• Halanay-type theory in the context of evolutionary equations with time-lag

      Baker, Christopher T. H.; University of Chester (University of Chester, 2009)
      We consider extensions and modifications of a theory due to Halanay, and the context in which such results may be applied. Our emphasis is on a mathematical framework for Halanay-type analysis of problems with time lag and simulations using discrete versions or numerical formulae. We present selected (linear and nonlinear, discrete and continuous) results of Halanay type that can be used in the study of systems of evolutionary equations with various types of delayed argument, and the relevance and application of our results is illustrated, by reference to delay-differential equations, difference equations, and methods.
    • Haptic feedback from human tissues of various stiffness and homogeneity.

      Vaughan, Neil; Dubey, Venketesh N.; Wee, Michael Y. K.; Isaacs, Richard; Bournemouth University; Poole Hospital NHS Foundation Trust (Techno-Press, 2014-07-01)
      This work presents methods for haptic modelling of soft and hard tissue with varying stiffness. The model provides visualization of deformation and calculates force feedback during simulated epidural needle insertion. A spring-mass-damper (SMD) network is configured from magnetic resonance image (MRI) slices of patient’s lumbar region to represent varying stiffness throughout tissue structure. Reaction force is calculated from the SMD network and a haptic device is configured to produce a needle insertion simulation. The user can feel the changing forces as the needle is inserted through tissue layers and ligaments. Methods for calculating the force feedback at various depths of needle insertion are presented. Voxelization is used to fill ligament surface meshes with spring mass damper assemblies for simulated needle insertion into soft and hard tissues. Modelled vertebrae cannot be pierced by the needle. Graphs were produced during simulated needle insertions to compare the applied force to haptic reaction force. Preliminary saline pressure measurements during Tuohy epidural needle insertion are also used as a basis for forces generated in the simulation.
    • High order algorithms for numerical solution of fractional differential equations

      Asl, Mohammad Shahbazi; Javidi, Mohammad; Yan, Yubin; University of Chester; University of Tabriz
      In this paper, two novel high order numerical algorithms are proposed for solving fractional differential equations where the fractional derivative is considered in the Caputo sense. The total domain is discretized into a set of small subdomains and then the unknown functions are approximated using the piecewise Lagrange interpolation polynomial of degree three and degree four. The detailed error analysis is presented, and it is analytically proven that the proposed algorithms are of orders 4 and 5. The stability of the algorithms is rigorously established and the stability region is also achieved. Numerical examples are provided to check the theoretical results and illustrate the efficiency and applicability of the novel algorithms.
    • A high order numerical method for solving nonlinear fractional differential equation with non-uniform meshes

      Fan, Lili; Yan, Yubin; University of Chester; Lvliang University (Springer Link, 2019-01-18)
      We introduce a high-order numerical method for solving nonlinear fractional differential equation with non-uniform meshes. We first transform the fractional nonlinear differential equation into the equivalent Volterra integral equation. Then we approximate the integral by using the quadratic interpolation polynomials. On the first subinterval $[t_{0}, t_{1}]$, we approximate the integral with the quadratic interpolation polynomials defined on the nodes $t_{0}, t_{1}, t_{2}$ and in the other subinterval $[t_{j}, t_{j+1}], j=1, 2, \dots N-1$, we approximate the integral with the quadratic interpolation polynomials defined on the nodes $t_{j-1}, t_{j}, t_{j+1}$. A high-order numerical method is obtained. Then we apply this numerical method with the non-uniform meshes with the step size $\tau_{j}= t_{j+1}- t_{j}= (j+1) \mu$ where $\mu= \frac{2T}{N (N+1)}$. Numerical results show that this method with the non-uniform meshes has the higher convergence order than the standard numerical methods obtained by using the rectangle and the trapzoid rules with the same non-uniform meshes.
    • High performing AgNW transparent conducting electrodes with a sheet resistance of 2.5 Ω Sq−1 based upon a roll-to-roll compatible post-processing technique

