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.

The importance of ecological scale for wildlife conservation in naturally fragmented environments: A case study of the Brush-Tailed Rock-Wallaby (Petrogale Penicillata)

Determining the ecologically relevant spatial scales for predicting species occurrences is an important concept when determining species–environment relationships. Therefore species distribution modelling should consider all ecologically relevant spatial scales. While several recent studies have addressed this problem in artificially fragmented landscapes, few studies have researched relevant ecological scales for organisms that also live in naturally fragmented landscapes. This situation is exemplified by the Australian rock-wallabies’ preference for rugged terrain and we addressed the issue of scale using the threatened brush-tailed rock-wallaby (Petrogale penicillata) in eastern Australia. We surveyed for brush-tailed rock-wallabies at 200 sites in southeast Queensland, collecting potentially influential site level and landscape level variables. We applied classification trees at either scale to capture a hierarchy of relationships between the explanatory variables and brush-tailed rock-wallaby presence/absence. Habitat complexity at the site level and geology at the landscape level were the best predictors of where we observed brush-tailed rock-wallabies. Our study showed that the distribution of the species is affected by both site scale and landscape scale factors, reinforcing the need for a multi-scale approach to understanding the relationship between a species and its environment. We demonstrate that careful design of data collection, using coarse scale spatial datasets and finer scale field data, can provide useful information for identifying the ecologically relevant scales for studying species–environment relationships. Our study highlights the need to determine patterns of environmental influence at multiple scales to conserve specialist species such as the brush-tailed rock-wallaby in naturally fragmented landscapes.

Numerical simulation of the Riesz fractional diffusion equation with a nonlinear source term

In this paper, A Riesz fractional diffusion equation with a nonlinear source term (RFDE-NST) is considered. This equation is commonly used to model the growth and spreading of biological species. According to the equivalent of the Riemann-Liouville(R-L) and Gr¨unwald-Letnikov(GL) fractional derivative definitions, an implicit difference approximation (IFDA) for the RFDE-NST is derived. We prove the IFDA is unconditionally stable and convergent. In order to evaluate the efficiency of the IFDA, a comparison with a fractional method of lines (FMOL) is used. Finally, two numerical examples are presented to show that the numerical results are in good agreement with our theoretical analysis.

Fundamental solution and discrete random walk model for time-space fractional diffusion equation

In this paper, we consider a time-space fractional diffusion equation of distributed order (TSFDEDO). The TSFDEDO is obtained from the standard advection-dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α∈(0,1], the first-order and second-order space derivatives by the Riesz fractional derivatives of orders β 1∈(0,1) and β 2∈(1,2], respectively. We derive the fundamental solution for the TSFDEDO with an initial condition (TSFDEDO-IC). The fundamental solution can be interpreted as a spatial probability density function evolving in time. We also investigate a discrete random walk model based on an explicit finite difference approximation for the TSFDEDO-IC.

Bacterial proteases from the intracellular vacuole niche ; protease conservation and adaptation for pathogenic advantage

Proteases with important roles for bacterial pathogens which specifically reside within intracellular vacuoles are frequently homologous to those which have important virulence functions for other bacteria. Research has identified that some of these conserved proteases have evolved specialised functions for intracellular vacuole residing bacteria. Unique proteases with pathogenic functions have also been described from Chlamydia, Mycobacteria, and Legionella. These findings suggest that there are further novel functions for proteases from these bacteria which remain to be described. This review summarises recent findings of novel protease functions from the intracellular human pathogenic bacteria which reside exclusively in vacuoles.