• Bifurcations in approximate solutions of stochastic delay differential equations

      Baker, Christopher T. H.; Ford, Judith M.; Ford, Neville J.; University College Chester/UMIST ; UMIST; University College Chester (World Scientific Publishing Company, 2004)
    • Bifurcations in numerical methods for volterra integro-differential equations

      Edwards, John T.; Ford, Neville J.; Roberts, Jason A. (World Scientific Publishing Company, 2003)
      This article discusses changes in bifurcations in the solutions. It extends the work of Brunner and Lambert and Matthys to consider other bifurcations.
    • A black box at the end of the rainbow: Searching for the perfect preconditioner

      Ford, Judith M.; Chester College of Higher Education (Royal Society, 2003-11-03)
    • Blending low-order stabilised finite element methods: a positivity preserving local projection method for the convection-diffusion equation

      Barrenechea, Gabriel; Burman, Erik; Karakatsani, Fotini; University of Strathclyde; UCL; University of Chester (Elsevier, 2017-01-20)
      In this work we propose a nonlinear blending of two low-order stabilisation mechanisms for the convection–diffusion equation. The motivation for this approach is to preserve monotonicity without sacrificing accuracy for smooth solutions. The approach is to blend a first-order artificial diffusion method, which will be active only in the vicinity of layers and extrema, with an optimal order local projection stabilisation method that will be active on the smooth regions of the solution. We prove existence of discrete solutions, as well as convergence, under appropriate assumptions on the nonlinear terms, and on the exact solution. Numerical examples show that the discrete solution produced by this method remains within the bounds given by the continuous maximum principle, while the layers are not smeared significantly.
    • Bordered Constructions of Self-Dual Codes from Group Rings and New Extremal Binary Self-Dual Codes

      Dougherty, Steven; Gildea, Joe; Kaya, Abidin; Korban, Adrian; Tylyshchak, Alexander; Yildiz, Bahattin; University of Scranton; University of Chester; Sampoerna Academy; Uzhgorod State University; Northern Arizona University (Elsevier, 2019-02-22)
      We introduce a bordered construction over group rings for self-dual codes. We apply the constructions over the binary field and the ring $\F_2+u\F_2$, using groups of orders 9, 15, 21, 25, 27, 33 and 35 to find extremal binary self-dual codes of lengths 20, 32, 40, 44, 52, 56, 64, 68, 88 and best known binary self-dual codes of length 72. In particular we obtain 41 new binary extremal self-dual codes of length 68 from groups of orders 15 and 33 using neighboring and extensions. All the numerical results are tabulated throughout the paper.
    • Boundedness and stability of solutions to difference equations

      Edwards, John T.; Ford, Neville J. (Elsevier Science, 2002-03-01)
      This article discusses the qualitative behaviour of solutions to difference equations, focusing on boundedness and stability of solutions. Examples demonstrate how the use of Lipschintz constants can provide insights into the qualitative behaviour of solutions to some nonlinear problems.
    • Boundness and stability of differential equations

      Edwards, John T.; Ford, Neville J. (Manchester Centre for Computational Mathematics, 2003-05-23)
      This paper discusses the qualitative behaviour of solutions to difference equations, focusing on boundedness and stability of solutions. Examples demonstrate how the use of Lipschintz constants can provide insights into the qualitative behaviour of solutions to some nonlinear problems.
    • Characterising small solutions in delay differential equations through numerical approximations

      Ford, Neville J.; Lunel, Sjoerd M. V. (Manchester Centre for Computational Mathematics, 2003-05-23)
      This paper discusses how the existence of small solutions for delay differential equations can be predicted from the behaviour of the spectrum of the finite dimensional approximations.
    • Characterising small solutions in delay differential equations through numerical approximations

      Ford, Neville J.; Lunel, Sjoerd M. V. (Elsevier Science, 2002-09-25)
      This article discusses how the existence of small solutions for delay differential equations can be predicted from the behaviour of the spectrum of the finite dimensional approximations.
    • Characteristic functions of differential equations with deviating arguments

