Computationally efficient methods for solving time-variable-order time-space fractional reaction-diffusion equation

Fractional differential equations are becoming more widely accepted as a powerful tool in modelling anomalous diffusion, which is exhibited by various materials and processes. Recently, researchers have suggested that rather than using constant order fractional operators, some processes are more accurately modelled using fractional orders that vary with time and/or space. In this paper we develop computationally efficient techniques for solving time-variable-order time-space fractional reaction-diffusion equations (tsfrde) using the finite difference scheme. We adopt the Coimbra variable order time fractional operator and variable order fractional Laplacian operator in space where both orders are functions of time. Because the fractional operator is nonlocal, it is challenging to efficiently deal with its long range dependence when using classical numerical techniques to solve such equations. The novelty of our method is that the numerical solution of the time-variable-order tsfrde is written in terms of a matrix function vector product at each time step. This product is approximated efficiently by the Lanczos method, which is a powerful iterative technique for approximating the action of a matrix function by projecting onto a Krylov subspace. Furthermore an adaptive preconditioner is constructed that dramatically reduces the size of the required Krylov subspaces and hence the overall computational cost. Numerical examples, including the variable-order fractional Fisher equation, are presented to demonstrate the accuracy and efficiency of the approach.

A gust-attenuation controller for fixed-wing UAVs during collision avoidance course

This paper presents a nonlinear gust-attenuation controller to stabilize velocities, attitudes and angular rates of a fixed-wing unmanned aerial vehicle (UAV) in the presence of wind gusts. The proposed controller aims to achieve a steady-state flight condition such that the host UAV can avoid airspace collision with other UAVs during the cruise flight. Based on the typical UAV model capturing flight aerodynamics, a nonlinear Hinf controller is developed with rapid response property in consideration of actuator constraints. Simulations are conducted for the Shadow UAV to verify performance of the proposed controller. Comparative studies with the proportional-integral derivative (PID) controllers demonstrate that the proposed controller exhibits great performance improvement in a gusty environment, making it suitable for integration into the design of flight control systems for cruise flight with safety guarantees.

Dynamic covalent chemistry on surfaces employing highly reactive cyclopentadienyl moieties

Silicon substrates coated with a bromide-terminated silane are transformed into highly reactive, cyclopentadiene covered analogues. These surfaces undergo rapid cycloaddition reactions with various dienophile-capped polymers. Mild heating of the substrates causes the retro-Diels-Alder reaction to occur, thus reforming the reactive cyclopentadiene surface, generating an efficiently switchable surface.

(Ultra)fast catalyst-free macromolecular conjugation in aqueous environment at ambient temperature

Tailor-made water-soluble macromolecules, including a glycopolymer, obtained by living/controlled RAFT-mediated polymerization are demonstrated to react in water with diene-functionalized poly(ethylene glycol)s without pre- or post-functionalization steps or the need for a catalyst at ambient temperature. As previously observed in organic solvents, hetero-Diels-Alder (HDA) conjugations reached quantitative conversion within minutes when cyclopentadienyl moieties were involved. However, while catalysts and elevated temperatures were previously necessary for open-chain diene conjugation, additive-free HDA cycloadditions occur in water within a few hours at ambient temperature. Experimental evidence for efficient conjugations is provided via unambiguous ESI-MS, UV/vis, NMR, and SEC data.

Alternating direction implicit numerical method for the space and time fractional Bloch-Torrey equation in 3-D

Recently, some authors have considered a new diffusion model–space and time fractional Bloch-Torrey equation (ST-FBTE). Magin et al. (2008) have derived analytical solutions with fractional order dynamics in space (i.e., _ = 1, β an arbitrary real number, 1 < β ≤ 2) and time (i.e., 0 < α < 1, and β = 2), respectively. Yu et al. (2011) have derived an analytical solution and an effective implicit numerical method for solving ST-FBTEs, and also discussed the stability and convergence of the implicit numerical method. However, due to the computational overheads necessary to perform the simulations for nuclear magnetic resonance (NMR) in three dimensions, they present a study based on a two-dimensional example to confirm their theoretical analysis. Alternating direction implicit (ADI) schemes have been proposed for the numerical simulations of classic differential equations. The ADI schemes will reduce a multidimensional problem to a series of independent one-dimensional problems and are thus computationally efficient. In this paper, we consider the numerical solution of a ST-FBTE on a finite domain. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. A fractional alternating direction implicit scheme (FADIS) for the ST-FBTE in 3-D is proposed. Stability and convergence properties of the FADIS are discussed. Finally, some numerical results for ST-FBTE are given.

