An implicit RBF meshless approach for time fractional diffusion equations

This paper aims to develop an implicit meshless approach based on the radial basis function (RBF) for numerical simulation of time fractional diffusion equations. The meshless RBF interpolation is firstly briefed. The discrete equations for two-dimensional time fractional diffusion equation (FDE) are obtained by using the meshless RBF shape functions and the strong-forms of the time FDE. The stability and convergence of this meshless approach are discussed and theoretically proven. Numerical examples with different problem domains and different nodal distributions are studied to validate and investigate accuracy and efficiency of the newly developed meshless approach. It has proven that the present meshless formulation is very effective for modeling and simulation of fractional differential equations.

Novel numerical methods for time-space fractional reaction diffusion equations in two dimensions

We consider time-space fractional reaction diffusion equations in two dimensions. This equation is obtained from the standard reaction diffusion equation by replacing the first order time derivative with the Caputo fractional derivative, and the second order space derivatives with the fractional Laplacian. Using the matrix transfer technique proposed by Ilic, Liu, Turner and Anh [Fract. Calc. Appl. Anal., 9:333--349, 2006] and the numerical solution strategy used by Yang, Turner, Liu, and Ilic [SIAM J. Scientific Computing, 33:1159--1180, 2011], the solution of the time-space fractional reaction diffusion equations in two dimensions can be written in terms of a matrix function vector product $f(A)b$ at each time step, where $A$ is an approximate matrix representation of the standard Laplacian. We use the finite volume method over unstructured triangular meshes to generate the matrix $A$, which is therefore non-symmetric. However, the standard Lanczos method for approximating $f(A)b$ requires that $A$ is symmetric. We propose a simple and novel transformation in which the standard Lanczos method is still applicable to find $f(A)b$, despite the loss of symmetry. Numerical results are presented to verify the accuracy and efficiency of our newly proposed numerical solution strategy.

An analytical solution for diffusion and nonlinear uptake of oxygen in a spherical cell

We develop a new analytical solution for a reactive transport model that describes the steady-state distribution of oxygen subject to diffusive transport and nonlinear uptake in a sphere. This model was originally reported by Lin (Journal of Theoretical Biology, 1976 v60, pp449–457) to represent the distribution of oxygen inside a cell and has since been studied extensively by both the numerical analysis and formal analysis communities. Here we extend these previous studies by deriving an analytical solution to a generalized reaction-diffusion equation that encompasses Lin’s model as a particular case. We evaluate the solution for the parameter combinations presented by Lin and show that the new solutions are identical to a grid-independent numerical approximation.

An advanced meshless method for time fractional diffusion equation

Recently, because of the new developments in sustainable engineering and renewable energy, which are usually governed by a series of fractional partial differential equations (FPDEs), the numerical modelling and simulation for fractional calculus are attracting more and more attention from researchers. The current dominant numerical method for modeling FPDE is Finite Difference Method (FDM), which is based on a pre-defined grid leading to inherited issues or shortcomings including difficulty in simulation of problems with the complex problem domain and in using irregularly distributed nodes. Because of its distinguished advantages, the meshless method has good potential in simulation of FPDEs. This paper aims to develop an implicit meshless collocation technique for FPDE. The discrete system of FPDEs is obtained by using the meshless shape functions and the meshless collocation formulation. The stability and convergence of this meshless approach are investigated theoretically and numerically. The numerical examples with regular and irregular 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 fractional partial differential equations.

Collision warning dissemination in vehicles strings : an empirical measurement

Inter-Vehicular Communications (IVC) are considered a promising technological approach for enhancing transportation safety and improving highway efficiency. Previous theoretical work has demonstrated the benefits of IVC in vehicles strings. Simulations of partially IVC-equipped vehicles strings showed that only a small equipment ratio is sufficient to drastically reduce the number of head on collisions. However, these results are based on the assumptions that IVC exhibit lossless and instantaneous messages transmission. This paper presents the research design of an empirical measurement of a vehicles string, with the goal of highlighting the constraints introduced by the actual characteristics of communication devices. A warning message diffusion system based on IEEE 802.11 wireless technology was developed for an emergency breaking scenario. Preliminary results are presented as well, showing the latencies introduced by using 802.11a and discussing early findings and experimental limitations

The Biodex Isokinetic Dynamometer for knee strength assessment in children : advantages and limitations

Objective: To critically appraise the Biodex System 4 isokinetic dynamometer for strength assessment of children. Methods: Appraisal was based on experiences from two independent laboratories involving testing of 213 children. Issues were recorded and the manufacturer was consulted regarding appropriate solutions. Results: The dynamometer had insufficient height adjustment for alignment of the knee for some children, requiring the construction of padding to better fit the child within the dynamometer. Potential for entrapment of the non-testing leg was evident in the passive and eccentric modes and a leg bracket restraint was constructed. Automated gravity correction did not operate when protocols were linked or data was exported to an external device. Conclusions: Limitations were noted, some of which were applicable to knee strength testing in general and others which were specific to use with children. However, most of these obstacles could be overcome, making the Biodex System 4 suitable for assessment of knee strength in children.