• The effect of irradiation impinging on disparate anchoring configurations of polar-organic molecules adsorbed on bulk and thin-film metal surfaces

      Papadopoulos, Theodoros A.; Metz, Sebastian; Tang, Shu-Jung; University of Chester; Daresbury Laboratory; National Tsing-Hua University (Elsevier, 2019-07-11)
      The behavior of polar metal organic molecules, chloroaluminum phthalocyanine (ClAlPc), upon ultraviolet (UV) irradiation was investigated to evaluate the stability of the adsorption process on the Ag(111) thin film and bulk crystal. Angle-resolved photoelectron spectroscopy (ARPES) was mainly employed to measure the molecular energy states (MES) and vacuum level (VL) shift for 1-ML ClAlPc in the Cl-down configuration. A consistent trend was observed showing that ClAlPc in the Cl-down configuration is energetically more stable on the Ag thin-film surface than on the corresponding surface of the Ag bulk crystal. The intermediate adsorption state in tilted configuration during the irradiation impinging is identified by large positive VL shifts and broad spectra line shapes to infer a flipping mechanism from Cl-down to Cl-up configuration. Strain on the Ag thin films from the underlying mismatched Ge(111) substrate is considered to cause enlarged hollow sites on the Ag(111) thin-films, that anchor the Cl-down configuration more tightly on the thin-film surfaces, as confirmed by density functional theory (DFT) calculations.
    • Effect of laser treatment on the attachment and viability of mesenchymal stem cell responses on shape memory NiTi alloy

      Chan, Chi-Wai; Hussain, Issam; Waugh, David G.; Lawrence, Jonathan; Man, Hau-Chung; Queen's University, Belfast ; University of Lincoln ; University of Chester ; University of Chester ; Hong Kong Polytechnic University (Elsevier, 2014-05-22)
      The objectives of this study were to investigate the effect of laser-induced surface features on the morphology, attachment and viability of mesenchymal stem cells (MSCs) at different periods of time, and to evaluate the biocompatibility of different zones: laser-melted zone (MZ), heat-affected zone (HAZ) and base metal (BM) in laser-treated NiTi alloy.
    • Effect of surface micro-pits on mode-II fracture toughness of Ti-6Al-4V/PEEK interface

      Pan, Lei; Pang, Xiaofei; Wang, Fei; Huang, Haiqiang; Shi, Yu; Tao, Jie; Nanjing University of Aeronautics and Astronautics; University of Chester (Elsevier, 2019-08-17)
      Herein, the delamination issue of TiGr(TC4/PEEK/Cf) laminate is addressed by investigating the influence of TC4(Ti-6Al-4V) surface micro-pits on mode-II interfacial fracture toughness of TC4/PEEK interface through experimental and finite element modeling. The micro-pits unit cell, unit strip and the end notched flexure (ENF) models are established based on the finite element simulations and the effect of micro-pit size parameters is studied in detail. The results of micro-pits unit cell model reveal that the presence of micro-pits can effectively buffer the interfacial stress concentration under mode-II loading conditions. Furthermore, the micro-pits unit strip model, with different micro-pit sizes, is analyzed to obtain the interface parameters, which are converted and used in the ENF model. Both the unit strip and ENF models conclude that the presence of interfacial micro-pits effectively improves the mode-II fracture toughness. It is worth mentioning that the utilization of converted interface parameters in ENF model avoided the limitation of micro-pit size and reduced the workload. Finally, the experimental and computational ENF results exhibited excellent consistency and confirmed the reliability of the proposed finite element models. The current study provides useful guidelines for the design and manufacturing of high-performance TC4/PEEK interfaces for a wide range of applications.
    • Effects of inkjet printed toughener on delamination suppression in drilling of carbon fibre reinforced plastics (CFRPs)

