Numerical analysis of the time variable fractional order mobile-immobile advection-dispersion model

Many physical processes exhibit fractional order behavior that varies with time 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 the time variable fractional order mobile-immobile advection-dispersion model. Numerical methods and analyses of stability and convergence for the fractional partial differential equations are quite limited and difficult to derive. This motivates us to develop efficient numerical methods as well as stability and convergence of the implicit numerical methods for the fractional order mobile immobile advection-dispersion model. In the paper, we use the Coimbra variable time fractional derivative which is more efficient from the numerical standpoint and is preferable for modeling dynamical systems. An implicit Euler approximation for the equation is proposed and then the stability of the approximation are investigated. As for the convergence of the numerical scheme we only consider a special case, i.e. the time fractional derivative is independent of time variable t. The case where the time fractional derivative depends both the time variable t and the space variable x will be considered in the future work. Finally, numerical examples are provided to show that the implicit Euler approximation is computationally efficient.

Numerical techniques for the variable order time fractional diffusion equation

In this paper we consider the variable order time fractional diffusion equation. We adopt the Coimbra variable order (VO) time fractional operator, which defines a consistent method for VO differentiation of physical variables. The Coimbra variable order fractional operator also can be viewed as a Caputo-type definition. Although this definition is the most appropriate definition having fundamental characteristics that are desirable for physical modeling, numerical methods for fractional partial differential equations using this definition have not yet appeared in the literature. Here an approximate scheme is first proposed. The stability, convergence and solvability of this numerical scheme are discussed via the technique of Fourier analysis. Numerical examples are provided to show that the numerical method is computationally efficient. Crown Copyright © 2012 Published by Elsevier Inc. All rights reserved.

Numerical methods for solving a two-dimensional variable-order anomalous subdiffusion equation

Anomalous subdiffusion equations have in recent years received much attention. In this paper, we consider a two-dimensional variable-order anomalous subdiffusion equation. Two numerical methods (the implicit and explicit methods) are developed to solve the equation. Their stability, convergence and solvability are investigated by the Fourier method. Moreover, the effectiveness of our theoretical analysis is demonstrated by some numerical examples. © 2011 American Mathematical Society.

The Roots of Danger: Violent Crime in Global Perspective

Prentice Hall’s Masters Series in Criminology brings the work of true masters to life for a new audience of readers, presenting brief and accessible introductions to crime and criminology topics from some of the leading scholars in criminology today. All authors in the series are true academic pioneers, and each book in the series pulls from the authors’ decades of research and writing in their fields. The first and only series of its kind, Prentice Hall’s Masters Series in Criminology introduces readers to the scholars and issues that are at the core of modern criminology.

Creating Institutions: Linking the Local and the Global in the Travel of Crime Policies

Institutions represent the ‘technologies of the social.’ They are increasingly modelled and transported to other cultures and societies, and criminal justice institutions—traditional, parochial, and local as they are—are no exception to this. Problems of crime and insecurity have engendered the travelling of institutions from the centre to the periphery and vice versa. This paper will explore the problems which arise from travelling and modelling, and from the transport and creation of institutions in the area of criminal justice. An important feature in the travel of criminal justice institutions is the use of ‘local knowledge’ and its role in this process.