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.

A new implicit numerical method for the space and time fractional Bloch-Torrey equation

In recent years, it has been found that many phenomena in engineering, physics, chemistry and other sciences can be described very successfully by models using mathematical tools from fractional calculus. Recently, noted a new space and time fractional Bloch-Torrey equation (ST-FBTE) has been proposed (see Magin et al. (2008)), and successfully applied to analyse diffusion images of human brain tissues to provide new insights for further investigations of tissue structures. In this paper, we consider the 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. Firstly, we propose a new effective implicit numerical method (INM) for the STFBTE whereby we discretize the Riesz fractional derivative using a fractional centered difference. Secondly, we prove that the implicit numerical method for the ST-FBTE is unconditionally stable and convergent, and the order of convergence of the implicit numerical method is ( T2 - α + h2 x + h2 y + h2 z ). Finally, some numerical results are presented to support our theoretical analysis.

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.

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.

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 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.

Finite element modelling of load bearing steel stud walls under real building fires

Fire safety has become an important part in structural design due to the ever increasing loss of properties and lives during fires. Conventionally the fire rating of load bearing wall systems made of Light gauge Steel Frames (LSF) is determined using fire tests based on the standard time-temperature curve given in ISO 834 (ISO, 1999). The standard time-temperature curve given in ISO 834 (ISO, 1999) originated from the application of wood burning furnaces in the early 1900s. However, modern commercial and residential buildings make use of thermoplastic materials, which mean considerably high fuel loads. Hence a detailed fire research study into the performance of LSF walls was undertaken using the developed real fire curves based on Eurocode parametric curves (ECS, 2002) and Barnett’s BFD curves (Barnett, 2002) using both full scale fire tests and numerical studies. It included LSF walls without any insulation, and the recently developed externally insulated composite panel system. This paper presents the details of the numerical studies and the results. It also includes brief details of the development of real building fire curves and experimental studies.

Performance evaluation of onsite sewage treatment : Evaluation of current research