      Shi, Yu; Wang, Xiaonan; Wang, Fuji; Gu, Tianyu; Xie, Pengheng; Jia, Yu; University of Chester; Dalian University of Technology; Aston University
      Delamination has been recognised as the predominant damage induced during the drilling of carbon fibre reinforced plastics (CFRPs). It could significantly reduce the bearing capacity and shorten the service life of the designed component. To enhance the delamination resistance of CFRPs for different applications, great affords have been done to improve their interlaminar fracture toughness. However, due to the difficulty in accurately controlling the amount of the toughener applied in the interface, effect of the toughener content on the toughening efficiency is rarely studied. In this work, an experimental research was developed to investigate the performance of the toughener on the improvement of delamination resistance in the drilling of CFRPs and parametrically optimise the toughener content with the consideration of different feed rates. Specifically, poly(methyl methacrylate) (PMMA) solutions with various concentrations were selected to add on the CFRP prepreg, and co-cured together with layups. The inkjet printing technology was adopted to deposit the PMMA solutions for precisely controlled toughener contents. Through drilling experiments on the toughened CFRPs, it was found that the optimal content of the PMMA solution was 10 wt% to offer the least delamination, in particular, for the situation under the highest feed rate condition. The toughing mechanisms were also concluded by analysing the histories of the thrust force and torque in the drilling process. The results of this study is significantly contribute to the locally toughening of the composite interfaces and the improvement of the drilling quality, which is specifically helpful to strengthen the joint property for the structural design stage for the aircraft.
    • The electrochemical characterisation of graphite felts

      Smith, Robert E. G.; Davies, Trevor J.; Baynes, Nicholas de B.; Nichols, Richard J.; University of Liverpool ; University of Chester/ACAL Energy Limited ; ACAL Energy Limited ; University of Liverpool (Elsevier, 2015-03-25)
      With the increasing demand for advanced energy storage, redox flow batteries (RFBs) are a rapidly growing market and research area. Carbon fibre based materials like graphite felts (GFs) are commonly used in RFBs as electrodes due to their electrochemical activity, high permeability for liquids, high surface area and relatively low cost. Previous characterisation of GFs has involved flow systems, which can be complicated to replicate and expensive. Using pre-treatment techniques to ensure sufficient wetting, GFs can be used effectively in quiescent electrochemistry. By combining the experimental voltammograms with digital simulations, key properties such as capacitance, electrode kinetics, average pore size and electrochemical surface area can be determined. This allows for the rapid and low cost examination of multiple GFs and pre-treatment methods, providing a useful tool for RFB development. In addition, the GF electrodes used in the study show promising limit of detection values, with initial experiments suggesting these electrodes may have applications in electroanalysis.
    • Enhanced design of an offgrid PV-battery-methanation hybrid energy system for power/gas supply

      Xu, Xiao; Hu, Weihao; Cao, Di; Liu, Wen; Huang, Qi; Hu, Yanting; Chen, Zhe
      Extensive studies have been carried out on various hybrid energy systems (HESs) for providing electricity to off-grid areas. However, a standalone HES that is capable of providing power and gas, has been less studied. In this paper, a standalone Photovoltaic (PV)-battery-methanation HES is proposed to provide adequate, reliable and cost-effective electricity and gas to the local consumers. Identifying a potential solution to maximize the reliability of the system, asked by consumers, and to minimize costs required by the investors is challenging. Bi-level programming is adopted in this study to tackle the pre-mentioned issue. In the outer layer, an optimal design is obtained by means of particle swarm optimization. In the inner layer, an optimal operation strategy is found under the optimal design of the outer layer using sequential quadratic programming. The results indicate that 1) The bi-level programming used in this study can find the optimal solution; 2) The proposed HES is proved to be able to supply power and gas simultaneously. 3) Compared with the right most and leftmost points on Pareto set, the total costs are reduced by 17.77% and 2.16%.
    • Enhanced design of an offgrid PV-battery-methanation hybrid energy system for power/gas supply

      Xu, x; Hu, W; Cao, D; Liu, W; Huang, Q; Hu, Y; Chen, Z; University of ESTC; Utrecht University; University of Chester; Aalborg University
      Extensive studies have been carried out on various hybrid energy systems (HESs) for providing electricity to off-grid areas. However, a standalone HES that is capable of providing power and gas, has been less studied. In this paper, a standalone Photovoltaic (PV)-battery-methanation HES is proposed to provide adequate, reliable and cost-effective electricity and gas to the local consumers. Identifying a potential solution to maximize the reliability of the system, asked by consumers, and to minimize costs required by the investors is challenging. Bi-level programming is adopted in this study to tackle the pre-mentioned issue. In the outer layer, an optimal design is obtained by means of particle swarm optimization. In the inner layer, an optimal operation strategy is found under the optimal design of the outer layer using sequential quadratic programming. The results indicate that 1) The bi-level programming used in this study can find the optimal solution; 2) The proposed HES is proved to be able to supply power and gas simultaneously. 3) Compared with the right most and leftmost points on Pareto set, the total costs are reduced by 17.77% and 2.16%.
    • Enhancement of Wear and Corrosion Resistance of Beta Titanium Alloy by Laser Technology

