Analytical solutions of the multi-term modified power law wave equations in a finite domain

Fractional partial differential equations with more than one fractional derivative term in time, such as the Szabo wave equation, or the power law wave equation, describe important physical phenomena. However, studies of these multi-term time-space or time fractional wave equations are still under development. In this paper, multi-term modified power law wave equations in a finite domain are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals (1, 2], [2, 3), [2, 4) or (0, n) (n > 2), respectively. Analytical solutions of the multi-term modified power law wave equations are derived. These new techniques are based on Luchko’s Theorem, a spectral representation of the Laplacian operator, a method of separating variables and fractional derivative techniques. Then these general methods are applied to the special cases of the Szabo wave equation and the power law wave equation. These methods and techniques can also be extended to other kinds of the multi term time-space fractional models including fractional Laplacian.

Analytical solutions for the multi-term time-fractional diffusion-wave/diffusion equations in a finite domain

Multi-term time-fractional differential equations have been used for describing important physical phenomena. However, studies of the multi-term time-fractional partial differential equations with three kinds of nonhomogeneous boundary conditions are still limited. In this paper, a method of separating variables is used to solve the multi-term time-fractional diffusion-wave equation and the multi-term time-fractional diffusion equation in a finite domain. In the two equations, the time-fractional derivative is defined in the Caputo sense. We discuss and derive the analytical solutions of the two equations with three kinds of nonhomogeneous boundary conditions, namely, Dirichlet, Neumann and Robin conditions, respectively.

Simulation of AE wave propagation in thin plate for source identification

In most materials, short stress waves are generated during the process of plastic deformation, phase transformation, crack formation and crack growth. These phenomena are applied in acoustic emission (AE) for the detection of material defects in wide spectrum areas, ranging from non-destructive testing for the detection of materials defects to monitoring of microeismical activity. AE technique is also used for defect source identification and for failure detection. AE waves consist of P waves (primary/longitudinal waves), S waves (shear/transverse waves) and Rayleight (surface) waves as well as reflected and diffracted waves. The propagation of AE waves in various modes has made the determination of source location difficult. In order to use the acoustic emission technique for accurate identification of source location, an understanding of wave propagation of the AE signals at various locations in a plate structure is essential. Furthermore, an understanding of wave propagation can also assist in sensor location for optimum detection of AE signals. In real life, as the AE signals radiate from the source it will result in stress waves. Unless the type of stress wave is known, it is very difficult to locate the source when using the classical propagation velocity equations. This paper describes the simulation of AE waves to identify the source location in steel plate as well as the wave modes. The finite element analysis (FEA) is used for the numerical simulation of wave propagation in thin plate. By knowing the type of wave generated, it is possible to apply the appropriate wave equations to determine the location of the source. For a single plate structure, the results show that the simulation algorithm is effective to simulate different stress waves.

Electromechanical wave in power systems : theory and applications

The continuum model is a key paradigm describing the behavior of electromechanical transients in power systems. In the past two decades, much research work has been done on applying the continuum model to analyze the electromechanical wave in power systems. In this work, the uniform and non-uniform continuum models are first briefly described, and some explanations borrowing concepts and tools from other fields are given. Then, the existing approaches of investigating the resulting wave equations are summarized. An application named the zero reflection controller based on the idea of the wave equations is next presented.

Adapting to the work-life interface : the influence of individual differences, work and family on well-being, mental health and work engagement

