Anomalous subdiffusion equations by the meshless collocation method

This paper aims to develop an implicit meshless collocation technique based on the moving least squares approximation for numerical simulation of the anomalous subdiffusion equation(ASDE). The discrete system of equations is obtained by using the MLS meshless shape functions and the meshless collocation formulation. The stability and convergence of this meshless approach related to the time discretization are investigated theoretically and numerically. The numerical examples with regular and irregular nodal distributions are used to the newly developed meshless formulation. It is concluded that the present meshless formulation is very effective for the modeling of ASDEs.

Tracking the magmatic evolution of island arc volcanism : insights from a high-precision Pb isotope record of Montserrat, Lesser Antilles

The volcanic succession on Montserrat provides an opportunity to examine the magmatic evolution of island arc volcanism over a ∼2.5 Ma period, extending from the andesites of the Silver Hills center, to the currently active Soufrière Hills volcano (February 2010). Here we present high-precision double-spike Pb isotope data, combined with trace element and Sr-Nd isotope data throughout this period of Montserrat's volcanic evolution. We demonstrate that each volcanic center; South Soufrière Hills, Soufrière Hills, Centre Hills and Silver Hills, can be clearly discriminated using trace element and isotopic parameters. Variations in these parameters suggest there have been systematic and episodic changes in the subduction input. The SSH center, in particular, has a greater slab fluid signature, as indicated by low Ce/Pb, but less sediment addition than the other volcanic centers, which have higher Th/Ce. Pb isotope data from Montserrat fall along two trends, the Silver Hills, Centre Hills and Soufrière Hills lie on a general trend of the Lesser Antilles volcanics, whereas SSH volcanics define a separate trend. The Soufrière Hills and SSH volcanic centers were erupted at approximately the same time, but retain distinctive isotopic signatures, suggesting that the SSH magmas have a different source to the other volcanic centers. We hypothesize that this rapid magmatic source change is controlled by the regional transtensional regime, which allowed the SSH magma to be extracted from a shallower source. The Pb isotopes indicate an interplay between subduction derived components and a MORB-like mantle wedge influenced by a Galapagos plume-like source.

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.

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.

Orthopedic bone plates : evolution in structure implementation technique and biomaterial

With many important developments over the last century, nowadays orthopedic bone plate now excels over other types of internal fixators in bone fracture fixation. The developments involve the design, material and implementation techniques of the plates. This paper aims to review the evolution in implementation technique and biomaterial of the orthopedic bone plates. Plates were initially used to fix the underlying bones firmly. Accordingly, Compression plate (CP), Dynamic compression plate (DCP), Limited contact dynamic compression plate (LC-DCP) and Point contact fixator (PC-Fix) were developed. Later, the implementation approach was changed to locking, and the Less Invasive Stabilization System (LISS) plate was introduced as a result. Finally, a combination of both of these approaches has been used by introducing the Locking Compression Plate (LCP). Currently, precontoured LCPs are mainly used for bone fracture fixation. In parallel with structure and implementation techniques, numerous advances have occurred in biomaterials of the plates. Titanium and stainless steel alloys are now the most common biomaterials in production of orthopedic bone plates. However, regarding the biocompatibility, bioactivity and biodegradability characteristics of Mg alloys, Ta alloys, SMAs, carbon fiber composites and bioceramics, these materials are considered as potentially suitable for plates. However, due to poor mechanical properties, they have very limited applications. Therefore, further studies are required in future to solve these problems and make them feasible for heavy-duty bone plates.

Mixed-dimensional consistent coupling by multi-point constraint equations for efficient multi-scale modeling

Finding an appropriate linking method to connect different dimensional element types in a single finite element model is a key issue in the multi-scale modeling. This paper presents a mixed dimensional coupling method using multi-point constraint equations derived by equating the work done on either side of interface connecting beam elements and shell elements for constructing a finite element multiscale model. A typical steel truss frame structure is selected as case example and the reduced scale specimen of this truss section is then studied in the laboratory to measure its dynamic and static behavior in global truss and local welded details while the different analytical models are developed for numerical simulation. Comparison of dynamic and static response of the calculated results among different numerical models as well as the good agreement with those from experimental results indicates that the proposed multi-scale model is efficient and accurate.

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.

Insights into historical drainage evolution based on the phylogeography of the chevron snakehead fish (Channa striata) in the Mekong Basin

1. The phylogeography of freshwater taxa is often integrally linked with landscape changes such as drainage re-alignments that may present the only avenue for historical dispersal for these taxa. Classical models of gene flow do not account for landscape changes and so are of little use in predicting phylogeography in geologically young freshwater landscapes. When the history of drainage formation is unknown, phylogeographical predictions can be based on current freshwater landscape structure, proposed historical drainage geomorphology, or from phylogeographical patterns of co-distributed taxa. 2. This study describes the population structure of a sedentary freshwater fish, the chevron snakehead (Channa striata), across two river drainages on the Indochinese Peninsula. The phylogeographical pattern recovered for C. striata was tested against seven hypotheses based on contemporary landscape structure, proposed history and phylogeographical patterns of codistributed taxa. 3. Consistent with the species ecology, analysis of mitochondrial and microsatellite loci revealed very high differentiation among all sampled sites. A strong signature of historical population subdivision was also revealed within the contemporary Mekong River Basin (MRB). Of the seven phylogeographical hypotheses tested, patterns of co-distributed taxa proved to be the most adequate for describing the phylogeography of C. striata. 4. Results shed new light on SE Asian drainage evolution, indicating that the Middle MRB probably evolved via amalgamation of at least three historically independent drainage sections and in particular that the Mekong River section centred around the northern Khorat Plateau in NE Thailand was probably isolated from the greater Mekong for an extensive period of evolutionary time. In contrast, C. striata populations in the Lower MRB do not show a phylogeographical signature of evolution in historically isolated drainage lines, suggesting drainage amalgamation has been less important for river landscape formation in this region.

Emergent innovation through the co-evolution of informal and formal media economies

Well-established distinctions between amateur and professional are blurring as the impact of social media, changes in cultural consumption, and crises in copyright industries’ business models are felt across society and economy. I call this the increasingly rapid co-evolution of the formal market and informal household sectors and analyse it through the concept of ‘social network markets’ – individual choices are made on the basis of other’s choices and such networked preferencing is enhanced by the growing ubiquity of social media platforms. This may allow us better to understand sources of disruption and innovation in audiovisual production and distribution in wealthy Western markets which are as significant as those posed by informal practices outside the West. I examine what is happening around the monetization and professionalization of online video (YouTube, for example) and the socialization of professional production strategies (transmedia, for example) as innovation from the margins.

Solution methods for advection-diffusion-reaction equations on growing domains and subdomains, with application to modelling skin substitutes

Problems involving the solution of advection-diffusion-reaction equations on domains and subdomains whose growth affects and is affected by these equations, commonly arise in developmental biology. Here, a mathematical framework for these situations, together with methods for obtaining spatio-temporal solutions and steady states of models built from this framework, is presented. The framework and methods are applied to a recently published model of epidermal skin substitutes. Despite the use of Eulerian schemes, excellent agreement is obtained between the numerical spatio-temporal, numerical steady state, and analytical solutions of the model.