on a linear quadratic regulator, minimizing the cost function of the system, with integral action used to impede voltage degradation from a reference voltage for outside disturbances of the system, such as abrupt load change. The combined integral term assists to recover the voltage difference between grid and reference grid voltage. The validity of the proposed controller has been tested with linear and non-linear loads under various conditions. In both cases, the effectiveness of the controller has been proved. The result of the proposed controller is good to track the instantaneous reference grid voltage.