Numerical algorithms for time-fractional subdiffusion equation with second-order accuracy

This article aims to fill in the gap of the second-order accurate schemes for the time-fractional subdiffusion equation with unconditional stability. Two fully discrete schemes are first proposed for the time-fractional subdiffusion equation with space discretized by finite element and time discretized by the fractional linear multistep methods. These two methods are unconditionally stable with maximum global convergence order of $O(\tau+h^{r+1})$ in the $L^2$ norm, where $\tau$ and $h$ are the step sizes in time and space, respectively, and $r$ is the degree of the piecewise polynomial space. The average convergence rates for the two methods in time are also investigated, which shows that the average convergence rates of the two methods are $O(\tau^{1.5}+h^{r+1})$. Furthermore, two improved algorithms are constrcted, they are also unconditionally stable and convergent of order $O(\tau^2+h^{r+1})$. Numerical examples are provided to verify the theoretical analysis. The comparisons between the present algorithms and the existing ones are included, which show that our numerical algorithms exhibit better performances than the known ones.

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.

A fast semi-implicit difference method for a nonlinear two-sided space-fractional diffusion equation with variable diffusivity coefficients

In this paper, we derive a new nonlinear two-sided space-fractional diffusion equation with variable coefficients from the fractional Fick’s law. A semi-implicit difference method (SIDM) for this equation is proposed. The stability and convergence of the SIDM are discussed. For the implementation, we develop a fast accurate iterative method for the SIDM by decomposing the dense coefficient matrix into a combination of Toeplitz-like matrices. This fast iterative method significantly reduces the storage requirement of O(n2)O(n2) and computational cost of O(n3)O(n3) down to n and O(nlogn)O(nlogn), where n is the number of grid points. The method retains the same accuracy as the underlying SIDM solved with Gaussian elimination. Finally, some numerical results are shown to verify the accuracy and efficiency of the new method.

A Crank--Nicolson ADI spectral method for a two-dimensional riesz space fractional nonlinear reaction-diffusion equation

In this paper, a new alternating direction implicit Galerkin--Legendre spectral method for the two-dimensional Riesz space fractional nonlinear reaction-diffusion equation is developed. The temporal component is discretized by the Crank--Nicolson method. The detailed implementation of the method is presented. The stability and convergence analysis is strictly proven, which shows that the derived method is stable and convergent of order $2$ in time. An optimal error estimate in space is also obtained by introducing a new orthogonal projector. The present method is extended to solve the fractional FitzHugh--Nagumo model. Numerical results are provided to verify the theoretical analysis.

Stability and convergence of a new finite volume method for a two-sided space-fractional diffusion equation

In this paper, we consider a two-sided space-fractional diffusion equation with variable coefficients on a finite domain. Firstly, based on the nodal basis functions, we present a new fractional finite volume method for the two-sided space-fractional diffusion equation and derive the implicit scheme and solve it in matrix form. Secondly, we prove the stability and convergence of the implicit fractional finite volume method and conclude that the method is unconditionally stable and convergent. Finally, some numerical examples are given to show the effectiveness of the new numerical method, and the results are in excellent agreement with theoretical analysis.

A Novel High Order Space-Time Spectral Method for the Time Fractional Fokker--Planck Equation

The fractional Fokker-Planck equation is an important physical model for simulating anomalous diffusions with external forces. Because of the non-local property of the fractional derivative an interesting problem is to explore high accuracy numerical methods for fractional differential equations. In this paper, a space-time spectral method is presented for the numerical solution of the time fractional Fokker-Planck initial-boundary value problem. The proposed method employs the Jacobi polynomials for the temporal discretization and Fourier-like basis functions for the spatial discretization. Due to the diagonalizable trait of the Fourier-like basis functions, this leads to a reduced representation of the inner product in the Galerkin analysis. We prove that the time fractional Fokker-Planck equation attains the same approximation order as the time fractional diffusion equation developed in [23] by using the present method. That indicates an exponential decay may be achieved if the exact solution is sufficiently smooth. Finally, some numerical results are given to demonstrate the high order accuracy and efficiency of the new numerical scheme. The results show that the errors of the numerical solutions obtained by the space-time spectral method decay exponentially.

