• Periodic solutions of discrete Volterra equations

      Baker, Christopher T. H.; Song, Yihong; University College Chester ; Suzhou University (Elsevier, 2004-02-25)
      This article investigates periodic solutions of linear and nonlinear discrete Volterra equations of convolution or non-convolution type with unbounded memory. For linear discrete Volterra equations of convolution type, we establish Fredholm’s alternative theorem and for equations of non-convolution type, and we prove that a unique periodic solution exists for a particular bounded initial function under appropriate conditions. Further, this unique periodic solution attracts all other solutions with bounded initial function. All solutions of linear discrete Volterra equations with bounded initial functions are asymptotically periodic under certain conditions. A condition for periodic solutions in the nonlinear case is established.
    • Perturbation of Volterra difference equations

      Song, Yihong; Baker, Christopher T. H.; Suzhou University ; University College Chester (2004)
      A fixed point theorem is used to investigate nonlinear Volterra difference equations that are perturbed versions of linear equations. Sufficient conditions are established to ensure that the stability properties of linear Volterra difference equations are preserved under perturbation. The existence of asymptotically periodic solutions of perturbed Volterra difference equations is also proved.
    • Pitfalls in fast numerical solvers for fractional differential equations

      Diethelm, Kai; Ford, Judith M.; Ford, Neville J.; Weilbeer, Marc (Elsevier, 2006-02-15)
      This preprint discusses the properties of high order methods for the solution of fractional differential equations. A number of fractional multistep methods are are discussed.
    • A Posteriori Analysis for Space-Time, discontinuous in time Galerkin approximations for parabolic equations in a variable domain

      Antonopoulou, Dimitra; Plexousakis, Michael; University of Chester; University of Crete (ECP sciences, 2019-04-24)
      This paper presents an a posteriori error analysis for the discontinuous in time space-time scheme proposed by Jamet for the heat equation in multi-dimensional, non-cylindrical domains [25]. Using a Cl ement-type interpolant, we prove abstract a posteriori error bounds for the numerical error. Furthermore, in the case of two-dimensional spatial domains we transform the problem into an equivalent one, of parabolic type, with space-time dependent coe cients but posed on a cylindrical domain. We formulate a discontinuous in time space{time scheme and prove a posteriori error bounds of optimal order. The a priori estimates of [19] for general parabolic initial and boundary value problems are used in the derivation of the upper bound. Our lower bound coincides with that of Picasso [36], proposed for adaptive, Runge-Kutta finite element methods for linear parabolic problems. Our theoretical results are verified by numerical experiments.
    • A posteriori error estimates for fully discrete fractional-step ϑ-approximations for parabolic equations

      Karakatsani, Fotini; University of Chester (Oxford University Press, 2015-07-22)
      We derive optimal order a posteriori error estimates for fully discrete approximations of initial and boundary value problems for linear parabolic equations. For the discretisation in time we apply the fractional-step #-scheme and for the discretisation in space the finite element method with finite element spaces that are allowed to change with time.
    • A posteriori error estimates for fully discrete schemes for the time dependent Stokes problem

      Baensch, Eberhard; Karakatsani, Fotini; Makridakis, Charalambos; University of Erlangen; University of Chester; University of Crete; Foundation for Research & Technology, Greece; University of Sussex (Springer, 2018-05-02)
      This work is devoted to a posteriori error analysis of fully discrete finite element approximations to the time dependent Stokes system. The space discretization is based on popular stable spaces, including Crouzeix–Raviart and Taylor–Hood finite element methods. Implicit Euler is applied for the time discretization. The finite element spaces are allowed to change with time steps and the projection steps include alternatives that is hoped to cope with possible numerical artifices and the loss of the discrete incompressibility of the schemes. The final estimates are of optimal order in L∞(L2) for the velocity error.
    • Predicting changes in dynamical behaviour in solutions to stochastic delay differential equations

      Norton, Stewart J.; Ford, Neville J.; University of Chester (AIMS Press, 2006-06)
      This article considers numerical approximations to parameter-dependent linear and logistic stochastic delay differential equations with multiplicative noise. The aim of the investigation is to explore the parameter values at which there are changes in qualitative behaviour of the solutions. One may use a phenomenological approach but a more analytical approach would be attractive. A possible tool in this analysis is the calculation of the approximate local Lyapunov exponents. In this paper we show that the phenomenological approach can be used effectively to estimate bifurcation parameters for deterministic linear equations but one needs to use the dynamical approach for stochastic equations.
    • A predictor corrector approach for the numerical solution of fractional differential equations

