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.

Child characteristics associated with delay of gratification in children with Down syndrome

Aim: Children with Down syndrome have been identified as having difficulty delaying gratification when compared to mental age matched children who are developing typically. This study investigated the association between individual characteristics hypopthesized to be associated with ability to delay as well as the strategies children used in a waiting task. Method: Thirty-two children with Down syndrome and 50 typically developing children matched for mental age completed the tasks. Observations of their behaviour while waiting were video-recorded for later analysis. In addition, parents completed questionnaires with respect to their child’s personality and behaviour. Results: Children with Down syndrome were significantly less able to delay gratification than the comparison group. Different patterns of association were found for the two groups between the observational and questionnaire measures and delay time. Conclusions: Children with Down syndrome have greater difficulty delaying gratification than would be predicted on the basis of their mental age. The contributions to delay appear to differ from those for typically developing children and these differences need to be considered when planning interventions for developing this skill

Strategy use of children with Down syndrome in a delay of gratification situation

Background: The capacity to delay gratification has been shown to be a very important developmental task for children who are developing typically. There is evidence that children with Down syndrome have more difficulty with a delay of gratification task than typically developing children of the same mental age. This study focused on the strategies children with Down syndrome use while in a delay of gratification situation to ascertain if these contribute to the differences in delay times from those of typically developing children. Method: Thirty-two children with Down syndrome (15 females) and 50 typically developing children participated in the study. Children with Down syndrome had a mental age, as measured by the Stanford-Binet IV, between 36 and 66 months (M = 45.66). The typically developing children had a mean chronological age of 45.76 months. Children participated in a delay of gratification task where they were offered two or one small treats and asked which they preferred. They were then told that they could have the two treats if they waited for the researcher to return (an undisclosed time of 15 min). If they did not want to wait any longer they could call the researcher back but then they could have only one treat. Twenty-two of the children with Down syndrome and 43 of the typically developing children demonstrated understanding of the task and their data are included here. Sessions were videotaped for later analysis. Results: There were significant differences in the mean waiting times of the two groups. The mean of the waiting times for children with Down syndrome was 181.32 s (SD = 347.62) and was 440.21 s (SD = 377.59) for the typically developing children. Eighteen percent of the group with Down syndrome waited for the researcher to return in comparison to 35% of the typically developing group. Sixty-four percent of children with Down syndrome called the researcher back and the remainder (18%) violated. In the typically developing group 37% called the researcher back and 28% violated. The mean waiting time for the group of children with Down syndrome who called the researcher back was 24 s. Examination of strategy use in this group was therefore very limited. There appeared to be quite similar strategy use across the groups who waited the full 15 min. Conclusions: These results confirm the difficulty children with Down syndrome have in delaying gratification. Teaching strategies for waiting, using information drawn from the behaviours of children who are developing typically may be a useful undertaking. Examination of other contributors to delay ability (e.g., language skills) is also likely to be helpful in understanding the difficulties demonstrated in delaying gratification.

Analytical and numerical solutions of the space and time fractional bloch-torrey equation

Fractional order dynamics in physics, particularly when applied to diffusion, leads to an extension of the concept of Brown-ian motion through a generalization of the Gaussian probability function to what is termed anomalous diffusion. As MRI is applied with increasing temporal and spatial resolution, the spin dynamics are being examined more closely; such examinations extend our knowledge of biological materials through a detailed analysis of relaxation time distribution and water diffusion heterogeneity. Here the dynamic models become more complex as they attempt to correlate new data with a multiplicity of tissue compartments where processes are often anisotropic. Anomalous diffusion in the human brain using fractional order calculus has been investigated. Recently, a new diffusion model was proposed by solving the Bloch-Torrey equation using fractional order calculus with respect to time and space (see R.L. Magin et al., J. Magnetic Resonance, 190 (2008) 255-270). However effective numerical methods and supporting error analyses for the fractional Bloch-Torrey equation are still limited. In this paper, the space and time fractional Bloch-Torrey equation (ST-FBTE) is considered. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. Firstly, we derive an analytical solution for the ST-FBTE with initial and boundary conditions on a finite domain. Secondly, we propose an implicit numerical method (INM) for the ST-FBTE, and the stability and convergence of the INM are investigated. We prove that the implicit numerical method for the ST-FBTE is unconditionally stable and convergent. Finally, we present some numerical results that support our theoretical analysis.

Stability and convergence of implicit numerical methods for a class of fractional advection-dispersion models

In this paper, a class of fractional advection-dispersion models (FADM) is investigated. These models include five fractional advection-dispersion models: the immobile, mobile/immobile time FADM with a temporal fractional derivative 0 < γ < 1, the space FADM with skewness, both the time and space FADM and the time fractional advection-diffusion-wave model with damping with index 1 < γ < 2. They describe nonlocal dependence on either time or space, or both, to explain the development of anomalous dispersion. These equations can be used to simulate regional-scale anomalous dispersion with heavy tails, for example, the solute transport in watershed catchments and rivers. We propose computationally effective implicit numerical methods for these FADM. The stability and convergence of the implicit numerical methods are analyzed and compared systematically. Finally, some results are given to demonstrate the effectiveness of our theoretical analysis.