      Baker, Christopher T. H.; Ford, Neville J.; University of Manchester; University of Chester (Elsevier, 2019-04-24)
      The material here is motivated by the discussion of solutions of linear homogeneous and autonomous differential equations with deviating arguments. If $a, b, c$ and $\{\check{\tau}_\ell\}$ are real and ${\gamma}_\natural$ is real-valued and continuous, an example with these parameters is \begin{equation} u'(t) = \big\{a u(t) + b u(t+\check{\tau}_1) + c u(t+\check{\tau}_2) \big\} { \red +} \int_{\check{\tau}_3}^{\check{\tau}_4} {{\gamma}_\natural}(s) u(t+s) ds \tag{\hbox{$\rd{\star}$}} . \end{equation} A wide class of equations ($\rd{\star}$), or of similar type, can be written in the {\lq\lq}canonical{\rq\rq} form \begin{equation} u'(t) =\DSS \int_{\tau_{\rd \min}}^{\tau_{\rd \max}} u(t+s) d\sigma(s) \quad (t \in \Rset), \hbox{ for a suitable choice of } {\tau_{\rd \min}}, {\tau_{\rd \max}} \tag{\hbox{${\rd \star\star}$}} \end{equation} where $\sigma$ is of bounded variation and the integral is a Riemann-Stieltjes integral. For equations written in the form (${\rd{\star\star}}$), there is a corresponding characteristic function \begin{equation} \chi(\zeta) ):= \zeta - \DSS \int_{\tau_{\rd \min}}^{\tau_{\rd \max}} \exp(\zeta s) d\sigma(s) \quad (\zeta \in \Cset), \tag{\hbox{${\rd{\star\star\star}}$}} \end{equation} %%($ \chi(\zeta) \equiv \chi_\sigma (\zeta)$) whose zeros (if one considers appropriate subsets of equations (${\rd \star\star}$) -- the literature provides additional information on the subsets to which we refer) play a r\^ole in the study of oscillatory or non-oscillatory solutions, or of bounded or unbounded solutions. We show that the related discussion of the zeros of $\chi$ is facilitated by observing and exploiting some simple and fundamental properties of characteristic functions.
    • Combining Kronecker product approximation with discrete wavelet transforms to solve dense, function-related linear systems

      Ford, Judith M.; Tyrtyshnikov, Eugene E.; Chester College of Higher Education ; Russian Academy of Sciences (Society for Industrial and Applied Mathematics, 2003-11)
    • Comparison of numerical methods for fractional differential equations

      Ford, Neville J.; Connolly, Joseph A. (American Institute of Mathematical Sciences/Shanghai Jiao Tong University, 2006-06)
      This article discusses and evaluates the merits of five numerical methods for the solution of single term fractional differential equations.
    • Composite Constructions of Self-Dual Codes from Group Rings and New Extremal Self-Dual Binary Codes of Length 68

      Dougherty, Steven; Gildea, Joe; Kaya, Abidin; Korban, Adrian; University of Scranton; University of Chester; Sampoerna University ; University of Chester (American Institute of Mathematical Sciences, 2019-11-30)
      We describe eight composite constructions from group rings where the orders of the groups are 4 and 8, which are then applied to find self-dual codes of length 16 over F4. These codes have binary images with parameters [32, 16, 8] or [32, 16, 6]. These are lifted to codes over F4 + uF4, to obtain codes with Gray images extremal self-dual binary codes of length 64. Finally, we use a building-up method over F2 + uF2 to obtain new extremal binary self-dual codes of length 68. We construct 11 new codes via the building-up method and 2 new codes by considering possible neighbors.
    • Computational and mathematical modelling of plant species interactions in a harsh climate

      Lumb, Patricia M.; Potter, Jacqueline; Ford, Neville J.; Ekaka-A, Enu-Obari N. (University of Liverpool (University of Chester), 2009-07)
      This thesis will consider the following assumptions which are based on a few insights about the artic climate: (1)the artic climate can be characterised by a growing season called summer and a dormat season called winter (2)in the summer season growing conditions are reasonably favourable and species are more likely to compete for plentiful resources (3)in the winter season there would be no further growth and the plant populations would instead by subjected to fierce weather events such as storms which is more likely to lead to the destruction of some or all of the biomass. Under these assumptions, is it possible to find those change in the environment that might cause mutualism (see section 1.9.2) from competition (see section 1.9.1) to change? The primary aim of this thesis to to provide a prototype simulation of growth of two plant species in the artic that: (1)take account of different models for summer and winter seasons (2)permits the effects of changing climate to be seen on each type of plant species interaction.
    • Computational approaches to parameter estimation and model selection in immunology

      Baker, Christopher T. H.; Bocharov, Gennady; Ford, Judith M.; Lumb, Patricia M.; Norton, Stewart J.; Paul, C. A. H.; Junt, Tobias; Krebs, Philippe; Ludewig, Burkhard (Elsevier, 2005-12-01)
      This article seeks to illustrate the computational implementation of an information-theoretic approach (associated with a maximum likelihood treatment) to modelling in immunology.
    • Computational aspects of time-lag models of Marchuk type that arise in immunology