Efficient management of domestic wastewater is a primary requirement for human well being. Failure to adequately address issues of wastewater collection, treatment and disposal can lead to adverse public health and environmental impacts. The increasing spread of urbanisation has led to the conversion of previously rural land into urban developments and the more intensive development of semi urban areas. However the provision of reticulated sewerage facilities has not kept pace with this expansion in urbanisation. This has resulted in a growing dependency on onsite sewage treatment. Though considered only as a temporary measure in the past, these systems are now considered as the most cost effective option and have become a permanent feature in some urban areas. This report is the first of a series of reports to be produced and is the outcome of a research project initiated by the Brisbane City Council. The primary objective of the research undertaken was to relate the treatment performance of onsite sewage treatment systems with soil conditions at site, with the emphasis being on septic tanks. This report consists of a ‘state of the art’ review of research undertaken in the arena of onsite sewage treatment. The evaluation of research brings together significant work undertaken locally and overseas. It focuses mainly on septic tanks in keeping with the primary objectives of the project. This report has acted as the springboard for the later field investigations and analysis undertaken as part of the project. Septic tanks still continue to be used widely due to their simplicity and low cost. Generally the treatment performance of septic tanks can be highly variable due to numerous factors, but a properly designed, operated and maintained septic tank can produce effluent of satisfactory quality. The reduction of hydraulic surges from washing machines and dishwashers, regular removal of accumulated septage and the elimination of harmful chemicals are some of the practices that can improve system performance considerably. The relative advantages of multi chamber over single chamber septic tanks is an issue that needs to be resolved in view of the conflicting research outcomes. In recent years, aerobic wastewater treatment systems (AWTS) have been gaining in popularity. This can be mainly attributed to the desire to avoid subsurface effluent disposal, which is the main cause of septic tank failure. The use of aerobic processes for treatment of wastewater and the disinfection of effluent prior to disposal is capable of producing effluent of a quality suitable for surface disposal. However the field performance of these has been disappointing. A significant number of these systems do not perform to stipulated standards and quality can be highly variable. This is primarily due to houseowner neglect or ignorance of correct operational and maintenance procedures. The other problems include greater susceptibility to shock loadings and sludge bulking. As identified in literature a number of design features can also contribute to this wide variation in quality. The other treatment processes in common use are the various types of filter systems. These include intermittent and recirculating sand filters. These systems too have their inherent advantages and disadvantages. Furthermore as in the case of aerobic systems, their performance is very much dependent on individual houseowner operation and maintenance practices. In recent years the use of biofilters has attracted research interest and particularly the use of peat. High removal rates of various wastewater pollutants have been reported in research literature. Despite these satisfactory results, leachate from peat has been reported in various studies. This is an issue that needs further investigations and as such biofilters can still be considered to be in the experimental stage. The use of other filter media such as absorbent plastic and bark has also been reported in literature. The safe and hygienic disposal of treated effluent is a matter of concern in the case of onsite sewage treatment. Subsurface disposal is the most common and the only option in the case of septic tank treatment. Soil is an excellent treatment medium if suitable conditions are present. The processes of sorption, filtration and oxidation can remove the various wastewater pollutants. The subsurface characteristics of the disposal area are among the most important parameters governing process performance. Therefore it is important that the soil and topographic conditions are taken into consideration in the design of the soil absorption system. Seepage trenches and beds are the common systems in use. Seepage pits or chambers can be used where subsurface conditions warrant, whilst above grade mounds have been recommended for a variety of difficult site conditions. All these systems have their inherent advantages and disadvantages and the preferable soil absorption system should be selected based on site characteristics. The use of gravel as in-fill for beds and trenches is open to question. It does not contribute to effluent treatment and has been shown to reduce the effective infiltrative surface area. This is due to physical obstruction and the migration of fines entrained in the gravel, into the soil matrix. The surface application of effluent is coming into increasing use with the advent of aerobic treatment systems. This has the advantage that treatment is undertaken on the upper soil horizons, which is chemically and biologically the most effective in effluent renovation. Numerous research studies have demonstrated the feasibility of this practice. However the overriding criteria is the quality of the effluent. It has to be of exceptionally good quality in order to ensure that there are no resulting public health impacts due to aerosol drift. This essentially is the main issue of concern, due to the unreliability of the effluent quality from aerobic systems. Secondly, it has also been found that most householders do not take adequate care in the operation of spray irrigation systems or in the maintenance of the irrigation area. Under these circumstances surface disposal of effluent should be approached with caution and would require appropriate householder education and stringent compliance requirements. However despite all this, the efficiency with which the process is undertaken will ultimately rest with the individual householder and this is where most concern rests. Greywater too should require similar considerations. Surface irrigation of greywater is currently being permitted in a number of local authority jurisdictions in Queensland. Considering the fact that greywater constitutes the largest fraction of the total wastewater generated in a household, it could be considered to be a potential resource. Unfortunately in most circumstances the only pretreatment that is required to be undertaken prior to reuse is the removal of oil and grease. This is an issue of concern as greywater can considered to be a weak to medium sewage as it contains primary pollutants such as BOD material and nutrients and may also include microbial contamination. Therefore its use for surface irrigation can pose a potential health risk. This is further compounded by the fact that most householders are unaware of the potential adverse impacts of indiscriminate greywater reuse. As in the case of blackwater effluent reuse, there have been suggestions that greywater should also be subjected to stringent guidelines. Under