Establishing initial trust in autonomous delay tolerant networks without centralised PKI

A Delay Tolerant Network (DTN) is one where nodes can be highly mobile, with long message delay times forming dynamic and fragmented networks. Traditional centralised network security is difficult to implement in such a network, therefore distributed security solutions are more desirable in DTN implementations. Establishing effective trust in distributed systems with no centralised Public Key Infrastructure (PKI) such as the Pretty Good Privacy (PGP) scheme usually requires human intervention. Our aim is to build and compare different de- centralised trust systems for implementation in autonomous DTN systems. In this paper, we utilise a key distribution model based on the Web of Trust principle, and employ a simple leverage of common friends trust system to establish initial trust in autonomous DTN’s. We compare this system with two other methods of autonomously establishing initial trust by introducing a malicious node and measuring the distribution of malicious and fake keys. Our results show that the new trust system not only mitigates the distribution of fake malicious keys by 40% at the end of the simulation, but it also improved key distribution between nodes. This paper contributes a comparison of three de-centralised trust systems that can be employed in autonomous DTN systems.

Stability and convergence of a finite volume method for the space fractional advection-dispersion equation

We consider the space fractional advection–dispersion equation, which is obtained from the classical advection–diffusion equation by replacing the spatial derivatives with a generalised derivative of fractional order. We derive a finite volume method that utilises fractionally-shifted Grünwald formulae for the discretisation of the fractional derivative, to numerically solve the equation on a finite domain with homogeneous Dirichlet boundary conditions. We prove that the method is stable and convergent when coupled with an implicit timestepping strategy. Results of numerical experiments are presented that support the theoretical analysis.

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 banded preconditioner for the two-sided, nonlinear space-fractional diffusion equation

The method of lines is a standard method for advancing the solution of partial differential equations (PDEs) in time. In one sense, the method applies equally well to space-fractional PDEs as it does to integer-order PDEs. However, there is a significant challenge when solving space-fractional PDEs in this way, owing to the non-local nature of the fractional derivatives. Each equation in the resulting semi-discrete system involves contributions from every spatial node in the domain. This has important consequences for the efficiency of the numerical solver, especially when the system is large. First, the Jacobian matrix of the system is dense, and hence methods that avoid the need to form and factorise this matrix are preferred. Second, since the cost of evaluating the discrete equations is high, it is essential to minimise the number of evaluations required to advance the solution in time. In this paper, we show how an effective preconditioner is essential for improving the efficiency of the method of lines for solving a quite general two-sided, nonlinear space-fractional diffusion equation. A key contribution is to show, how to construct suitable banded approximations to the system Jacobian for preconditioning purposes that permit high orders and large stepsizes to be used in the temporal integration, without requiring dense matrices to be formed. The results of numerical experiments are presented that demonstrate the effectiveness of this approach.

A preconditioned Lanczos method for space-fractional reaction-diffusion equations on two-dimensional unstructured meshes

We consider a two-dimensional space-fractional reaction diffusion equation with a fractional Laplacian operator and homogeneous Neumann boundary conditions. The finite volume method is used with the matrix transfer technique of Ilić et al. (2006) to discretise in space, yielding a system of equations that requires the action of a matrix function to solve at each timestep. Rather than form this matrix function explicitly, we use Krylov subspace techniques to approximate the action of this matrix function. Specifically, we apply the Lanczos method, after a suitable transformation of the problem to recover symmetry. To improve the convergence of this method, we utilise a preconditioner that deflates the smallest eigenvalues from the spectrum. We demonstrate the efficiency of our approach for a fractional Fisher’s equation on the unit disk.

Numerical techniques for simulating a fractional mathematical model of epidermal wound healing

A number of mathematical models investigating certain aspects of the complicated process of wound healing are reported in the literature in recent years. However, effective numerical methods and supporting error analysis for the fractional equations which describe the process of wound healing are still limited. In this paper, we consider the numerical simulation of a fractional mathematical model of epidermal wound healing (FMM-EWH), which is based on the coupled advection-diffusion equations for cell and chemical concentration in a polar coordinate system. The space fractional derivatives are defined in the Left and Right Riemann-Liouville sense. Fractional orders in the advection and diffusion terms belong to the intervals (0, 1) or (1, 2], respectively. Some numerical techniques will be used. Firstly, the coupled advection-diffusion equations are decoupled to a single space fractional advection-diffusion equation in a polar coordinate system. Secondly, we propose a new implicit difference method for simulating this equation by using the equivalent of Riemann-Liouville and Grünwald-Letnikov fractional derivative definitions. Thirdly, its stability and convergence are discussed, respectively. Finally, some numerical results are given to demonstrate the theoretical analysis.

A novel numerical approximation for the space fractional advection-dispersion equation

In this paper, we consider a space fractional advection–dispersion equation. The equation is obtained from the standard advection–diffusion equation by replacing the first- and second-order space derivatives by the Riesz fractional derivatives of order β1 ∈ (0, 1) and β2 ∈ (1, 2], respectively. The fractional advection and dispersion terms are approximated by using two fractional centred difference schemes. A new weighted Riesz fractional finite-difference approximation scheme is proposed. When the weighting factor θ equals 12, a second-order accuracy scheme is obtained. The stability, consistency and convergence of the numerical approximation scheme are discussed. A numerical example is given to show that the numerical results are in good agreement with our theoretical analysis.