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.

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.

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.

Regulating Prostitution: Different Problems, Different Solutions, Same Old Story

This article looks at three main models of intervention that have informed recent policy and practice with people involved in the sex trade. It reveals the inherent contradictions within attempts to both help and punish workers in the existing prostitution strategy.

Unexpected issues and solutions: Emerge from environmental manipulation strategy

Environmental manipulation removes students from their everyday worlds to unfamiliar worlds, to facil- itate learning. This article reports that this strategy was effective when applied in a university design unit, using the tactic of immersion in the Second Life online virtual environment. The objective was for teams of stu- dents each to design a series of modules for an orbiting space station using supplied data. The changed and futuristic environment led the students to an important but previously unconsidered design decision which they were able to address in novel ways because of, rather than in spite of, the Second Life immersion.

Fast tessellated solid navigation in GEANT4

Navigation through tessellated solids in GEANT4 can degrade computational performance, especially if the tessellated solid is large and is comprised of many facets. Redefining a tessellated solid as a mesh of tetrahedra is common in other computational techniques such as finite element analysis as computations need only consider local tetrahedrons rather than the tessellated solid as a whole. Here within we describe a technique that allows for automatic tetrahedral meshing of tessellated solids in GEANT4 and the subsequent loading of these meshes as assembly volumes; loading nested tessellated solids and tetrahedral meshes is also examined. As the technique makes the geometry suitable for automatic optimisation using smartvoxels, navigation through a simple tessellated volume has been found to be more than two orders of magnitude faster than that through the equivalent tessellated solid. Speed increases of more than two orders of magnitude were also observed for a more complex tessellated solid with voids and concavities. The technique was benchmarked for geometry load time, simulation run time and memory usage. Source code enabling the described functionality in GEANT4 has been made freely available on the Internet.

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.

Valorization of solid tire wastes available in Bangladesh by thermal treatment

In this study available solid tire wastes in Bangladesh were characterized through proximate and ultimate analyses, gross calorific values and thermogravimetric analysis to investigate their suitability as feedstock for thermal recycling by pyrolysis technology. A new approach in heating system, fixedbed fire-tube heating pyrolysis reactor has been designed and fabricated for the recovery of liquid hydrocarbons from solid tire wastes. The tire wastes were pyrolysed in the internally heated fixed-bed fire-tube heating reactor and maximum liquid yield of 46-55 wt% of solid tire waste was obtained at a temperature of 475 oC, feed size 4 cm3, with a residence time of 5 s under N2 atmosphere. The liquid products were characterized by physical properties, elemental analysis, FT-IR, 1H-NMR, GC MS techniques and distillation. The results show that the liquid products are comparable to petroleum fuels whereas fractional distillations and desulphurization are essential to be used as alternative for diesel engine fuels.

Literacy in Early Childhood and Primary Education : Issues, Challenges and Solutions

Literacy in Early Childhood and Primary Education provides a comprehensive introduction to literacy teaching and learning. The book explores the continuum of literacy learning and children’s transitions from early childhood settings to junior primary classrooms, and then to senior primary and beyond. Reader-friendly and accessible, this book equips pre-service teachers with the theoretical underpinnings and practical strategies and skills needed to teach literacy. It places the ‘reading wars’ firmly in the past as it examines contemporary research and practices. The book covers important topics such as literacy acquisition, family literacies and multiliteracies, foundation skills for literacy learning, reading difficulties, assessment, and supporting diverse literacy learners in early childhood and primary classrooms. It also addresses some of the challenges that teachers may face in the classroom and provides solutions to these. Each chapter includes learning objectives, reflective questions and definitions to key terms to engage and assist readers. Further resources are also available at www.cambridge.edu.au/academic/literacy. Written by an expert author team and featuring real-world examples from literacy teachers and learners. Literacy in Early Childhood and Primary Education will help pre-service teachers feel confident teaching literacy to diverse age groups and abilities.

An assessment tool for selection of appropriate daylighting solutions for buildings in tropical and subtropical regions : Validation using radiance simulation

A new decision-making tool that will assist designers in the selection of appropriate daylighting solutions for buildings in tropical locations has been previously proposed by the authors. Through an evaluation matrix that prioritizes the parameters that best respond to the needs of tropical climates (e.g. reducing solar gain and protection from glare) the tool determines the most appropriate devices for specific climate and building inputs. The tool is effective in demonstrating the broad benefits and limitations of the different daylight strategies for buildings in the tropics. However for thorough analysis and calibration of the tool, validation is necessary. This paper presents a first step in the validation process. RADIANCE simulations were conducted to compare simulation performance with the performance predicted by the tool. To this end, an office building case study in subtropical Brisbane, Australia, and five different daylighting devices including openings, light guiding systems and light transport systems were simulated. Illuminance, light uniformity, daylight penetration and glare analysis were assessed for each device. The results indicate the tool can appropriately rank and recommend daylighting strategies based on specific building inputs for tropical and subtropical regions, making it a useful resource for designers.

A coupled SPH-DEM model for fluid and solid mechanics of apple parenchyma cells during drying

A coupled SPH-DEM based two-dimensional (2-D) micro-scale single cell model is developed to predict basic cell-level shrinkage effects of apple parenchyma cells during air drying. In this newly developed drying model, Smoothed Particle Hydrodynamics (SPH) is used to model the low Reynolds Number fluid motions of the cell protoplasm, and a Discrete Element Method (DEM) is employed to simulate the polymer-like cell wall. Simulations results reasonably agree with published experimental drying results on cellular shrinkage properties such as cellular area, diameter and perimeter. These preliminary results indicate that the model is effective for the modelling and simulation of apple parenchyma cells during air drying.

The El Chichon stratospheric cloud : solid particulates and settling rates

Sampling of the El Chichón stratospheric cloud in early May and in late July, 1982, showed that a significant proportion of the cloud consisted of solid particles between 2 μm and 40 μm size. In addition, many particles may have been part of larger aggregates or clusters that ranged in size from < 10 μm to > 50 μm. The majority of individual grains were angular aluminosilicate glass shards with various amounts of smaller, adhering particles. Surface features on individual grains include sulfuric acid droplets and larger (0.5 μm to 1 μm) sulfate gel droplets with various amounts of Na, Mg, Ca and Fe. The sulfate gels probably formed by the interaction of sulfur-rich gases and solid particles within the cloud soon after eruption. Ca-sulfate laths may have formed by condensation within the plume during eruption, or alternatively, at a later stage by the reaction of sulfuric acid aerosols with ash fragments within the stratospheric cloud. A Wilson-Huang formulation for the settling rate of individual particles qualitatively agrees with the observed particle-size distribution for a period at least four months after injection of material into the stratosphere. This result emphasizes the importance of particle shape in controlling the settling rate of volcanic ash from the stratosphere.