      Kumar, Dinesh; Stoichkov, Vasil; Brousseau, Emmanuel; Smith, Graham C.; Kettle, Jeff; Bangor University; University of Chester; Cardiff University (Royal Society of Chemistry, 2019-02-12)
      The report of transparent and conducting silver nanowires (AgNWs) that produce remarkable electrical performance, surface planarity and environmental stability is given. This research presents an innovative process that relies on three sequential steps, which are roll-to-roll (R2R) compatible; thermal embossing, infrared sintering and plasma treatment. This process leads to the demonstration of a conductive film with a sheet resistance of 2.5Ω/sq and high transmittance, thus demonstrating the highest reported figure-of-merit in AgNWs to date (FoM = 933). A further benefit of the process is that the surface roughness is substantially reduced compared to traditional AgNW processing techniques. Finally, consideration of the long-term stability is given by developing an accelerated life test process that simultaneously stresses the applied bias and temperature. Regression line fitting shows that a ∼150-times improvement in stability is achieved at ‘normal operational conditions’ when compared to traditionally deposited AgNW films. X-ray photoelectron spectroscopy (XPS) is used to understand the root cause of the improvement in long-term stability, which is related to reduced chemcial changes in the AgNWs.
    • High speed CO2 laser surface modification of iron/cobalt co-doped boroaluminosilicate glass

      Hodgson, Simon D.; Waugh, David G.; Gillett, Alice R.; Lawrence, Jonathan; University of Chester (IOP Publishing, 2016-06-10)
      A preliminary study into the impact of high speed laser processing on the surface of iron and cobalt co-doped glass substrates using a 60 W continuous wave (cw) CO2 laser. Two types of processing, termed fill-processing and line-processing, were trialled. In fill-processed samples the surface roughness of the glass was found to increase linearly with laser power from an Sa value of 20.8 nm–2.1 μm at a processing power of 54 W. With line processing, a more exponential-like increase was observed with a roughness of 4 μm at 54 W. The change in surface properties of the glass, such as gloss and wettability, have also been measured. The contact angle of water was found to increase after laser processing by up to 64°. The surface gloss was varied between 45 and 100 gloss units (GUs).
    • High temperature performance of a piezoelectric micro cantilever for vibration energy harvesting

      Arroyo, Emmanuelle; Jia, Yu; Du, Sijun; Chen, Shao-Tuan; Seshia, Ashwin A. (IOP Publishing, 2016-12-06)
      Energy harvesters withstanding high temperatures could provide potentially unlimited energy to sensor nodes placed in harsh environments, where manual maintenance is difficult and costly. Experimental results on a classical microcantilever show a 67% drop of the maximum power when the temperature is increased up to 160 °C. This decrease is investigated using a lumped-parameters model which takes into account variations in material parameters with temperature, damping increase and thermal stresses induced by mismatched thermal coefficients in a composite cantilever. The model allows a description of the maximum power evolution as a function of temperature and input acceleration. Simulation results further show that an increase in damping and the apparition of thermal stresses are contributing to the power drop at 59% and 13% respectively.
    • High-Order Numerical Methods for Solving Time Fractional Partial Differential Equations

      Li, Zhiqiang; Liang, Zongqi; Yan, Yubin; Luliang University, P. R. China, Jimei University, P. R. China, University of Chester, UK (Springer Link, 2016-11-15)
      In this paper we introduce a new numerical method for solving time fractional partial differential equation. The time discretization is based on Diethelm’s method where the Hadamard finite-part integral is approximated by using the piecewise quadratic interpolation polynomials. The space discretization is based on the standard finite element method. The error estimates with the convergence order O(τ^(3−α) +h^2 ),0
    • 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.
    • A higher order numerical method for time fractional partial differential equations with nonsmooth data

