Discrete phase-locked loop systems and spreadsheets

This paper demonstrates the use of a spreadsheet in exploring non-linear difference equations that describe digital control systems used in radio engineering, communication and computer architecture. These systems, being the focus of intensive studies of mathematicians and engineers over the last 40 years, may exhibit extremely complicated behaviour interpreted in contemporary terms as transition from global asymptotic stability to chaos through period-doubling bifurcations. The authors argue that embedding advanced mathematical ideas in the technological tool enables one to introduce fundamentals of discrete control systems in tertiary curricula without learners having to deal with complex machinery that rigorous mathematical methods of investigation require. In particular, in the appropriately designed spreadsheet environment, one can effectively visualize a qualitative difference in the behviour of systems with different types of non-linear characteristic.

Renormalization group interpretation of the Born and Rytov approximations

In this paper the method of renormalization group (RG) [Phys. Rev. E 54, 376 (1996)] is related to the well-known approximations of Rytov and Born used in wave propagation in deterministic and random media. Certain problems in linear and nonlinear media are examined from the viewpoint of RG and compared with the literature on Born and Rytov approximations. It is found that the Rytov approximation forms a special case of the asymptotic expansion generated by the RG, and as such it gives a superior approximation to the exact solution compared with its Born counterpart. Analogous conclusions are reached for nonlinear equations with an intensity-dependent index of refraction where the RG recovers the exact solution. © 2008 Optical Society of America.

Butterfly catastrophe for fronts in a three-component reaction–diffusion system

We study the dynamics of front solutions in a three-component reaction–diffusion system via a combination of geometric singular perturbation theory, Evans function analysis, and center manifold reduction. The reduced system exhibits a surprisingly complicated bifurcation structure including a butterfly catastrophe. Our results shed light on numerically observed accelerations and oscillations and pave the way for the analysis of front interactions in a parameter regime where the essential spectrum of a single front approaches the imaginary axis asymptotically.

Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation

In this paper, we consider the variable-order nonlinear fractional diffusion equation View the MathML source where xRα(x,t) is a generalized Riesz fractional derivative of variable order View the MathML source and the nonlinear reaction term f(u,x,t) satisfies the Lipschitz condition |f(u1,x,t)-f(u2,x,t)|less-than-or-equals, slantL|u1-u2|. A new explicit finite-difference approximation is introduced. The convergence and stability of this approximation are proved. Finally, some numerical examples are provided to show that this method is computationally efficient. The proposed method and techniques are applicable to other variable-order nonlinear fractional differential equations.

Time-dependent fractional advection–diffusion equations by an implicit MLS meshless method

Recently, many new applications in engineering and science are governed by a series of fractional partial differential equations (FPDEs). Unlike the normal partial differential equations (PDEs), the differential order in a FPDE is with a fractional order, which will lead to new challenges for numerical simulation, because most existing numerical simulation techniques are developed for the PDE with an integer differential order. The current dominant numerical method for FPDEs is Finite Difference Method (FDM), which is usually difficult to handle a complex problem domain, and also hard to use irregular nodal distribution. This paper aims to develop an implicit meshless approach based on the moving least squares (MLS) approximation for numerical simulation of fractional advection-diffusion equations (FADE), which is a typical FPDE. The discrete system of equations is obtained by using the MLS meshless shape functions and the meshless strong-forms. The stability and convergence related to the time discretization of this approach are then discussed and theoretically proven. Several numerical examples with different problem domains and different nodal distributions are used to validate and investigate accuracy and efficiency of the newly developed meshless formulation. It is concluded that the present meshless formulation is very effective for the modeling and simulation of the FADE.

Numerical methods for solving the multi-term time-fractional wave-diffusion equations

In this paper, the multi-term time-fractional wave diffusion equations are considered. The multiterm time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and techniques can also be extended to other kinds of the multi-term fractional time-space models with fractional Laplacian.

