Analytical solutions of the multi-term modified power law wave equations in a finite domain

Fractional partial differential equations with more than one fractional derivative term in time, such as the Szabo wave equation, or the power law wave equation, describe important physical phenomena. However, studies of these multi-term time-space or time fractional wave equations are still under development. In this paper, multi-term modified power law wave equations in a finite domain are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals (1, 2], [2, 3), [2, 4) or (0, n) (n > 2), respectively. Analytical solutions of the multi-term modified power law wave equations are derived. These new techniques are based on Luchko’s Theorem, a spectral representation of the Laplacian operator, a method of separating variables and fractional derivative techniques. Then these general methods are applied to the special cases of the Szabo wave equation and the power law wave equation. These methods and techniques can also be extended to other kinds of the multi term time-space fractional models including fractional Laplacian.

Numerical methods for solving the multi-term time-fractional wave-diffusion equations

In this paper, the multi-term time-fractional wave diffusion equations are considered. The multiterm time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and techniques can also be extended to other kinds of the multi-term fractional time-space models with fractional Laplacian.

A finite volume method for solving the two-sided time-space fractional advection-dispersion equation

The field of fractional differential equations provides a means for modelling transport processes within complex media which are governed by anomalous transport. Indeed, the application to anomalous transport has been a significant driving force behind the rapid growth and expansion of the literature in the field of fractional calculus. In this paper, we present a finite volume method to solve the time-space two-sided fractional advection dispersion equation on a one-dimensional domain. Such an equation allows modelling different flow regime impacts from either side. The finite volume formulation provides a natural way to handle fractional advection-dispersion equations written in conservative form. The novel spatial discretisation employs fractionally-shifted Gr¨unwald formulas to discretise the Riemann-Liouville fractional derivatives at control volume faces in terms of function values at the nodes, while the L1-algorithm is used to discretise the Caputo time fractional derivative. Results of numerical experiments are presented to demonstrate the effectiveness of the approach.

An implicit RBF meshless method for a fractal mobile/immobile transport model

Fractional differential equation is used to describe a fractal model of mobile/immobile transport with a power law memory function. This equation is the limiting equation that governs continuous time random walks with heavy tailed random waiting times. In this paper, we firstly propose a finite difference method to discretize the time variable and obtain a semi-discrete scheme. Then we discuss its stability and convergence. Secondly we consider a meshless method based on radial basis functions (RBF) to discretize the space variable. By contrast to conventional FDM and FEM, the meshless method is demonstrated to have distinct advantages: calculations can be performed independent of a mesh, it is more accurate and it can be used to solve complex problems. Finally the convergence order is verified from a numerical example is presented to describe the fractal model of mobile/immobile transport process with different problem domains. The numerical results indicate that the present meshless approach is very effective for modeling and simulating of fractional differential equations, and it has good potential in development of a robust simulation tool for problems in engineering and science that are governed by various types of fractional differential equations.

A second-order accurate numerical approximation for the Riesz space fractional advection-dispersion equation

In this paper, we consider a space Riesz fractional advection-dispersion equation. The equation is obtained from the standard advection-diffusion equation by replacing the ¯rst-order and second-order space derivatives by the Riesz fractional derivatives of order β 1 Є (0; 1) and β2 Є(1; 2], respectively. Riesz fractional advection and dispersion terms are approximated by using two fractional centered difference schemes, respectively. A new weighted Riesz fractional ¯nite difference approximation scheme is proposed. When the weighting factor Ѳ = 1/2, a second- order accurate numerical approximation scheme for the Riesz fractional advection-dispersion equation is obtained. 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.

Numerical simulation of a fractional mathematical model for 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 numerical simulation of fractional model 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 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 the Riemann-Liouville and Gr¨unwald-Letnikov fractional derivative definitions. Thirdly, its stability and convergence are discussed, respectively. Finally, some numerical results are given to demonstrate the theoretical analysis.

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.

Analytical solutions for the two- and three-dimensional time-fractional telegraph equations by the method of separating variables

In this paper, a method of separating variables is effectively implemented for solving a time-fractional telegraph equation (TFTE) in two and three dimensions. We discuss and derive the analytical solution of the TFTE in two and three dimensions with nonhomogeneous Dirichlet boundary condition. This method can be extended to other kinds of the boundary conditions.

Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain

Generalized fractional partial differential equations have now found wide application for describing important physical phenomena, such as subdiffusive and superdiffusive processes. However, studies of generalized multi-term time and space fractional partial differential equations are still under development. In this paper, the multi-term time-space Caputo-Riesz fractional advection diffusion equations (MT-TSCR-FADE) with Dirichlet nonhomogeneous boundary conditions are considered. The multi-term time-fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0, 1], [1, 2] and [0, 2], respectively. These are called respectively the multi-term time-fractional diffusion terms, the multi-term time-fractional wave terms and the multi-term time-fractional mixed diffusion-wave terms. The space fractional derivatives are defined as Riesz fractional derivatives. Analytical solutions of three types of the MT-TSCR-FADE are derived with Dirichlet boundary conditions. By using Luchko's Theorem (Acta Math. Vietnam., 1999), we proposed some new techniques, such as a spectral representation of the fractional Laplacian operator and the equivalent relationship between fractional Laplacian operator and Riesz fractional derivative, that enabled the derivation of the analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations. © 2012.

