A Petrov-Galerkin method for a singularly perturbed ordinary differential equation with non-smooth data

In this paper, a singularly perturbed ordinary differential equation with non-smooth data is considered. The numerical method is generated by means of a Petrov-Galerkin finite element method with the piecewise-exponential test function and the piecewise-linear trial function. At the discontinuous point of the coefficient, a special technique is used. The method is shown to be first-order accurate and singular perturbation parameter uniform convergence. Finally, numerical results are presented, which are in agreement with theoretical results.

Numerical approximation of a fractional-in-space diffusion equation (II) - with nonhomogeneous boundary conditions

In this paper, a space fractional di®usion equation (SFDE) with non- homogeneous boundary conditions on a bounded domain is considered. A new matrix transfer technique (MTT) for solving the SFDE is proposed. The method is based on a matrix representation of the fractional-in-space operator and the novelty of this approach is that a standard discretisation of the operator leads to a system of linear ODEs with the matrix raised to the same fractional power. Analytic solutions of the SFDE are derived. Finally, some numerical results are given to demonstrate that the MTT is a computationally e±cient and accurate method for solving SFDE.

Report : stochastic analysis of road asset data for maintenance cost prediction

Reliable budget/cost estimates for road maintenance and rehabilitation are subjected to uncertainties and variability in road asset condition and characteristics of road users. The CRC CI research project 2003-029-C ‘Maintenance Cost Prediction for Road’ developed a method for assessing variation and reliability in budget/cost estimates for road maintenance and rehabilitation. The method is based on probability-based reliable theory and statistical method. The next stage of the current project is to apply the developed method to predict maintenance/rehabilitation budgets/costs of large networks for strategic investment. The first task is to assess the variability of road data. This report presents initial results of the analysis in assessing the variability of road data. A case study of the analysis for dry non reactive soil is presented to demonstrate the concept in analysing the variability of road data for large road networks. In assessing the variability of road data, large road networks were categorised into categories with common characteristics according to soil and climatic conditions, pavement conditions, pavement types, surface types and annual average daily traffic. The probability distributions, statistical means, and standard deviation values of asset conditions and annual average daily traffic for each type were quantified. The probability distributions and the statistical information obtained in this analysis will be used to asset the variation and reliability in budget/cost estimates in later stage. Generally, we usually used mean values of asset data of each category as input values for investment analysis. The variability of asset data in each category is not taken into account. This analysis method demonstrated that it can be used for practical application taking into account the variability of road data in analysing large road networks for maintenance/rehabilitation investment analysis.

Increasing stochastic perturbations enhances acquisition and learning of complex sport movements

Traditionally, the aquisition of skills and sport movement has been characterised by numerous repetitions of presumed model movement pattern to be acquired by learners. This approach has been questioned by research identifying the presence of individualised movement patterns and the low probability of occurrence of two identical movements within and between individuals. In contrast, the differential learning approach claims advantage for incurring variability in the learning process by adding stochastic perturbations during practice. These ideas are exemplified by data from a high jump experiment which compared the effectiveness of classical and a differential training approach with pre-post test design. Results showed clear advantages for the group with additional stochastic perturbation during the aquisition phase in comparison to classically trained athletes. Analogies to similar phenomenological effects in the neurobiological literature are discussed.

Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term

In this paper, we consider a modified anomalous subdiffusion equation with a nonlinear source term for describing processes that become less anomalous as time progresses by the inclusion of a second fractional time derivative acting on the diffusion term. A new implicit difference method is constructed. The stability and convergence are discussed using a new energy method. Finally, some numerical examples are given. The numerical results demonstrate the effectiveness of theoretical analysis

Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term

In this paper, we consider a variable-order fractional advection-diffusion equation with a nonlinear source term on a finite domain. Explicit and implicit Euler approximations for the equation are proposed. Stability and convergence of the methods are discussed. Moreover, we also present a fractional method of lines, a matrix transfer technique, and an extrapolation method for the equation. Some numerical examples are given, and the results demonstrate the effectiveness of theoretical analysis.

Numerical schemes and multivariate extrapolation of a two-dimensional anomalous sub-diffusion equation

Anomalous dynamics in complex systems have gained much interest in recent years. In this paper, a two-dimensional anomalous subdiffusion equation (2D-ASDE) is considered. Two numerical methods for solving the 2D-ASDE are presented. Their stability, convergence and solvability are discussed. A new multivariate extrapolation is introduced to improve the accuracy. Finally, numerical examples are given to demonstrate the effectiveness of the schemes and confirm the theoretical analysis.

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.