Numeracy counts in the statistical reasoning equation

The relationship between mathematics and statistical reasoning frequently receives comment (Vere-Jones 1995, Moore 1997); however most of the research into the area tends to focus on maths anxiety. Gnaldi (Gnaldi 2003) showed that in a statistics course for psychologists, the statistical understanding of students at the end of the course depended on students’ basic numeracy, rather than the number or level of previous mathematics courses the student had undertaken. As part of a study into the development of statistical thinking at the interface between secondary and tertiary education, students enrolled in an introductory data analysis subject were assessed regarding their statistical reasoning ability, basic numeracy skills and attitudes towards statistics. This work reports on the relationships between these factors and in particular the importance of numeracy to statistical reasoning.

Investigating the impact of project definition clarity on project performance: Structural equation modeling study

Having a clear project definition is crucial for successful construction projects. It affects design quality, project communication between stakeholders and final project performance in terms of cost, schedule and quality. This study examines the relationship between project definition and final project performance through a structural equation model comprising 4 latent constructs and 6 path hypotheses using data from a questionnaire survey of 120 general contractors in the Malaysian construction industry. The results show that in the study population, all three items impact the project performance, but the link between design quality and project performance is indirect. Instead, the clarity of project definition affects project performance indirectly through design quality and project communication and design quality affects project performance indirectly through project communication. The primary contribution is to provide quantitative confirmation of the more general statements made in the literature from around the world and therefore adds to and consolidates existing knowledge. Practical implications derived from the finding are also proposed for various project stakeholders. Furthermore, as lack of the clarity of project definition is a very common occurrence in construction projects globally, these findings have important ramifications for all construction projects in expanding and clarifying existing knowledge on what is needed for the successful delivery of construction projects.

Development of height-weight based equation for assessment of body composition in Sri Lankan children

Objective To develop a height and weight based equation to estimate total body water (TBW) in Sri Lankan children. Methods Cross sectional descriptive study done involving 5–15 year old healthy children. Height and weight were measured. TBW was assessed using isotope dilution method (D2O) and fat free mass (FFM) calculated. Multiple regression analysis was used to develop prediction equation and validated using PRESS statistical technique. Height, weight and sex code (male=1; female=0) were used as prediction variables. Results This study provides height and weight equation for the prediction of TBW in Sri Lankan children. To the best of our knowledge there are no published height weight prediction equations validated on South Asian populations. Conclusion Results of this study need to be affirmed by more studies on other closely related populations by using multicomponent body composition.

Numerical analysis of a new space-time variable fractional order advection-dispersion equation

Many physical processes appear to exhibit fractional order behavior that may vary with time and/or space. The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable. In this paper, we consider a new space–time variable fractional order advection–dispersion equation on a finite domain. The equation is obtained from the standard advection–dispersion equation by replacing the first-order time derivative by Coimbra’s variable fractional derivative of order α(x)∈(0,1]α(x)∈(0,1], and the first-order and second-order space derivatives by the Riemann–Liouville derivatives of order γ(x,t)∈(0,1]γ(x,t)∈(0,1] and β(x,t)∈(1,2]β(x,t)∈(1,2], respectively. We propose an implicit Euler approximation for the equation and investigate the stability and convergence of the approximation. Finally, numerical examples are provided to show that the implicit Euler approximation is computationally efficient.

Spatially explicit structural equation modeling

Structural equation modeling (SEM) is a powerful statistical approach for the testing of networks of direct and indirect theoretical causal relationships in complex data sets with intercorrelated dependent and independent variables. SEM is commonly applied in ecology, but the spatial information commonly found in ecological data remains difficult to model in a SEM framework. Here we propose a simple method for spatially explicit SEM (SE-SEM) based on the analysis of variance/covariance matrices calculated across a range of lag distances. This method provides readily interpretable plots of the change in path coefficients across scale and can be implemented using any standard SEM software package. We demonstrate the application of this method using three studies examining the relationships between environmental factors, plant community structure, nitrogen fixation, and plant competition. By design, these data sets had a spatial component, but were previously analyzed using standard SEM models. Using these data sets, we demonstrate the application of SE-SEM to regularly spaced, irregularly spaced, and ad hoc spatial sampling designs and discuss the increased inferential capability of this approach compared with standard SEM. We provide an R package, sesem, to easily implement spatial structural equation modeling.