      Chan, Chi-Wai; Lee, Seunghwan; Smith, Graham C.; Sarri, Gianluca; Ng, Chi-Ho; Sharba, Ahmed; Man, Hau-Chung; Queens University Belfast (Chan, Sharba); Technical University of Denmark (Lee); University of Chester (Smith, Ng); Hong Kong Polytechnic University (Man) (Elsevier, 2016-03-30)
      The relatively high elastic modulus coupled with the presence of toxic vanadium (V) in Ti6Al4V alloy has long been a concern in orthopaedic applications. To solve the problem, a variety of non-toxic and low modulus beta-titanium (beta-Ti) alloys have been developed. Among the beta-Ti alloy family, the quaternary Ti-Nb-Zr-Ta (TZNT) alloys have received the highest attention as a promising replacement for Ti6Al4V due to their lower elastic modulus and outstanding long term stability against corrosion in biological environments. However, the inferior wear resistance of TNZT is still a problem that must be resolved before commercialising in the orthopaedic market. In this work, a newly-developed laser surface treatment technique was employed to improve the surface properties of Ti-35.3Nb-7.3Zr-5.7Ta alloy. The surface microstructure and composition of the laser-treated TNZT surface were examined by grazing incidence x-ray diffraction (GI-XRD) and x-ray photoelectron spectroscopy (XPS). The wear and corrosion resistance were evaluated by pin-on-plate sliding test and anodic polarisation test in Hanks’ solution. The experimental results were compared with the untreated (or base) TNZT material. The research findings showed that the laser surface treatment technique reported in this work can effectively improve the wear and corrosion resistance of TNZT. The enhancement of such surface properties was due to the formation of a smooth and hard layer on the substrate surface. The laser-formed layer was metallurgically bonded to the substrate, and had no concern of coating delamination or peel-off.
    • Enhancing interfacial strength between AA5083 and cryogenic adhesive via anodic oxidation and silanization

      Lei, Pan; Zhang, Aiai; Zheng, Zengmin; Duan, Lixiang; Zhang, Lei; Shi, Yu; Tao, Jie; Nanjing University of Aeronautics and Astronautics; University of Chester (Elsevier, 2018-04-27)
      AA5083 aluminum alloy was treated in turn with phosphoric-sulfuric acid anodic oxidation and then with silanization using the silane coupling agent KH560. A chemical bond (Si-O-Al) was created between the aluminum alloy and silane film, and a dehydration condensation reaction occurred between the silane film and cryogenic adhesive to enhance the bonding strength between the aluminum alloy and the cryogenic adhesive. Scanning electron microscopy, Energy dispersive spectroscopy, and Fourier transform infrared spectroscopy were used to explore the interfacial characteristics of the aluminum alloy both with and without the applied treatment. Furthermore, single lap shear tests and durability tests were performed to assess the adhesive strength of the interface between the aluminum alloy and the cryogenic adhesive at low temperature. The most improved interfacial strength using the anodic oxidation and the silanization treatments reached 33.96 MPa at −60 °C. The interface strength with the same treatments after the durability test was 25.4 MPa.
    • Enhancing the antibacterial performance of orthopaedic implant materials by fibre laser surface engineering

      Chan, Chi-Wai; Carson, Louise; Smith, Graham C.; Morelli, Alessio; Lee, Seunghwan; Queens University Belfast (Chan, Carson, Morelli); University of Chester (Smith); Technical University of Denmark (Lee) (Elsevier, 2017-05-15)
      Implant failure caused by bacterial infection is extremely difficult to treat and usually requires the removal of the infected components. Despite the severe consequence of bacterial infection, research into bacterial infection of orthopaedic implants is still at an early stage compared to the effort on enhancing osseointegration, wear and corrosion resistance of implant materials. In this study, the effects of laser surface treatment on enhancing the antibacterial properties of commercially pure (CP) Ti (Grade 2), Ti6Al4V (Grade 5) and CoCrMo alloy implant materials were studied and compared for the first time. Laser surface treatment was performed by a continuous wave (CW) fibre laser with a near-infrared wavelength of 1064 nm in a nitrogen-containing environment. Staphylococcus aureus, commonly implicated in infection associated with orthopaedic implants, was used to investigate the antibacterial properties of the laser-treated surfaces. The surface roughness and topography of the laser-treated materials were analysed by a 2D roughness testing and by AFM. The surface morphologies before and after 24 h of bacterial cell culture were captured by SEM, and bacterial viability was determined using live/dead staining. Surface chemistry was analysed by XPS and surface wettability was measured using the sessile drop method. The findings of this study indicated that the laser-treated CP Ti and Ti6Al4V surfaces exhibited a noticeable reduction in bacterial adhesion and possessed a bactericidal effect. Such properties were attributable to the combined effects of reduced hydrophobicity, thicker and stable oxide films and presence of laser-induced nano-features. No similar antibacterial effect was observed in the laser-treated CoCrMo.
    • Evaluation of the performance and economic viability of a novel low temperature carbon capture process