      Diethelm, Kai; Ford, Neville J.; Freed, Alan D. (Springer, 2002-07)
      This article discusses an Adams-type predictor-corrector method for the numerical solution of fractional differential equations.
    • Quadruple Bordered Constructions of Self-Dual Codes from Group Rings

      Dougherty, Steven; Gildea, Joe; Kaya, Abidin; University of Scranton; University of Chester; Sampoerna University (Springer Verlag, 2019-07-05)
      In this paper, we introduce a new bordered construction for self-dual codes using group rings. We consider constructions over the binary field, the family of rings Rk and the ring F4 + uF4. We use groups of order 4, 12 and 20. We construct some extremal self-dual codes and non-extremal self-dual codes of length 16, 32, 48, 64 and 68. In particular, we construct 33 new extremal self-dual codes of length 68.
    • Qualitative behaviour of numerical approximations to Volterra integro-differential equations

      Song, Yihong; Baker, Christopher T. H.; Suzhou University ; University College Chester (Elsevier, 2004-11-01)
      This article investigates the qualitative behaviour of numerical approximations to a nonlinear Volterra integro-differential equation with unbounded delay.
    • Quenching solutions of a stochastic parabolic problem arising in electrostatic MEMS control

      Kavallaris, Nikos I.; University of Chester (Wiley, 2016-09-15)
      In the current paper, we consider a stochastic parabolic equation which actually serves as a mathematical model describing the operation of an electrostatic actuated micro-electro-mechanical system (MEMS). We first present the derivation of the mathematical model. Then after establishing the local well-posedeness of the problem we investigate under which circumstances a {\it finite-time quenching} for this SPDE, corresponding to the mechanical phenomenon of {\it touching down}, occurs. For that purpose the Kaplan's eigenfunction method adapted in the context of SPDES is employed.
    • Recombination: Multiply infected spleen cells in HIV patients

      Jung, Andreas; Maier, Reinhard; Vartanian, Jean-Pierre; Bocharov, Gennady; Jung, Volker; Fischer, Ulrike; Meese, Eckart; Wain-Hobson, Simon; Meyerhans, Andreas (Nature Publishing Group, 2002-07-11)
      The genome of the human immunodeficiency virus is highly prone to recombination, although it is not obvious whether recombinants arise infrequently or whether they are constantly being spawned but escape identification because of the massive and rapid turnover of virus particles. Here we use fluorescence in situ hybridization to estimate the number of proviruses harboured by individual splenocytes from two HIV patients, and determine the extent of recombination by sequencing amplified DNA from these cells. We find an average of three or four proviruses per cell and evidence for huge numbers of recombinants and extensive genetic variation. Although this creates problems for phylogenetic analyses, which ignore recombination effects, the intracellular variation may help to broaden immune recognition.
    • Self-Dual Codes using Bisymmetric Matrices and Group Rings

      Gildea, Joe; Kaya, Abidin; Korban, Adrian; Tylyshchak, Alexander; University of Chester ; Sampoerna University ; University of Chester: Uzhgorod National University
      In this work, we describe a construction in which we combine together the idea of a bisymmetric matrix and group rings. Applying this construction over the ring F4 + uF4 together with the well known extension and neighbour methods, we construct new self-dual codes of length 68: In particular, we find 41 new codes of length 68 that were not known in the literature before.
    • The sharp interface limit for the stochastic Cahn-Hilliard Equation

      Antonopoulou, Dimitra; Bloemker, Dirk; Karali, Georgia D.; Universiy of Chester (IMS Journals, 2018-02-19)
      We study the two and three dimensional stochastic Cahn-Hilliard equation in the sharp interface limit, where the positive parameter \eps tends to zero, which measures the width of transition layers generated during phase separation. We also couple the noise strength to this parameter. Using formal asymptotic expansions, we identify the limit. In the right scaling we indicate that the solutions of stochastic Cahn-Hilliard converge to a solution of a Hele-Shaw problem with stochastic forcing. In the case when the noise is sufficiently small, we rigorously prove that the limit is a deterministic Hele-Shaw problem. Finally, we discuss which estimates are necessary in order to extend the rigorous result to larger noise strength.
    • Simulation of grain boundary diffusion creep: Analysis of some new numerical techniques

      Ford, Judith M.; Ford, Neville J.; Wheeler, John (Royal Society, 2004)
    • Solution map methods for stability analysis of linear and nonlinear Volterra difference equations