Comparing measures of fat-free mass in overweight older adults using three different bioelectrical impedance devices and three prediction equations

Objectives: To compare measures of fat-free mass (FFM) by three different bioelectrical impedance analysis (BIA) devices and to assess the agreement between three different equations validated in older adult and/or overweight populations. Design: Cross-sectional study. Setting: Orthopaedics ward of Brisbane public hospital, Australia. Participants: Twenty-two overweight, older Australians (72 yr ± 6.4, BMI 34 kg/m2 ± 5.5) with knee osteoarthritis. Measurements: Body composition was measured using three BIA devices: Tanita 300-GS (foot-to-foot), Impedimed DF50 (hand-to-foot) and Impedimed SFB7 (bioelectrical impedance spectroscopy (BIS)). Three equations for predicting FFM were selected based on their ability to be applied to an older adult and/ or overweight population. Impedance values were extracted from the hand-to-foot BIA device and included in the equations to estimate FFM. Results: The mean FFM measured by BIS (57.6 kg ± 9.1) differed significantly from those measured by foot-to-foot (54.6 kg ± 8.7) and hand-to-foot BIA (53.2 kg ± 10.5) (P < 0.001). The mean ± SD FFM predicted by three equations using raw data from hand-to-foot BIA were 54.7 kg ± 8.9, 54.7 kg ± 7.9 and 52.9 kg ± 11.05 respectively. These results did not differ from the FFM predicted by the hand-to-foot device (F = 2.66, P = 0.118). Conclusions: Our results suggest that foot-to-foot and hand-to-foot BIA may be used interchangeably in overweight older adults at the group level but due to the large limits of agreement may lead to unacceptable error in individuals. There was no difference between the three prediction equations however these results should be confirmed within a larger sample and against a reference standard.

Efficient solution of two-sided nonlinear space-fractional diffusion equations using fast Poisson preconditioners

We develop a fast Poisson preconditioner for the efficient numerical solution of a class of two-sided nonlinear space fractional diffusion equations in one and two dimensions using the method of lines. Using the shifted Gr¨unwald finite difference formulas to approximate the two-sided(i.e. the left and right Riemann-Liouville) fractional derivatives, the resulting semi-discrete nonlinear systems have dense Jacobian matrices owing to the non-local property of fractional derivatives. We employ a modern initial value problem solver utilising backward differentiation formulas and Jacobian-free Newton-Krylov methods to solve these systems. For efficient performance of the Jacobianfree Newton-Krylov method it is essential to apply an effective preconditioner to accelerate the convergence of the linear iterative solver. The key contribution of our work is to generalise the fast Poisson preconditioner, widely used for integer-order diffusion equations, so that it applies to the two-sided space fractional diffusion equation. A number of numerical experiments are presented to demonstrate the effectiveness of the preconditioner and the overall solution strategy.

A comparison of finite difference and finite volume methods for solving the space-fractional advection-dispersion equation with variable coefficients

Transport processes within heterogeneous media may exhibit non- classical diffusion or dispersion which is not adequately described by the classical theory of Brownian motion and Fick’s law. We consider a space-fractional advection-dispersion equation based on a fractional Fick’s law. Zhang et al. [Water Resources Research, 43(5)(2007)] considered such an equation with variable coefficients, which they dis- cretised using the finite difference method proposed by Meerschaert and Tadjeran [Journal of Computational and Applied Mathematics, 172(1):65-77 (2004)]. For this method the presence of variable coef- ficients necessitates applying the product rule before discretising the Riemann–Liouville fractional derivatives using standard and shifted Gru ̈nwald formulas, depending on the fractional order. As an alternative, we propose using a finite volume method that deals directly with the equation in conservative form. Fractionally-shifted Gru ̈nwald formulas are used to discretise the Riemann–Liouville fractional derivatives at control volume faces, eliminating the need for product rule expansions. We compare the two methods for several case studies, highlighting the convenience of the finite volume approach.

Predictive validity of accelerometer prediction equations for energy expenditure (EE) during overland walking and running in children and adolescents