      Wilson, Paul; Lychnos, George; Clements, Alastair; Michailos, Stavros; Font Palma, Carolina; Diego, Maria E.; Pourkashanian, Mohamed; Howe, Joseph; PMW Technology Ltd; University of Sheffield; University of Chester (Elsevier, 2019-04-22)
      A novel Advanced Cryogenic Carbon Capture (A3C) process is being developed using low cost but high intensity heat transfer to achieve high CO2 capture efficiencies with a much reduced energy consumption and process equipment size. These characteristics, along with the purity of CO2 product and absence of process chemicals, offer the potential for application across a range of sectors. This work presents a techno-economic evaluation for applications ranging from 3% to 35%vol. CO2 content. The A3C process is evaluated against an amine-based CO2 capture process for three applications; an oil-fired boiler, a combined cycle gas turbine (CCGT) and a biogas upgrading plant. The A3C process has shown a modest life cost advantage over the mature MEA technology for the larger selected applications, and substantially lower costs in the smaller biogas application. Enhanced energy recovery and optimization offer significant opportunities for further reductions in cost.
    • Evaporation of liquid nitrogen droplets in superheated immiscible liquids

      Rebelo, Neville; Zhao, Huayong; Nadal, Francois; Garner, Colin; Williams, Andy; Loughborough University; University of Chester (Elsevier, 2019-08-22)
      Liquid nitrogen or other cryogenic liquids have the potential to replace or augment current energy sources in cooling and power applications. This can be done by the rapid evaporation and expansion processes that occur when liquid nitrogen is injected into hotter fluids in mechanical expander systems. In this study, the evaporation process of single liquid nitrogen droplets when submerged into n-propanol, methanol, n-hexane, and n-pentane maintained at 294 K has been investigated experimentally and numerically. The evaporation process is quantified by tracking the growth rate of the resulting nitrogen vapour bubble that has an interface with the bulk liquid. The experimental data suggest that the bubble volume growth is proportional to the time and the bubble growth rate is mainly determined by the initial droplet size. A comparison between the four different bulk liquids indicates that the evaporation rate in n-pentane is the highest, possibly due to its low surface tension. A scaling law based on the pure diffusion-controlled evaporation of droplet in open air environment has been successfully implemented to scale the experimental data. The deviation between the scaling law predictions and the experimental data for 2-propanol, methanol and n-hexane vary between 4% and 30% and the deviation for n-pentane was between 24% and 65%. The more detailed bubble growth rates have been modelled by a heuristic one-dimensional, spherically symmetric quasi-steady-state confined model, which can predict the growth trend well but consistently underestimate the growth rate. A fixed effective thermal conductivity is then introduced to account for the complex dynamics of the droplet inside the bubble and the subsequent convective processes in the surrounding vapour, which leads to a satisfactory quantitative prediction of the growth rate.
    • Existence and regularity of solution for a Stochastic CahnHilliard / Allen-Cahn equation with unbounded noise diffusion

      Antonopoulou, Dimitra; Karali, Georgia D.; Millet, Annie; University of Chester (Elsevier, 2015-10-24)
      The Cahn-Hilliard/Allen-Cahn equation with noise is a simplified mean field model of stochastic microscopic dynamics associated with adsorption and desorption-spin flip mechanisms in the context of surface processes. For such an equation we consider a multiplicative space-time white noise with diffusion coefficient of linear growth. Applying technics from semigroup theory, we prove local existence and uniqueness in dimensions d = 1,2,3. Moreover, when the diffusion coefficient satisfies a sub-linear growth condition of order α bounded by 1 3, which is the inverse of the polynomial order of the nonlinearity used, we prove for d = 1 global existence of solution. Path regularity of stochastic solution, depending on that of the initial condition, is obtained a.s. up to the explosion time. The path regularity is identical to that proved for the stochastic Cahn-Hilliard equation in the case of bounded noise diffusion. Our results are also valid for the stochastic Cahn-Hilliard equation with unbounded noise diffusion, for which previous results were established only in the framework of a bounded diffusion coefficient. As expected from the theory of parabolic operators in the sense of Petrovsk˘ıı, the bi-Laplacian operator seems to be dominant in the combined model.
    • Experimental and process modelling study of integration of a micro-turbine with an amine plant