      Edwards, John T.; Ford, Neville J. (Institute of Applied Science & Computations, 2004)
    • Solution of a singular integral equation by a split-interval method

      Diogo, Teresa; Ford, Neville J.; Lima, Pedro M.; Thomas, Sophy M. (2007)
      This article discusses a new numerical method for the solution of a singular integral equation of Volterra type that has an infinite class of solutions. The split-interval method is discussed and examples demonstrate its effectiveness.
    • Some time stepping methods for fractional diffusion problems with nonsmooth data

      Yang, Yan; Yan, Yubin; Ford, Neville J.; Lvliang University; University of Chester (De Gruyter, 2017-09-02)
      We consider error estimates for some time stepping methods for solving fractional diffusion problems with nonsmooth data in both homogeneous and inhomogeneous cases. McLean and Mustapha \cite{mclmus} (Time-stepping error bounds for fractional diffusion problems with non-smooth initial data, Journal of Computational Physics, 293(2015), 201-217) established an $O(k)$ convergence rate for the piecewise constant discontinuous Galerkin method with nonsmooth initial data for the homogeneous problem when the linear operator $A$ is assumed to be self-adjoint, positive semidefinite and densely defined in a suitable Hilbert space, where $k$ denotes the time step size. In this paper, we approximate the Riemann-Liouville fractional derivative by Diethelm's method (or $L1$ scheme) and obtain the same time discretisation scheme as in McLean and Mustapha \cite{mclmus}. We first prove that this scheme has also convergence rate $O(k)$ with nonsmooth initial data for the homogeneous problem when $A$ is a closed, densely defined linear operator satisfying some certain resolvent estimates. We then introduce a new time discretization scheme for the homogeneous problem based on the convolution quadrature and prove that the convergence rate of this new scheme is $O(k^{1+ \alpha}), 0<\alpha <1 $ with the nonsmooth initial data. Using this new time discretization scheme for the homogeneous problem, we define a time stepping method for the inhomogeneous problem and prove that the convergence rate of this method is $O(k^{1+ \alpha}), 0<\alpha <1 $ with the nonsmooth data. Numerical examples are given to show that the numerical results are consistent with the theoretical results.
    • Space-Time Discontinuous Galerkin Methods for the '\eps'-dependent Stochastic Allen-Cahn Equation with mild noise

      Antonopoulou, Dimitra; Department of Mathematics, University of Chester, UK (Oxford University Press, 2019-04-08)
      We consider the $\eps$-dependent stochastic Allen-Cahn equation with mild space- time noise posed on a bounded domain of R^2. The positive parameter $\eps$ is a measure for the inner layers width that are generated during evolution. This equation, when the noise depends only on time, has been proposed by Funaki in [15]. The noise although smooth becomes white on the sharp interface limit as $\eps$ tends to zero. We construct a nonlinear dG scheme with space-time finite elements of general type which are discontinuous in time. Existence of a unique discrete solution is proven by application of Brouwer's Theorem. We first derive abstract error estimates and then for the case of piece-wise polynomial finite elements we prove an error in expectation of optimal order. All the appearing constants are estimated in terms of the parameter $\eps$. Finally, we present a linear approximation of the nonlinear scheme for which we prove existence of solution and optimal error in expectation in piece-wise linear finite element spaces. The novelty of this work is based on the use of a finite element formulation in space and in time in 2+1-dimensional subdomains for a nonlinear parabolic problem. In addition, this problem involves noise. These type of schemes avoid any Runge-Kutta type discretization for the evolutionary variable and seem to be very effective when applied to equations of such a difficulty.
    • Stability analysis of a continuous model of mutualism with delay dynamics

      Roberts, Jason A.; Joharjee, Najwa G.; University of Chester; King Abdul Aziz University (Hikari Ltd, 2016-05-31)
      In this paper we introduce delay dynamics to a coupled system of ordinary differential equations which represent two interacting species exhibiting facultative mutualistic behaviour. The delays are represen- tative of the beneficial effects of the indirect, interspecies interactions not being realised immediately. We show that the system with delay possesses a continuous solution, which is unique. Furthermore we show that, for suitably-behaved, positive initial functions that this unique solution is bounded and remains positive, i.e. both of the components representing the two species remain greater than zero. We show that the system has a positive equilibrium point and prove that this point is asymptotically stable for positive solutions and that this stability property is not conditional upon the delays.