      Xing, Yanyuan; Yan, Yubin; Lvliang University; University of Chester (Elsevier, 2018-01-02)
      Gao et al. (2014) introduced a numerical scheme to approximate the Caputo fractional derivative with the convergence rate $O(k^{3-\alpha}), 0< \alpha <1$ by directly approximating the integer-order derivative with some finite difference quotients in the definition of the Caputo fractional derivative, see also Lv and Xu (2016), where $k$ is the time step size. Under the assumption that the solution of the time fractional partial differential equation is sufficiently smooth, Lv and Xu (2016) proved by using energy method that the corresponding numerical method for solving time fractional partial differential equation has the convergence rate $O(k^{3-\alpha}), 0< \alpha <1$ uniformly with respect to the time variable $t$. However, in general the solution of the time fractional partial differential equation has low regularity and in this case the numerical method fails to have the convergence rate $O(k^{3- \alpha}), 0 < \alpha <1$ uniformly with respect to the time variable $t$. In this paper, we first obtain a similar approximation scheme to the Riemann-Liouville fractional derivative with the convergence rate $O(k^{3- \alpha}), 0 < \alpha <1$ as in Gao \et \cite{gaosunzha} (2014) by approximating the Hadamard finite-part integral with the piecewise quadratic interpolation polynomials. Based on this scheme, we introduce a time discretization scheme to approximate the time fractional partial differential equation and show by using Laplace transform methods that the time discretization scheme has the convergence rate $O(k^{3- \alpha}), 0 < \alpha <1$ for any fixed $t_{n}>0$ for smooth and nonsmooth data in both homogeneous and inhomogeneous cases. Numerical examples are given to show that the theoretical results are consistent with the numerical results.
    • Higher Order Numerical Methods for Fractional Order Differential Equations

      Pal, Kamal (University of Chester, 2015-08)
      This thesis explores higher order numerical methods for solving fractional differential equations.
    • Higher order numerical methods for solving fractional differential equations

      Yan, Yubin; Pal, Kamal; Ford, Neville J.; University of Chester (Springer, 2013-10-05)
      In this paper we introduce higher order numerical methods for solving fractional differential equations. We use two approaches to this problem. The first approach is based on a direct discretisation of the fractional differential operator: we obtain a numerical method for solving a linear fractional differential equation with order 0 < α < 1. The order of convergence of the numerical method is O(h^(3−α)). Our second approach is based on discretisation of the integral form of the fractional differential equation and we obtain a fractional Adams-type method for a nonlinear fractional differential equation of any order α >0. The order of convergence of the numerical method is O(h^3) for α ≥ 1 and O(h^(1+2α)) for 0 < α ≤ 1 for sufficiently smooth solutions. Numerical examples are given to show that the numerical results are consistent with the theoretical results.
    • Higher Order Time Stepping Methods for Subdiffusion Problems Based on Weighted and Shifted Grünwald–Letnikov Formulae with Nonsmooth Data

      Yan, yubin; Wang, Yanyong; Yan, Yuyuan; Pani, Amiya K.; University of Chester, Lvliang University, Jimei University, Indian Institute of Technology Bombay (Springer Link, 2020-05-19)
      Two higher order time stepping methods for solving subdiffusion problems are studied in this paper. The Caputo time fractional derivatives are approximated by using the weighted and shifted Gr\"unwald-Letnikov formulae introduced in Tian et al. [Math. Comp. 84 (2015), pp. 2703-2727]. After correcting a few starting steps, the proposed time stepping methods have the optimal convergence orders $O(k^2)$ and $ O(k^3)$, respectively for any fixed time $t$ for both smooth and nonsmooth data. The error estimates are proved by directly bounding the approximation errors of the kernel functions. Moreover, we also present briefly the applicabilities of our time stepping schemes to various other fractional evolution equations. Finally, some numerical examples are given to show that the numerical results are consistent with the proven theoretical results.
    • High‐order ADI orthogonal spline collocation method for a new 2D fractional integro‐differential problem

      Yan, Yubin; Qiao, Leijie; Xu, Da; University of Chester, UK; Guangdong University of Technology, PR. China; Hunan Normal University, P. R. China (John Wiley & Sons Ltd, 2020-02-05)
      We use the generalized L1 approximation for the Caputo fractional deriva-tive, the second-order fractional quadrature rule approximation for the inte-gral term, and a classical Crank-Nicolson alternating direction implicit (ADI)scheme for the time discretization of a new two-dimensional (2D) fractionalintegro-differential equation, in combination with a space discretization by anarbitrary-order orthogonal spline collocation (OSC) method. The stability of aCrank-Nicolson ADI OSC scheme is rigourously established, and error estimateis also derived. Finally, some numerical tests are given
    • How do numerical methods perform for delay differential equations undergoing a Hopf bifurcation?