Numerical methods and analysis for a class of fractional advection-dispersion models

In this paper, a class of fractional advection–dispersion models (FADMs) is considered. These models include five fractional advection–dispersion models, i.e., the time FADM, the mobile/immobile time FADM with a time Caputo fractional derivative 0 < γ < 1, the space FADM with two sides Riemann–Liouville derivatives, the time–space FADM and the time fractional advection–diffusion-wave model with damping with index 1 < γ < 2. These equations can be used to simulate the regional-scale anomalous dispersion with heavy tails. We propose computationally effective implicit numerical methods for these FADMs. The stability and convergence of the implicit numerical methods are analysed and compared systematically. Finally, some results are given to demonstrate the effectiveness of theoretical analysis.

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.

Dynamic compression improves biosynthesis of human zonal chondrocytes from osteoarthritis patients

Objective: We hypothesize that chondrocytes from distinct zones of articular cartilage respond differently to compressive loading, and that zonal chondrocytes from osteoarthritis (OA) patients can benefit from optimized compressive stimulation. Therefore, we aimed to determine the transcriptional response of superficial (S) and middle/deep (MD) zone chondrocytes to varying dynamic compressive strain and loading duration. To confirm effects of compressive stimulation on overall matrix production, we subjected zonal chondrocytes to compression for 2 weeks. Design: Human S and MD chondrocytes from osteoarthritic joints were encapsulated in 2% alginate, pre-cultured, and subjected to compression with varying dynamic strain (5, 15, 50% at 1 Hz) and loading duration (1, 3, 12 h). Temporal changes in cartilage-specific, zonal, and dedifferentiation genes following compression were evaluated using quantitative real-time reverse transcriptase polymerase chain reaction (qRT-PCR). The benefits of long-term compression (50% strain, 3 h/day, for 2 weeks) were assessed by measuring construct glycosaminoglycan (GAG) content and compressive moduli, as well as immunostaining. Results: Compressive stimulation significantly induced aggrecan (ACAN), COL2A1, COL1A1, proteoglycan 4 (PRG4), and COL10A1 gene expression after 2 h of unloading, in a zone-dependent manner (P < 0.05). ACAN and PRG4 mRNA levels depended on strain and load duration, with 50% and 3 h loading resulting in highest levels (P < 0.05). Long-term compression increased collagen type II and ACAN immunostaining and total GAG (P < 0.05), but only S constructs showed more PRG4 stain, retained more GAG (P < 0.01), and developed higher compressive moduli than non-loaded controls. Conclusions: The biosynthetic activity of zonal chondrocytes from osteoarthritis joints can be enhanced with selected compression regimes, indicating the potential for cartilage tissue engineering applications. © 2012 Osteoarthritis Research Society International.

District Court of Queensland Act 1968 s 68 : jurisdiction in pre-proceedings applications under Personal Injuries Proceedings Act 2002

In Woolworths Ltd v Graham [2007] QDC 301 Searles DCJ struck out a pre-proceedings application under the Personal Injuries Proceedings Act 2002 (Qld)on the basis that the material before the Court was not sufficient to attract the jurisdiction of the District Court.The decision serves more broadly as a reminder that the District Court is an inferior court of defined and limited jurisdiction and that any proceedings brought in it must be demonstrably within the jurisdiction conferred on that court by legislation.

Democracy, Crime and Justice. Annals of the American Academy of Political and Social Sciences

As a growing number of nations embark on a path to democracy, criminologists have become increasingly interested and engaged in the challenges, concerns, and questions connecting democracy with both crime and criminal justice. Rising levels of violence and street crime, white collar crime and corruption both in countries where democracy is securely in place and where it is struggling, have fuelled a deepening skepticism as to the capacity of democracy to deliver on its promise of security and justice for all citizens. What role does crime and criminal justice play in the future of democracy and for democratic political development on a global level? The editors of this special volume of The Annals realized the importance of collecting research from a broad spectrum of countries and covering a range of problems that affect citizens, politicians, and criminal justice officials. The articles here represent a solid balance between mature democracies like the U.S. and U.K. as well as emerging democracies around the globe – specifically in Latin America, Africa and Eastern Europe. They are based on large and small cross-national samples, regional comparisons, and case studies. Each contribution addresses a seminal question for the future of democratic political development across the globe. What is the role of criminal justice in the process of building democracy and instilling confidence in its institutions? Is there a role for unions in democratizing police forces? What is the impact of widespread disenfranchisement of felons on democratic citizenship and the life of democratic institutions? Under what circumstances do mature democracies adopt punitive sentencing regimes? Addressing sensitive topics such as relations between police and the Muslim communities of Western Europe in the wake of terrorist attacks, this volume also sheds light on the effects of terrorism on mature democracies under increasing pressure to provide security for their citizens. By taking a broad vantage point, this collection of research delves into complex topics such as the relationship between the process of democratization and violent crime waves; the impact of rising crime rates on newly established as well as secure democracies; how crime may endanger the transition to democracy; and how existing practices of criminal justice in mature democracies affect their core values and institutions. The collection of these insightful articles not only begins to fill a gap in criminological research but also addresses issues of critical interest to political scientists as well as other social and behavioral scientists and scholars. Taking a fresh approach to the intersection of crime, criminal justice, and democracy, this volume of The Annals is a must-read for criminologists and political scientists and provides a solid foundation for further interdisciplinary research.

A proposed validation framework for expert elicited Bayesian Networks

The popularity of Bayesian Network modelling of complex domains using expert elicitation has raised questions of how one might validate such a model given that no objective dataset exists for the model. Past attempts at delineating a set of tests for establishing confidence in an entirely expert-elicited model have focused on single types of validity stemming from individual sources of uncertainty within the model. This paper seeks to extend the frameworks proposed by earlier researchers by drawing upon other disciplines where measuring latent variables is also an issue. We demonstrate that even in cases where no data exist at all there is a broad range of validity tests that can be used to establish confidence in the validity of a Bayesian Belief Network.