Bronfenbrenner.s Bioecological Model, expressed as the developmental equation, D f PPCT, is the theoretical framework for two studies that bring together diverse strands of psychology to study the work-life interface of working adults. Occupational and organizational psychology is focused on the demands and resources of work and family, without emphasising the individual in detail. Health and personality psychology examine the individual but without emphasis on the individual.s work and family roles. The current research used Bronfenbrenner.s theoretical framework to combine individual differences, work and family to understand how these factors influence the working adult.s psychological functioning. Competent development has been defined as high well-being (measured as life satisfaction and psychological well-being) and high work engagement (as work vigour, work dedication and absorption in work) and as the absence of mental illness (as depression, anxiety and stress) and the absence of burnout (as emotional exhaustion, cynicism and professional efficacy). Study 1 and 2 were linked, with Study 1 as a cross-sectional survey and Study 2, a prospective panel study that followed on from the data used in Study1. Participants were recruited from a university and from a large public hospital to take part in a 3-wave, online study where they completed identical surveys at 3-4 month intervals (N = 470 at Time 1 and N = 198 at Time 3). In Study 1, hierarchical multiple regressions were used to assess the effects of individual differences (Block 1, e.g. dispositional optimism, coping self-efficacy, perceived control of time, humour), work and family variables (Block 2, e.g. affective commitment, skill discretion, work hours, children, marital status, family demands) and the work-life interface (Block 3, e.g. direction and quality of spillover between roles, work-life balance) on the outcomes. There were a mosaic of predictors of the outcomes with a group of seven that were the most frequent significant predictors and which represented the individual (dispositional optimism and coping self-efficacy), the workplace (skill discretion, affective commitment and job autonomy) and the work-life interface (negative work-to-family spillover and negative family-to-work spillover). Interestingly, gender and working hours were not important predictors. The effects of job social support, generally and for work-life issues, perceived control of time and egalitarian gender roles on the outcomes were mediated by negative work-to-family spillover, particularly for emotional exhaustion. Further, the effect of negative spillover on depression, anxiety and work engagement was moderated by the individual.s personal and workplace resources. Study 2 modelled the longitudinal relationships between the group of the seven most frequent predictors and the outcomes. Using a set of non-nested models, the relative influences of concurrent functioning, stability and change over time were assessed. The modelling began with models at Time 1, which formed the basis for confirmatory factor analysis (CFA) to establish the underlying relationships between the variables and calculate the composite variables for the longitudinal models. The CFAs were well fitting with few modifications to ensure good fit. However, using burnout and work engagement together required additional analyses to resolve poor fit, with one factor (representing a continuum from burnout to work engagement) being the only acceptable solution. Five different longitudinal models were investigated as the Well-Being, Mental Distress, Well-Being-Mental Health, Work Engagement and Integrated models using differing combinations of the outcomes. The best fitting model for each was a reciprocal model that was trimmed of trivial paths. The strongest paths were the synchronous correlations and the paths within variables over time. The reciprocal paths were more variable with weak to mild effects. There was evidence of gain and loss spirals between the variables over time, with a slight net gain in resources that may provide the mechanism for the accumulation of psychological advantage over a lifetime. The longitudinal models also showed that there are leverage points at which personal, psychological and managerial interventions can be targeted to bolster the individual and provide supportive workplace conditions that also minimise negative spillover. Bronfenbrenner.s developmental equation has been a useful framework for the current research, showing the importance of the person as central to the individual.s experience of the work-life interface. By taking control of their own life, the individual can craft a life path that is most suited to their own needs. Competent developmental outcomes were most likely where the person was optimistic and had high self-efficacy, worked in a job that they were attached to and which allowed them to use their talents and without too much negative spillover between their work and family domains. In this way, individuals had greater well-being, better mental health and greater work engagement at any one time and across time.