      Ford, Neville J.; Wulf, Volker (Manchester Centre for Computational Mathematics, 1999-09-30)
      This paper discusses the numerical solution of delay differential equations undergoing a Hopf birufication. Three distinct and complementary approaches to the analysis are presented.
    • How effective is Ant Colony Optimisation at Robot Path Planning

      Wolfenden, A.; Vaughan, Neil; University of Chester (The Society for the Study of Artificial Intelligence and Simulation for Behaviour (AISB), 2018-04-06)
      This project involves investigation of the problem robot path planning using ant colony optimisation heuristics to construct the quickest path from the starting point to the end. The project has developed a simulation that successfully simulates as well as demonstrates visually through a graphical user interface, robot path planning using ant colony optimisation. The simulation shows an ability to traverse an unknown environment from a start point to an end and successfully construct a route for others to follow both when the terrain is dynamic and static
    • HOx cycling during the Cyprus Photochemistry Experiment

      Mallik, Chinmay; Harder, Hartwig; Tomsche, Laura; Novelli, Anna; Martinez, Monica; Meusel, Hannah; Su, Hang; Kessel, Stephan; Bourtsoukidis, Efstratios; Sauvage, Carina; et al. (Asia Oceania Geosciences Society, 2016-08-31)
      Abstract from attached
    • Hybrid Heat Pump for Micro Heat Network

      Counsell, John M.; Khalid, Yousaf; Stewart, M.; University of Chester (World Academy of Science, Engineering and Technology (WASET), 2017-10-17)
      Achieving nearly zero carbon heating continues to be identified by UK government analysis as an important feature of any lowest cost pathway to reducing greenhouse gas emissions. Heat currently accounts for 48% of UK energy consumption and approximately one third of UK’s greenhouse gas emissions. Heat Networks are being promoted by UK investment policies as one means of supporting hybrid heat pump based solutions. To this effect the RISE (Renewable Integrated and Sustainable Electric) heating system project is investigating how an all-electric heating sourceshybrid configuration could play a key role in long-term decarbonisation of heat. For the purposes of this study, hybrid systems are defined as systems combining the technologies of an electric driven air source heat pump, electric powered thermal storage, a thermal vessel and micro-heat network as an integrated system. This hybrid strategy allows for the system to store up energy during periods of low electricity demand from the national grid, turning it into a dynamic supply of low cost heat which is utilized only when required. Currently a prototype of such a system is being tested in a modern house integrated with advanced controls and sensors. This paper presents the virtual performance analysis of the system and its design for a micro heat network with multiple dwelling units. The results show that the RISE system is controllable and can reduce carbon emissions whilst being competitive in running costs with a conventional gas boiler heating system.
    • Identification of the initial function for discretized delay differential equations

      Baker, Christopher T. H.; Parmuzin, Evgeny I.; University College Chester ; Institute of Numerical Mathematics, Russian Academy of Sciences (Elsevier, 2005-09-15)
      In the present work, we analyze a discrete analogue for the problem of the identification of the initial function for a delay differential equation (DDE) discussed by Baker and Parmuzin in 2004. The basic problem consists of finding an initial function that gives rise to a solution of a discretized DDE, which is a close fit to observed data.
    • Identification of the initial function for nonlinear delay differential equations

      Baker, Christopher T. H.; Parmuzin, Evgeny I.; University College Chester ; Institute of Numerical Mathematics, Russian Academy of Sciences (de Gruyter, 2005)
      We consider a 'data assimilation problem' for nonlinear delay differential equations. Our problem is to find an initial function that gives rise to a solution of a given nonlinear delay differential equation, which is a close fit to observed data. A role for adjoint equations and fundamental solutions in the nonlinear case is established. A 'pseudo-Newton' method is presented. Our results extend those given by the authors in [(C. T. H. Baker and E. I. Parmuzin, Identification of the initial function for delay differential equation: Part I: The continuous problem & an integral equation analysis. NA Report No. 431, MCCM, Manchester, England, 2004.), (C. T. H. Baker and E. I. Parmuzin, Analysis via integral equations of an identification problem for delay differential equations. J. Int. Equations Appl. (2004) 16, 111–135.)] for the case of linear delay differential equations.