these circumstances the surface application of any wastewater requires careful consideration. The other option available for the disposal effluent is the use of evaporation systems. The use of evapotranspiration systems has been covered in this report. Research has shown that these systems are susceptible to a number of factors and in particular to climatic conditions. As such their applicability is location specific. Also the design of systems based solely on evapotranspiration is questionable. In order to ensure more reliability, the systems should be designed to include soil absorption. The successful use of these systems for intermittent usage has been noted in literature. Taking into consideration the issues discussed above, subsurface disposal of effluent is the safest under most conditions. This is provided the facility has been designed to accommodate site conditions. The main problem associated with subsurface disposal is the formation of a clogging mat on the infiltrative surfaces. Due to the formation of the clogging mat, the capacity of the soil to handle effluent is no longer governed by the soil’s hydraulic conductivity as measured by the percolation test, but rather by the infiltration rate through the clogged zone. The characteristics of the clogging mat have been shown to be influenced by various soil and effluent characteristics. Secondly, the mechanisms of clogging mat formation have been found to be influenced by various physical, chemical and biological processes. Biological clogging is the most common process taking place and occurs due to bacterial growth or its by-products reducing the soil pore diameters. Biological clogging is generally associated with anaerobic conditions. The formation of the clogging mat provides significant benefits. It acts as an efficient filter for the removal of microorganisms. Also as the clogging mat increases the hydraulic impedance to flow, unsaturated flow conditions will occur below the mat. This permits greater contact between effluent and soil particles thereby enhancing the purification process. This is particularly important in the case of highly permeable soils. However the adverse impacts of the clogging mat formation cannot be ignored as they can lead to significant reduction in the infiltration rate. This in fact is the most common cause of soil absorption systems failure. As the formation of the clogging mat is inevitable, it is important to ensure that it does not impede effluent infiltration beyond tolerable limits. Various strategies have been investigated to either control clogging mat formation or to remediate its severity. Intermittent dosing of effluent is one such strategy that has attracted considerable attention. Research conclusions with regard to short duration time intervals are contradictory. It has been claimed that the intermittent rest periods would result in the aerobic decomposition of the clogging mat leading to a subsequent increase in the infiltration rate. Contrary to this, it has also been claimed that short duration rest periods are insufficient to completely decompose the clogging mat, and the intermediate by-products that form as a result of aerobic processes would in fact lead to even more severe clogging. It has been further recommended that the rest periods should be much longer and should be in the range of about six months. This entails the provision of a second and alternating seepage bed. The other concepts that have been investigated are the design of the bed to meet the equilibrium infiltration rate that would eventuate after clogging mat formation; improved geometry such as the use of seepage trenches instead of beds; serial instead of parallel effluent distribution and low pressure dosing of effluent. The use of physical measures such as oxidation with hydrogen peroxide and replacement of the infiltration surface have been shown to be only of short-term benefit. Another issue of importance is the degree of pretreatment that should be provided to the effluent prior to subsurface application and the influence exerted by pollutant loadings on the clogging mat formation. Laboratory studies have shown that the total mass loadings of BOD and suspended solids are important factors in the formation of the clogging mat. It has also been found that the nature of the suspended solids is also an important factor. The finer particles from extended aeration systems when compared to those from septic tanks will penetrate deeper into the soil and hence will ultimately cause a more dense clogging mat. However the importance of improved pretreatment in clogging mat formation may need to be qualified in view of other research studies. It has also shown that effluent quality may be a factor in the case of highly permeable soils but this may not be the case with fine structured soils. The ultimate test of onsite sewage treatment system efficiency rests with the final disposal of effluent. The implication of system failure as evidenced from the surface ponding of effluent or the seepage of contaminants into the groundwater can be very serious as it can lead to environmental and public health impacts. Significant microbial contamination of surface and groundwater has been attributed to septic tank effluent. There are a number of documented instances of septic tank related waterborne disease outbreaks affecting large numbers of people. In a recent incident, the local authority was found liable for an outbreak of viral hepatitis A and not the individual septic tank owners as no action had been taken to remedy septic tank failure. This illustrates the responsibility placed on local authorities in terms of ensuring the proper operation of onsite sewage treatment systems. Even a properly functioning soil absorption system is only capable of removing phosphorus and microorganisms. The nitrogen remaining after plant uptake will not be retained in the soil column, but will instead gradually seep into the groundwater as nitrate. Conditions for nitrogen removal by denitrification are not generally present in a soil absorption bed. Dilution by groundwater is the only treatment available for reducing the nitrogen concentration to specified levels. Therefore based on subsurface conditions, this essentially entails a maximum allowable concentration of septic tanks in a given area. Unfortunately nitrogen is not the only wastewater pollutant of concern. Relatively long survival times and travel distances have been noted for microorganisms originating from soil absorption systems. This is likely to happen if saturated conditions persist under the soil absorption bed or due to surface runoff of effluent as a result of system failure. Soils have a finite capacity for the removal of phosphorus. Once this capacity is exceeded, phosphorus too will seep into the groundwater. The relatively high mobility of phosphorus in sandy soils have been noted in the literature. These issues have serious implications in the design and siting of soil absorption systems. It is not only important to ensure that the system design is based on subsurface conditions but also the density of these systems in given areas is a critical issue. This essentially involves the adoption of a land capability approach to determine the limitations of an individual site for onsite sewage disposal. The most limiting factor at a particular site would determine the overall capability classification for that site which would also dictate the type of effluent disposal method to be adopted.