Numerical simulation of a new two-dimensional variable-order fractional percolation equation in non-homogeneous porous media

Percolation flow problems are discussed in many research fields, such as seepage hydraulics, groundwater hydraulics, groundwater dynamics and fluid dynamics in porous media. Many physical processes appear to exhibit fractional-order behavior that may vary with time, or space, or space and time. The theory of pseudodifferential operators and equations has been used to deal with this situation. In this paper we use a fractional Darcys law with variable order Riemann-Liouville fractional derivatives, this leads to a new variable-order fractional percolation equation. In this paper, a new two-dimensional variable-order fractional percolation equation is considered. A new implicit numerical method and an alternating direct method for the two-dimensional variable-order fractional model is proposed. Consistency, stability and convergence of the implicit finite difference method are established. Finally, some numerical examples are given. The numerical results demonstrate the effectiveness of the methods. This technique can be used to simulate a three-dimensional variable-order fractional percolation equation.

The Galerkin finite element approximation of the fractional cable equation

The cable equation is one of the most fundamental equations for modeling neuronal dynamics. Cable equations with a fractional order temporal derivative have been introduced to model electrotonic properties of spiny neuronal dendrites. In this paper, the fractional cable equation involving two integro-differential operators is considered. The Galerkin finite element approximations of the fractional cable equation are proposed. The main contribution of this work is outlined as follow: • A semi-discrete finite difference approximation in time is proposed. We prove that the scheme is unconditionally stable, and the numerical solution converges to the exact solution with order O(Δt). • A semi-discrete difference scheme for improving the order of convergence for solving the fractional cable equation is proposed, and the numerical solution converges to the exact solution with order O((Δt)2). • Based on the above semi-discrete difference approximations, Galerkin finite element approximations in space for a full discretization are also investigated. • Finally, some numerical results are given to demonstrate the 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.

Entrepreneurship education in non-business schools : best practice for Australian contexts of knowledge and innovation communities (Final Report)

In the current era of global economic instability, business and industry have already identified a widening gap between graduate skills and employability. An important element of this is the lack of entrepreneurial skills in graduates. This Teaching Fellowship investigated two sides of a story about entrepreneurial skills and their teaching. Senior players in the innovation commercialisation industry, a high profile entrepreneurial sector, were surveyed to gauge their needs and experiences of graduates they employ. International contexts of entrepreneurship education were investigated to explore how their teaching programs impart the skills of entrepreneurship. Such knowledge is an essential for the design of education programs that can deliver the entrepreneurial skills deemed important by industry for future sustainability. Two programs of entrepreneurship education are being implemented at QUT that draw on the best practice exemplars investigated during this Fellowship. The QUT Innovation Space (QIS) focuses on capturing the innovation and creativity of students, staff and others. The QIS is a physical and virtual meeting and networking space; a connected community enhancing the engagement of participants. The Q_Hatchery is still embryonic; but it is intended to be an innovation community that brings together nascent entrepreneurial businesses to collaborate, train and support each other. There is a niche between concept product and business incubator where an experiential learning environment for otherwise isolated ‘garage-at-home’ businesses could improve success rates. The QIS and the Q_Hatchery serve as living research laboratories to trial the concepts emerging from the skills survey. The survey of skills requirements of the innovation commercialisation industry has produced a large and high quality data set still being explored. Work experience as an employability factor has already emerged as an industry requirement that provides employee maturity. Exploratory factor analysis of the skills topics surveyed has led to a process-based conceptual model for teaching and learning higher-order entrepreneurial skills. Two foundational skills domains (Knowledge, Awareness) are proposed as prerequisites which allow individuals with a suite of early stage entrepreneurial and behavioural skills (Pre-leadership) to further leverage their careers into a leadership role in industry with development of skills around higher order elements of entrepreneurship, management in new business ventures and progressing winning technologies to market. The next stage of the analysis is to test the proposed model through structured equation modelling. Another factor that emerged quickly from the survey analysis broadens the generic concept of team skills currently voiced in Australian policy documents discussing the employability agenda. While there was recognition of the role of sharing, creating and using knowledge in a team-based interdisciplinary context, the adoption and adaptation of behaviours and attitudes of other team members of different disciplinary backgrounds (interprofessionalism) featured as an issue. Most undergraduates are taught and undertake teamwork in silos and, thus, seldom experience a true real-world interdisciplinary environment. Enhancing the entrepreneurial capacity of Australian industry is essential for the economic health of the country and can only be achieved by addressing the lack of entrepreneurial skills in graduates from the higher education system. This Fellowship has attempted to address this deficiency by identifying the skills requirements and providing frameworks for their teaching.