Purpose The aim of this study was to assess the predictive validity of three accelerometer prediction equations (Freedson et aL, 1997; Trost et aL, 1998; Puyau et al., 2002) for energy expenditure (EE) during overland walking and running in children and adolescents. Methods 45 healthy children and adolescents aged 10-18 completed the following protocol, each task 5-mins in duration, with a 5-min rest period in between; walking normally; walking briskly; running easily and running fast. During each task participants wore MTI (WAM 7164) Actigraphs on the left and right hips. VO2 was monitored breath by breath using the Cosmed K4b2 portable indirect calorimetry system. For each prediction equation, difference scores were calculated as EE measured minus EE predicted. The percentage of 1-min epochs correctly categorized as light (<3 METs), moderate (3-5.9 METs), and vigorous (≥6 METS) was also calculated. Results The Freedson and Trost equations consistently overestimated MET level. The level of overestimation was statistically significant across all tasks for the Freedson equation, and was significant for only the walking tasks for the Trost equation. The Puyau equation consistently underestimated AEE with the exception of the walking normally task. In terms of categorisation, the Freedson equation (72.8% agreement) demonstrated better agreement than the Puyau (60.6%). Conclusions These data suggest that the three accelerometer prediction equations do not accurately predict EE on a minute-by-minute basis in children and adolescents during overland walking and running. However, the cut points generated by these equations maybe useful for classifying activity as either, light, moderate, or vigorous.

High order unconditionally stable difference schemes for the Riesz space-fractional telegraph equation

In this paper, a class of unconditionally stable difference schemes based on the Pad´e approximation is presented for the Riesz space-fractional telegraph equation. Firstly, we introduce a new variable to transform the original dfferential equation to an equivalent differential equation system. Then, we apply a second order fractional central difference scheme to discretise the Riesz space-fractional operator. Finally, we use (1, 1), (2, 2) and (3, 3) Pad´e approximations to give a fully discrete difference scheme for the resulting linear system of ordinary differential equations. Matrix analysis is used to show the unconditional stability of the proposed algorithms. Two examples with known exact solutions are chosen to assess the proposed difference schemes. Numerical results demonstrate that these schemes provide accurate and efficient methods for solving a space-fractional hyperbolic equation.

Ability of commonly used prediction equations to predict resting energy expenditure in children with inflammatory bowel disease

Background: Paediatric onset inflammatory bowel disease (IBD) may cause alterations in energy requirements and invalidate the use of standard prediction equations. Our aim was to evaluate four commonly used prediction equations for resting energy expenditure (REE) in children with IBD. Methods: Sixty-three children had repeated measurements of REE as part of a longitudinal research study yielding a total of 243 measurements. These were compared with predicted REE from Schofield, Oxford, FAO/WHO/UNU, and Harris-Benedict equations using the Bland-Altman method. Results: Mean (±SD) age of the patients was 14.2 (2.4) years. Mean measured REE was 1566 (336) kcal per day compared with 1491 (236), 1441 (255), 1481 (232), and 1435 (212) kcal per day calculated from Schofield, Oxford, FAO/WHO/UNU, and Harris-Benedict, respectively. While the Schofield equation demonstrated the least difference between measured and predicted REE, it, along with the other equations tested, did not perform uniformly across all subjects, indicating greater errors at either end of the spectrum of energy expenditure. Smaller differences were found for all prediction equations for Crohn's disease compared with ulcerative colitis. Conclusions: Of the commonly used equations, the equation of Schofield should be used in pediatric patients with IBD when measured values are not able to be obtained. (Inflamm Bowel Dis 2010;) Copyright © 2010 Crohn's & Colitis Foundation of America, Inc.

Effective Cell-Centred Time-Domain Maxwell's Equations Numerical Solvers

This research work analyses techniques for implementing a cell-centred finite-volume time-domain (ccFV-TD) computational methodology for the purpose of studying microwave heating. Various state-of-the-art spatial and temporal discretisation methods employed to solve Maxwell's equations on multidimensional structured grid networks are investigated, and the dispersive and dissipative errors inherent in those techniques examined. Both staggered and unstaggered grid approaches are considered. Upwind schemes using a Riemann solver and intensity vector splitting are studied and evaluated. Staggered and unstaggered Leapfrog and Runge-Kutta time integration methods are analysed in terms of phase and amplitude error to identify which method is the most accurate and efficient for simulating microwave heating processes. The implementation and migration of typical electromagnetic boundary conditions. from staggered in space to cell-centred approaches also is deliberated. In particular, an existing perfectly matched layer absorbing boundary methodology is adapted to formulate a new cell-centred boundary implementation for the ccFV-TD solvers. Finally for microwave heating purposes, a comparison of analytical and numerical results for standard case studies in rectangular waveguides allows the accuracy of the developed methods to be assessed.