      Baker, Christopher T. H.; Bocharov, Gennady; University of Chester ; Institute of Numerical Mathematics, Russian Academy of Sciences (de Gryuter, 2005)
      In his book published in English translation in 1983, Marchuk proposed a set of evolutionary equations incorporating delay-differential equations, and the corresponding initial conditions as a model ('Marchuk's model') for infectious diseases. The parameters in this model (and its subsequent extensions) represent scientifically meaningful characteristics. For a given infection, the parameters can be estimated using observational data on the course of the infection. Sensitivity analysis is an important tool for understanding a particular model; this can be viewed as an issue of stability with respect to structural perturbations in the model. Examining the sensitivity of the models based on delay differential equations leads to systems of neutral delay differential equations. Below we formulate a general set of equations for the sensitivity coefficients for models comprising neutral delay differential equations. We discuss computational approaches to the sensitivity of solutions — (i) sensitivity to the choice of model, in particular, to the lag parameter τ > 0 and (ii) sensitivity to the initial function — of dynamical systems with time lag and illustrate them by considering the sensitivity of solutions of time-lag models of Marchuk type.
    • Computational methods for a mathematical model of propagation of nerve impulses in myelinated axons

      Lima, Pedro M.; Ford, Neville J.; Lumb, Patricia M.; CEMAT, IST, Lisbon ; University of Chester ; University of Chester (Elsevier, 2014-06-26)
      This paper is concerned with the approximate solution of a nonlinear mixed type functional differential equation (MTFDE) arising from nerve conduction theory. The equation considered describes conduction in a myelinated nerve axon. We search for a monotone solution of the equation defined in the whole real axis, which tends to given values at ±∞. We introduce new numerical methods for the solution of the equation, analyse their performance, and present and discuss the results of the numerical simulations.
    • Computational modelling with functional differential equations: Identification, selection, and sensitivity

      Baker, Christopher T. H.; Bocharov, Gennady; Paul, C. A. H.; Rihan, F. A. R.; University College Chester ; Institute of Numerical Mathematics, Russian Academy of Sciences ; University of Salford (Elsevier, 2005-05)
      Mathematical models based upon certain types of differential equations, functional differential equations, or systems of such equations, are often employed to represent the dynamics of natural, in particular biological, phenomena. We present some of the principles underlying the choice of a methodology (based on observational data) for the computational identification of, and discrimination between, quantitatively consistent models, using scientifically meaningful parameters. We propose that a computational approach is essential for obtaining meaningful models. For example, it permits the choice of realistic models incorporating a time-lag which is entirely natural from the scientific perspective. The time-lag is a feature that can permit a close reconciliation between models incorporating computed parameter values and observations. Exploiting the link between information theory, maximum likelihood, and weighted least squares, and with distributional assumptions on the data errors, we may construct an appropriate objective function to be minimized computationally. The minimizer is sought over a set of parameters (which may include the time-lag) that define the model. Each evaluation of the objective function requires the computational solution of the parametrized equations defining the model. To select a parametrized model, from amongst a family or hierarchy of possible best-fit models, we are able to employ certain indicators based on information-theoretic criteria. We can evaluate confidence intervals for the parameters, and a sensitivity analysis provides an expression for an information matrix, and feedback on the covariances of the parameters in relation to the best fit. This gives a firm basis for any simplification of the model (e.g., by omitting a parameter).
    • Concerning periodic solutions to non-linear discrete Volterra equations with finite memory

      Baker, Christopher T. H.; Song, Yihong; University of Chester (University of Chester, 2007)
      In this paper we discuss the existence of periodic solutions of discrete (and discretized) non-linear Volterra equations with finite memory. The literature contains a number of results on periodic solutions of non linear Volterra integral equations with finite memory, of a type that arises in biomathematics. The “summation” equations studied here can arise as discrete models in their own right but are (as we demonstrate) of a type that arise from the discretization of such integral equations. Our main results are in two parts: (i) results for discrete equations and (ii) consequences for quadrature methods applied to integral equations. The first set of results are obtained using a variety of fixed point theorems. The second set of results address the preservation of properties of integral equations on discretizing them. An expository style is adopted and examples are given to illustrate the discussion.
    • Constructing Self-Dual Codes from Group Rings and Reverse Circulant Matrices

      Gildea, Joe; Kaya, Abidin; Korban, Adrian; Yildiz, Bahattin; University of Chester; Sampoerna Academy; Northern Arizona University (American Institute of Mathematical Sciences, 2020-01-20)
      In this work, we describe a construction for self-dual codes in which we employ group rings and reverse circulant matrices. By applying the construction directly over different alphabets, and by employing the well known extension and neighbor methods we were able to obtain extremal binary self-dual codes of different lengths of which some have parameters that were not known in the literature before. In particular, we constructed three new codes of length 64, twenty-two new codes of length 68, twelve new codes of length 80 and four new codes of length 92.