Assessment of body composition in Sri Lankan children: Validation of a bioelectrical impedance prediction equation

Objective: To develop bioelectrical impedance analysis (BIA) equations to predict total body water (TBW) and fat-free mass (FFM) of Sri Lankan children. Subjects/Methods: Data were collected from 5- to 15-year-old healthy children. They were randomly assigned to validation (M/F: 105/83) and cross-validation (M/F: 53/41) groups. Height, weight and BIA were measured. TBW was assessed using isotope dilution method (D2 O). Multiple regression analysis was used to develop preliminary equations and cross-validated on an independent group. Final prediction equation was constructed combining the two groups and validated by PRESS (prediction of sum of squares) statistics. Impedance index (height2/impedance; cm2/Ω), weight and sex code (male = 1; female = 0) were used as variables. Results: Independent variables of the final prediction equation for TBW were able to predict 86.3% of variance with root means-squared error (RMSE) of 2.1l. PRESS statistics was 2.1l with press residuals of 1.2l. Independent variables were able to predict 86.9% of variance of FFM with RMSE of 2.7 kg. PRESS statistics was 2.8 kg with press residuals of 1.4 kg. Bland Altman technique showed that the majority of the residuals were within mean bias±1.96 s.d. Conclusions: Results of this study provide BIA equation for the prediction of TBW and FFM in Sri Lankan children. To the best of our knowledge there are no published BIA prediction equations validated on South Asian populations. Results of this study need to be affirmed by more studies on other closely related populations by using multi-component body composition assessment.

A comment on the consistency of truncated nonlinear integral equation based theories of freezing

We report the results of two studies of aspects of the consistency of truncated nonlinear integral equation based theories of freezing: (i) We show that the self-consistent solutions to these nonlinear equations are unfortunately sensitive to the level of truncation. For the hard sphere system, if the Wertheim–Thiele representation of the pair direct correlation function is used, the inclusion of part but not all of the triplet direct correlation function contribution, as has been common, worsens the predictions considerably. We also show that the convergence of the solutions found, with respect to number of reciprocal lattice vectors kept in the Fourier expansion of the crystal singlet density, is slow. These conclusions imply great sensitivity to the quality of the pair direct correlation function employed in the theory. (ii) We show the direct correlation function based and the pair correlation function based theories of freezing can be cast into a form which requires solution of isomorphous nonlinear integral equations. However, in the pair correlation function theory the usual neglect of the influence of inhomogeneity of the density distribution on the pair correlation function is shown to be inconsistent to the lowest order in the change of density on freezing, and to lead to erroneous predictions. The Journal of Chemical Physics is copyrighted by The American Institute of Physics.

A comment on the consistency of truncated nonlinear integral equation based theories of freezing

We report the results of two studies of aspects of the consistency of truncated nonlinear integral equation based theories of freezing: (i) We show that the self-consistent solutions to these nonlinear equations are unfortunately sensitive to the level of truncation. For the hard sphere system, if the Wertheim–Thiele representation of the pair direct correlation function is used, the inclusion of part but not all of the triplet direct correlation function contribution, as has been common, worsens the predictions considerably. We also show that the convergence of the solutions found, with respect to number of reciprocal lattice vectors kept in the Fourier expansion of the crystal singlet density, is slow. These conclusions imply great sensitivity to the quality of the pair direct correlation function employed in the theory. (ii) We show the direct correlation function based and the pair correlation function based theories of freezing can be cast into a form which requires solution of isomorphous nonlinear integral equations. However, in the pair correlation function theory the usual neglect of the influence of inhomogeneity of the density distribution on the pair correlation function is shown to be inconsistent to the lowest order in the change of density on freezing, and to lead to erroneous predictions. The Journal of Chemical Physics is copyrighted by The American Institute of Physics.

Analysis of Paschen Curves for air, N2 and SF6 Using the Townsend Breakdown Equation

It has been shown that it is possible to extend the validity of the Townsend breakdown criterion for evaluating the breakdown voltages in the complete pd range in which Paschen curves are available. Evaluation of the breakdown voltages for air (pd=0.0133 to 1400 kPa · cm), N2(pd=0.0313 to 1400 kPa · cm) and SF6 (pd=0.3000 to 1200 kPa · cm) has been done and in most cases the computed values are accurate to ±3% of the measured values. The computations show that it is also possible to estimate the secondary ionization coefficient ¿ in the pd ranges mentioned above.