      Agbonghae, Elvis O.; Best, Thom; Finney, Karen N.; Font Palma, Carolina; Hughes, Kevin J.; Pourkashanian, Mohamed; University of Leeds (Elsevier, 2014-12-31)
      An integrated model of a micro-turbine coupled to a CO2 capture plant has been developed with Aspen Plus, and validated with experimental data obtained from a Turbec T100 microturbine at the PACT facilities in the UKCCS Research Centre, Beighton, UK. Monoethanolamine (MEA) was used as solvent and experimental measurements from the CO2 capture plant have been used to validate the steady-state model developed with Aspen Plus®. The optimum liquid/gas ratio and the lean CO2 loading for 90% CO2 capture has been quantified for flue gases with CO2 concentrations ranging from 3 to 8 mol%.
    • Exponential stability in p-th mean of solutions, and of convergent Euler-type solutions, of stochastic delay differential equations

      Baker, Christopher T. H.; Buckwar, Evelyn; Univesity College Chester ; Humboldt-Universität zu Berlin (Elsevier, 2005-12-15)
      This article carries out an analysis which proceeds as follows: showing that an inequality of Halanay type (derivable via comparison theory) can be employed to derive conditions for p-th mean stability of a solution; producing a discrete analogue of the Halanay-type theory, that permits the development of a p-th mean stability analysis of analogous stochastic difference equations. The application of the theoretical results is illustrated by deriving mean-square stability conditions for solutions and numerical solutions of a constant-coefficient linear test equation.
    • Fibre laser joining of dissimilar materials: Commercially pure Ti and PET hybrid joint for medical device applications

      Chan, Chi-Wai; Smith, Graham C.; Queen's University Belfast; University of Chester (Elsevier, 2016-04-29)
      Laser transmission joining (LTJ) is growing in importance, and has the potential to become a niche technique for the fabrication of hybrid plastic-metal joints for medical device applications. The possibility of directly joining plastics to metals by LTJ has been demonstrated by a number of recent studies. However, a reliable and quantitative method for defining the contact area between the plastic and metal, facilitating calculation of the mechanical shear stress of the hybrid joints, is still lacking. A new method, based on image analysis using ImageJ, is proposed here to quantify the contact area at the joint interface. The effect of discolouration on the mechanical performance of the hybrid joints is also reported for the first time. Biocompatible polyethylene terephthalate (PET) and commercially pure titanium (Ti) were selected as materials for laser joining using a 200 W CW fibre laser system. The effect of laser power, scanning speed and stand-off distance between the nozzle tip and top surface of the plastic were studied and analysed by Taguchi L9 orthogonal array and ANOVA respectively. The surface morphology, structure and elemental composition on the PET and Ti surfaces after shearing/peeling apart were characterized by SEM, EDX, XRD and XPS.
    • Fibre Laser Nitriding of Titanium and its Alloy in Open Atmosphere for Orthopaedic Implant Applications: Investigations on Surface Quality, Microstructure and Tribological Properties

      Chan, Chi-Wai; Lee, Seunghwan; Smith, Graham C.; Donaghy, Clare L.; Queens University Belfast (Chan, Donaghy), Technical University Denmark (Lee), University of Chester (Smith) (Elsevier, 2016-12-20)
      Laser nitriding is known to be an effective method to improve the surface hardness and wear resistance of titanium and its alloys. However, the process requires a gas chamber and this greatly limits the practicability for treating orthopaedic implants which involve complex-shaped parts or curved surfaces, such as the tapered surface in a femoral stemor the ball-shaped surface in a femoral head. To tackle this problem, a direct laser nitriding process in open atmosphere was performed on commercially pure titanium (grade 2, TiG2) and Ti6Al4V alloy (grade 5, TiG5) using a continuous-wave (CW) fibre laser. The effects of varying process parameters, for instance laser power and nitrogen pressure on the surface quality, namely discolouration were quantified using ImageJ analysis. The optimised process parameters to produce the gold-coloured nitride surfaces were also identified: 40W(laser power), 25mm/s (scanning speed), 1.5mm(standoff distance) and 5 bar (N2 pressure). Particularly, N2 pressure at 5 barwas found to be the threshold above which significant discolouration will occur. The surface morphology, composition, microstructure, micro-hardness, and tribological properties, particularly hydrodynamic size distribution of wear debris, were carefully characterized and compared. The experimental results showed that TiG2 and TiG5 reacted differently with the laser radiation at 1.06 μm wavelength in laser nitriding as evidenced by substantial differences in the microstructure, and surface colour and morphology. Furthermore, both friction andwear properties were strongly affected by the hardness and microstructure of titaniumsamples and direct laser nitriding led to substantial improvements in their wear resistant properties. Between the two types of titanium samples, bare TiG2 showed higher friction forces and wear rates, but this trend was reversed after laser nitriding treatments.
    • Fibre laser treatment of martensitic NiTi alloys for load-bearing implant applications: Effects of surface chemistry on inhibiting Staphylococcus aureus biofilm formation