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.

Australian Learning and Teaching Council, Learning and Teaching Academic Standards Statement for Architecture

The development of the Learning and Teaching Academic Standards Statement for Architecture (the Statement) centred on requirements for the Master of Architecture and proceeded alongside similar developments in the building and construction discipline under the guidance and support of the Australian Deans of Built Environment and Design (ADBED). Through their representation of Australian architecture programs, ADBED have provided high-level leadership for the Learning and Teaching Academic Standards Project in Architecture (LTAS Architecture). The threshold learning outcomes (TLOs), the description of the nature and extent of the discipline, and accompanying notes were developed through wide consultation with the discipline and profession nationally. They have been considered and debated by ADBED on a number of occasions and have, in their fi nal form, been strongly endorsed by the Deans. ADBED formed the core of the Architecture Reference Group (chaired by an ADBED member) that drew together representatives of every peak organisation for the profession and discipline in Australia. The views of the architectural education community and profession have been provided both through individual submissions and the voices of a number of peak bodies. Over two hundred individuals from the practising profession, the academic workforce and the student cohort have worked together to build consensus about the capabilities expected of a graduate of an Australian Master of Architecture degree. It was critical from the outset that the Statement should embrace the wisdom of the greater ‘tribe’, should ensure that graduates of the Australian Master of Architecture were eligible for professional registration and, at the same time, should allow for scope and diversity in the shape of Australian architectural education. A consultation strategy adopted by the Discipline Scholar involved meetings and workshops in Perth, Melbourne, Sydney, Canberra and Brisbane. Stakeholders from all jurisdictions and most universities participated in the early phases of consultation through a series of workshops that concluded late in October 2010. The Draft Architecture Standards Statement was formed from these early meetings and consultation in respect of that document continued through early 2011. This publication represents the outcomes of work to establish an agreed standards statement for the Master of Architecture. Significant further work remains to ensure the alignment of professional accreditation and recognition procedures with emerging regulatory frameworks cascading from the establishment of the Tertiary Education Quality and Standards Agency (TEQSA). The Australian architecture community hopes that mechanisms can be found to integrate TEQSA’s quality assurance purpose with well-established and understood systems of professional accreditation to ensure the good standing of Australian architectural education into the future. The work to build renewed and integrated quality assurance processes and to foster the interests of this project will continue, for at least the next eighteen months, under the auspices of Australian Learning and Teaching Council (ALTC)-funded Architecture Discipline Network (ADN), led by ADBED and Queensland University of Technology. The Discipline Scholar gratefully acknowledges the generous contributions given by those in stakeholder communities to the formulation of the Statement. Professional and academic colleagues have travelled and gathered to shape the Standards Statement. Debate has been vigorous and spirited and the Statement is rich with the purpose, critical thinking and good judgement of the Australian architectural education community. The commitments made to the processes that have produced this Statement reflect a deep and abiding interest by the constituency in architectural education. This commitment bodes well for the vibrancy and productivity of the emergent Architecture Discipline Network (ADN). Endorsement, in writing, was received from the Australian Institute of Architects National Education Committee (AIA NEC): The National Education Committee (NEC) of the Australian Institute of Architects thank you for your work thus far in developing the Learning and Teaching Academic Standards for Architecture In particular, we acknowledge your close consultation with the NEC on the project along with a comprehensive cross-section of the professional and academic communities in architecture. The TLOs with the nuanced levels of capacities – to identify, develop, explain, demonstrate etc – are described at an appropriate level to be understood as minimum expectations for a Master of Architecture graduate. The Architects Accreditation Council of Australia (AACA) has noted: There is a clear correlation between the current processes for accreditation and what may be the procedures in the future following the current review. The requirement of the outcomes as outlined in the draft paper to demonstrate capability is an appropriate way of expressing the measure of whether the learning outcomes have been achieved. The measure of capability as described in the outcome statements is enhanced with explanatory descriptions in the accompanying notes.

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.