Simulation of inplane and out of plane AE source in thin plate

In most materials, short stress waves are generated during the process of plastic deformation, phase transformation, crack formation and crack growth. These phenomena are applied in acoustic emission (AE) for the detection of material defects in a wide spectrum of areas, ranging from nondestructive testing for the detection of materials defects to monitoring of microseismical activity. AE technique is also used for defect source identification and for failure detection. AE waves consist of P waves (primary longitudinal waves), S waves (shear/transverse waves) and Rayleigh (surface) waves as well as reflected and diffracted waves. The propagation of AE waves in various modes has made the determination of source location difficult. In order to use acoustic emission technique for accurate identification of source, an understanding of wave propagation of the AE signals at various locations in a plate structure is essential. Furthermore, an understanding of wave propagation can also assist in sensor location for optimum detection of AE signals along with the characteristics of the source. In real life, as the AE signals radiate from the source it will result in stress waves. Unless the type of stress wave is known, it is very difficult to locate the source when using the classical propagation velocity equations. This paper describes the simulation of AE waves to identify the source location and its characteristics in steel plate as well as the wave modes. The finite element analysis (FEA) is used for the numerical simulation of wave propagation in thin plate. By knowing the type of wave generated, it is possible to apply the appropriate wave equations to determine the location of the source. For a single plate structure, the results show that the simulation algorithm is effective to simulate different stress waves.

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.

Accurate series solutions for gravity-driven Stokes waves

In the past, high order series expansion techniques have been used to study the nonlinear equations that govern the form of periodic Stokes waves moving steadily on the surface of an inviscid fluid. In the present study, two such series solutions are recomputed using exact arithmetic, eliminating any loss of accuracy due to accumulation of round-off error, allowing a much greater number of terms to be found with confidence. It is shown that higher order behaviour of series generated by the solution casts doubt over arguments that rely on estimating the series’ radius of convergence. Further, the exact nature of the series is used to shed light on the unusual nature of convergence of higher order Pade approximants near the highest wave. Finally, it is concluded that, provided exact values are used in the series, these Pade approximants prove very effective in successfully predicting three turning points in both the dispersion relation and the total energy.

A simultaneous equations model of crash frequency by collision type for rural intersections

Safety at roadway intersections is of significant interest to transportation professionals due to the large number of intersections in transportation networks, the complexity of traffic movements at these locations that leads to large numbers of conflicts, and the wide variety of geometric and operational features that define them. A variety of collision types including head-on, sideswipe, rear-end, and angle crashes occur at intersections. While intersection crash totals may not reveal a site deficiency, over exposure of a specific crash type may reveal otherwise undetected deficiencies. Thus, there is a need to be able to model the expected frequency of crashes by collision type at intersections to enable the detection of problems and the implementation of effective design strategies and countermeasures. Statistically, it is important to consider modeling collision type frequencies simultaneously to account for the possibility of common unobserved factors affecting crash frequencies across crash types. In this paper, a simultaneous equations model of crash frequencies by collision type is developed and presented using crash data for rural intersections in Georgia. The model estimation results support the notion of the presence of significant common unobserved factors across crash types, although the impact of these factors on parameter estimates is found to be rather modest.

A parallel computation of power system equations

Streaming SIMD Extensions (SSE) is a unique feature embedded in the Pentium III and P4 classes of microprocessors. By fully exploiting SSE, parallel algorithms can be implemented on a standard personal computer and a theoretical speedup of four can be achieved. In this paper, we demonstrate the implementation of a parallel LU matrix decomposition algorithm for solving power systems network equations with SSE and discuss advantages and disadvantages of this approach.

SSE based parallel solution for power systems network equations

Streaming SIMD Extensions (SSE) is a unique feature embedded in the Pentium III class of microprocessors. By fully exploiting SSE, parallel algorithms can be implemented on a standard personal computer and a theoretical speedup of four can be achieved. In this paper, we demonstrate the implementation of a parallel LU matrix decomposition algorithm for solving power systems network equations with SSE and discuss advantages and disadvantages of this approach.

Seeing the Wood for the Trees: A Critical Evaluation of Methods to Estimate the Parameters of Stochastic Differential Equations

Maximum-likelihood estimates of the parameters of stochastic differential equations are consistent and asymptotically efficient, but unfortunately difficult to obtain if a closed-form expression for the transitional probability density function of the process is not available. As a result, a large number of competing estimation procedures have been proposed. This article provides a critical evaluation of the various estimation techniques. Special attention is given to the ease of implementation and comparative performance of the procedures when estimating the parameters of the Cox–Ingersoll–Ross and Ornstein–Uhlenbeck equations respectively.