      Smith, Graham C.; Chan, Chi-Wai; Carson, Louise; Queens University Belfast; University of Chester (Elsevier, 2018-06-15)
      Biofilm infection is one of the main reasons for implant failure. It is extremely difficult to cure due to its high resistance to antibiotic treatments, and can result in substantial healthcare costs. In this study, the important shape memory NiTi alloy, in its martensitic state, was laser-treated using our newly-developed surface modification technique, aiming to tackle the biofilm infection problem. Martensitic NiTi was chosen for investigation because of its potential advantages in terms of (i) lower elastic modulus and (ii) higher damping capacity over its austenitic counterpart, giving rise to a lower risk of stress shielding and maximum stress between bones and load-bearing implants. The surfaces after laser treatment were systemically analysed using a series of surface measurement (i.e. surface roughness and water contact angle) and material characterisation (i.e. SEM-EDX, XRD and XPS) techniques. The antibacterial performance of the laser-treated surfaces was evaluated using the Staphylococcus aureus (or S. aureus) cells in-vitro cultured at 37 oC for 24h. Fluorescence microscopy accompanied by Live/Dead staining was employed to analyse the cell culture results. The surfaces in their as-received states and after polishing were also tested and compared with the laser-treated surfaces in order to gain a deeper insight in how different surface conditions would influence biofilm formation. Our results indicate that the surfaces after laser treatment can mitigate bacterial attachment and biofilm formation effectively. The antibacterial performance was mainly attributable to the laser-formed oxides which brought desirable changes to the surface chemistry of NiTi. The laser-induced changes in surface roughness and topography, on a micrometre scale, only played a minor role in influencing bacterial attachment. The findings of this study demonstrated for the first time that martensitic NiTi with laser treatment could be a promising choice for the next-generation implants given its superior antimicrobial resistance and favourable mechanical properties for loading bearing applications.
    • Finite-time blow-up of a non-local stochastic parabolic problem

      Kavallaris, Nikos I.; Yan, Yubin; University of Chester (Elsevier, 2020-04-13)
      The main aim of the current work is the study of the conditions under which (finite-time) blow-up of a non-local stochastic parabolic problem occurs. We first establish the existence and uniqueness of the local-in-time weak solution for such problem. The first part of the manuscript deals with the investigation of the conditions which guarantee the occurrence of noise-induced blow-up. In the second part we first prove the $C^{1}$-spatial regularity of the solution. Then, based on this regularity result, and using a strong positivity result we derive, for first in the literature of SPDEs, a Hopf's type boundary value point lemma. The preceding results together with Kaplan's eigenfunction method are then employed to provide a (non-local) drift term induced blow-up result. In the last part of the paper, we present a method which provides an upper bound of the probability of (non-local) drift term induced blow-up.
    • A high-order scheme to approximate the Caputo fractional derivative and its application to solve the fractional diffusion wave equation

      Du, Ruilian; Yan, Yubin; Liang, Zongqi; Jimei University; University of Chester (Elsevier, 2018-10-05)
      A new high-order finite difference scheme to approximate the Caputo fractional derivative $\frac{1}{2} \big ( \, _{0}^{C}D^{\alpha}_{t}f(t_{k})+ \, _{0}^{C}D^{\alpha}_{t}f(t_{k-1}) \big ), k=1, 2, \dots, N, $ with the convergence order $O(\Delta t^{4-\alpha}), \, \alpha\in(1,2)$ is obtained when $f^{\prime \prime \prime} (t_{0})=0$, where $\Delta t$ denotes the time step size. Based on this scheme we introduce a finite difference method for solving fractional diffusion wave equation with the convergence order $O(\Delta t^{4-\alpha} + h^2)$, where $h$ denotes the space step size. Numerical examples are given to show that the numerical results are consistent with the theoretical results.