Critical timescales and time intervals for coupled linear processes

In 1991, McNabb introduced the concept of mean action time (MAT) as a finite measure of the time required for a diffusive process to effectively reach steady state. Although this concept was initially adopted by others within the Australian and New Zealand applied mathematics community, it appears to have had little use outside this region until very recently, when in 2010 Berezhkovskii and coworkers rediscovered the concept of MAT in their study of morphogen gradient formation. All previous work in this area has been limited to studying single–species differential equations, such as the linear advection–diffusion–reaction equation. Here we generalise the concept of MAT by showing how the theory can be applied to coupled linear processes. We begin by studying coupled ordinary differential equations and extend our approach to coupled partial differential equations. Our new results have broad applications including the analysis of models describing coupled chemical decay and cell differentiation processes, amongst others.

Residence time investigation in a co-axial dielectric barrier discharge reactor

The residence time distribution (RTD) is a crucial parameter when treating engine exhaust emissions with a Dielectric Barrier Discharge (DBD) reactor. In this paper, the residence time of such a reactor is investigated using a finite element based software: COMSOL Multiphysics 4.3. Non-thermal plasma (NTP) discharge is being introduced as a promising method for pollutant emission reduction. DBD is one of the most advantageous of NTP technologies. In a two cylinder co-axial DBD reactor, tubes are placed between two electrodes and flow passes through the annuals between these barrier tubes. If the mean residence time increases in a DBD reactor, there will be a corresponding increase in reaction time and consequently, the pollutant removal efficiency can increase. However, pollutant formation can occur during increased mean residence time and so the proportion of fluid that may remain for periods significantly longer than the mean residence time is of great importance. In this study, first, the residence time distribution is calculated based on the standard reactor used by the authors for ultrafine particle (10-500 nm) removal. Then, different geometrics and various inlet velocities are considered. Finally, for selected cases, some roughness elements added inside the reactor and the residence time is calculated. These results will form the basis for a COMSOL plasma and CFD module investigation.

Porohyperelastic analysis to explore mechanical properties of chondrocytes using numerical modeling and experiments : a finite element study

There are many continuum mechanical models have been developed such as liquid drop models, solid models, and so on for single living cell biomechanics studies. However, these models do not give a fully approach to exhibit a clear understanding of the behaviour of single living cells such as swelling behaviour, drag effect, etc. Hence, the porohyperelastic (PHE) model which can capture those aspects would be a good candidature to study cells behaviour (e.g. chondrocytes in this study). In this research, an FEM model of single chondrocyte cell will be developed by using this PHE model to simulate Atomic Force Microscopy (AFM) experimental results with the variation of strain rate. This material model will be compared with viscoelastic model to demonstrate the advantages of PHE model. The results have shown that the maximum value of force applied of PHE model is lower at lower strain rates. This is because the mobile fluid does not have enough time to exude in case of very high strain rate and also due to the lower permeability of the membrane than that of the protoplasm of chondrocyte. This behavior is barely observed in viscoelastic model. Thus, PHE model is the better model for cell biomechanics studies.

Funding : Australia’s grant system wastes time

We found that scientists in Australia spent more than five centuries' worth of time preparing research-grant proposals for consideration by the largest funding scheme of 2012. Because just 20.5% of these applications were successful, the equivalent of some four centuries of effort returned no immediate benefit to researchers and wasted valuable research time. The system needs reforming and alternative funding processes should be investigated...

The minority report : disproportionate representation in Australia’s largest education system

The overrepresentation of students from minority ethnic groups in separate special education settings has been extensively documented in North America, yet little research exists for Australian school systems. To address this gap, we systematically analyzed 13 years of enrolment data from the state of New South Wales. Stark differences are seen in patterns of enrolment between Indigenous students, students from a Language Background Other than English (LBOTE), and non-Indigenous English speaking students. Moreover, these differences are increasing. While enrollments of Indigenous students in separate settings increased faster across time than did enrollments of Indigenous students in mainstream, enrollments of LBOTE students in mainstream increased faster than did enrollments of LBOTE students in separate settings.

A computationally effective alternating direction method for the space and time fractional Bloch–Torrey equation in 3-D

The space and time fractional Bloch–Torrey equation (ST-FBTE) has been used to study anomalous diffusion in the human brain. Numerical methods for solving ST-FBTE in three-dimensions are computationally demanding. In this paper, we propose a computationally effective fractional alternating direction method (FADM) to overcome this problem. We consider ST-FBTE on a finite domain where the time and space derivatives are replaced by the Caputo–Djrbashian and the sequential Riesz fractional derivatives, respectively. The stability and convergence properties of the FADM are discussed. Finally, some numerical results for ST-FBTE are given to confirm our theoretical findings.