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.

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.

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.

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.

Tooth fracture risk analysis based on a new finite element dental structure models using micro-CT data

The finite element (FE) analysis is an effective method to study the strength and predict the fracture risk of endodontically-treated teeth. This paper presents a rapid method developed to generate a comprehensive tooth FE model using data retrieved from micro-computed tomography (μCT). With this method, the inhomogeneity of material properties of teeth was included into the model without dividing the tooth model into different regions. The material properties of the tooth were assumed to be related to the mineral density. The fracture risk at different tooth portions was assessed for root canal treatments. The micro-CT images of a tooth were processed by a Matlab software programme and the CT numbers were retrieved. The tooth contours were obtained with thresholding segmentation using Amira. The inner and outer surfaces of the tooth were imported into Solidworks and a three-dimensional (3D) tooth model was constructed. An assembly of the tooth model with the periodontal ligament (PDL) layer and surrounding bone was imported into ABAQUS. The material properties of the tooth were calculated from the retrieved CT numbers via ABAQUS user's subroutines. Three root canal geometries (original and two enlargements) were investigated. The proposed method in this study can generate detailed 3D finite element models of a tooth with different root canal enlargements and filling materials, and would be very useful for the assessment of the fracture risk at different tooth portions after root canal treatments.

Finite element analyses of lipped channel beams with Web openings in shear

Cold-formed steel members are increasingly used as primary structural elements in buildings due to the availability of thin and high strength steels and advanced cold-forming technologies. Cold-formed lipped channel beams (LCB) are commonly used as flexural members such as floor joists and bearers. Shear behaviour of LCBs with web openings is more complicated and their shear capacities are considerably reduced by the presence of web openings. However, limited research has been undertaken on the shear behaviour and strength of LCBs with web openings. Hence a numerical study was undertaken to investigate the shear behaviour and strength of LCBs with web openings. Finite element models of simply supported LCBs with aspect ratios of 1.0 and 1.5 were considered under a mid-span load. They were then validated by comparing their results with test results and used in a detailed parametric study. Experimental and numerical results showed that the current design rules in cold-formed steel structures design codes are very conservative for the shear design of LCBs with web openings. Improved design equations were therefore proposed for the shear strength of LCBs with web openings. This paper presents the details of this numerical study of LCBs with web openings, and the results.

Design and fabrication of tubular scaffolds via direct writing in a melt electrospinning mode

Flexible tubular structures fabricated from solution electrospun fibers are finding increasing use in tissue engineering applications. However it is difficult to control the deposition of fibers due to the chaotic nature of the solution electrospinning jet. By using non-conductive polymer melts instead of polymer solutions the path and collection of the fiber becomes predictable. In this work we demonstrate the melt electrospinning of polycaprolactone in a direct writing mode onto a rotating cylinder. This allows the design and fabrication of tubes using 20 μm diameter fibers with controllable micropatterns and mechanical properties. A key design parameter is the fiber winding angle, where it allows control over scaffold pore morphology (e.g. size, shape, number and porosity). Furthermore, the establishment of a finite element model as a predictive design tool is validated against mechanical testing results of melt electrospun tubes to show that a lesser winding angle provides improved mechanical response to uniaxial tension and compression. In addition, we show that melt electrospun tubes support the growth of three different cell types in vitro and are therefore promising scaffolds for tissue engineering applications.

Measuring the apparent size of the Moon with a digital camera

The Moon appears to be much larger closer to the horizon than when higher in the sky. This is called the ‘Moon Illusion’ since the observed size of the Moon is not actually larger when the Moon is just above the horizon. This article describes a technique for verifying that the observed size of the Moon in not larger on the horizon. The technique can be easily performed in a high school teaching environment. Moreover, the technique demonstrates the surprising fact that the observed size of the Moon is actually smaller on the horizon due to atmospheric refraction. For the purposes of this paper, several images of the moon were taken with the Moon close to the horizon and close to the zenith. Images were processed using a free program called ImageJ. The Moon was found to be 5.73 ±0.04% smaller in area on the horizon then at the zenith.

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.

A least squares based finite volume method for the Cahn-Hilliard and Cahn-Hilliard-reaction equations

A vertex-centred finite volume method (FVM) for the Cahn-Hilliard (CH) and recently proposed Cahn-Hilliard-reaction (CHR) equations is presented. Information at control volume faces is computed using a high-order least-squares approach based on Taylor series approximations. This least-squares problem explicitly includes the variational boundary condition (VBC) that ensures that the discrete equations satisfy all of the boundary conditions. We use this approach to solve the CH and CHR equations in one and two dimensions and show that our scheme satisfies the VBC to at least second order. For the CH equation we show evidence of conservative, gradient stable solutions, however for the CHR equation, strict gradient-stability is more challenging to achieve.

Does occupational aggregation matter in determining the size of the gender wage gap?

The traditional decomposition of the gender wage gap distinguishes between a component attributable to gender differences in productivity-related characteristics and a residual component that is often taken as a measure of discrimination. This study of data from the 1989 Canadian Labour Market Activity Survey shows that when occupation is treated as a productivity-related characteristic, the proportion of the gender wage gap labeled explained increases with the number of occupational classifications distinguished. However, on the basis of evidence that occupational differences reflect the presence of barriers faced by women attempting to enter male-dominated occupations, the authors conclude that occupation should not be treated as a productivity-related characteristic; and in a decomposition of the gender wage gap that treats occupation as endogenously determined, they find that the level of occupational aggregation has little effect on the size of the "explained" component of the gap.