• Neutral delay differential equations in the modelling of cell growth

      Baker, Christopher T. H.; Bocharov, Gennady; Rihan, F. A. R.; University of Chester (University of Chester, 2008)
      In this contribution, we indicate (and illustrate by example) roles that may be played by neutral delay differential equations in modelling of certain cell growth phenomena that display a time lag in reacting to events. We explore, in this connection, questions involving the sensitivity analysis of models and related mathematical theory; we provide some associated numerical results.
    • New binary self-dual codes via a generalization of the four circulant construction

      Gildea, Joe; Kaya, Abidin; Yildiz, Bahattin; University of Chester ; Sampoerna University ; Northern Arizona University (Croatian Mathematical Society, 2020-05-31)
      In this work, we generalize the four circulant construction for self-dual codes. By applying the constructions over the alphabets $\mathbb{F}_2$, $\mathbb{F}_2+u\mathbb{F}_2$, $\mathbb{F}_4+u\mathbb{F}_4$, we were able to obtain extremal binary self-dual codes of lengths 40, 64 including new extremal binary self-dual codes of length 68. More precisely, 43 new extremal binary self-dual codes of length 68, with rare new parameters have been constructed.
    • New Extremal binary self-dual codes of length 68 from generalized neighbors

      Gildea, Joe; Kaya, Abidin; Korban, Adrian; Yildiz, Bahattin; University of Chester; Sampoerna University; Northern Arizona University
      In this work, we use the concept of distance between self-dual codes, which generalizes the concept of a neighbor for self-dual codes. Using the $k$-neighbors, we are able to construct extremal binary self-dual codes of length 68 with new weight enumerators. We construct 143 extremal binary self-dual codes of length 68 with new weight enumerators including 42 codes with $\gamma=8$ in their $W_{68,2}$ and 40 with $\gamma=9$ in their $W_{68,2}$. These examples are the first in the literature for these $\gamma$ values. This completes the theoretical list of possible values for $\gamma$ in $W_{68,2}$.
    • New Extremal Self-Dual Binary Codes of Length 68 via Composite Construction, F2 + uF2 Lifts, Extensions and Neighbors

      Dougherty, Steven; Gildea, Joe; Korban, Adrian; Kaya, Abidin; University of Scranton; University of Chester; University of Chester; Sampoerna Academy; (Inderscience, 2020-02-29)
      We describe a composite construction from group rings where the groups have orders 16 and 8. This construction is then applied to find the extremal binary self-dual codes with parameters [32, 16, 8] or [32, 16, 6]. We also extend this composite construction by expanding the search field which enables us to find more extremal binary self-dual codes with the above parameters and with different orders of automorphism groups. These codes are then lifted to F2 + uF2, to obtain extremal binary images of codes of length 64. Finally, we use the extension method and neighbor construction to obtain new extremal binary self-dual codes of length 68. As a result, we obtain 28 new codes of length 68 which were not known in the literature before.
    • New Self-Dual and Formally Self-Dual Codes from Group Ring Constructions

      Dougherty, Steven; Gildea, Joe; Kaya, Abidin; Yildiz, Bahattin; University of Scranton; University of Chester; Sampoerna Academy; University of Chester; Northern Arizona University (American Institute of Mathematical Sciences, 2019-08-31)
      In this work, we study construction methods for self-dual and formally self-dual codes from group rings, arising from the cyclic group, the dihedral group, the dicyclic group and the semi-dihedral group. Using these constructions over the rings $_F2 +uF_2$ and $F_4 + uF_4$, we obtain 9 new extremal binary self-dual codes of length 68 and 25 even formally self-dual codes with parameters [72,36,14].
    • New Self-Dual Codes of Length 68 from a 2 × 2 Block Matrix Construction and Group Rings

      Bortos, Maria; Gildea, Joe; Kaya, Abidin; Korban, Adrian; Tylyshchak, Alexander; Uzhgorod National University, University of Chester, Harmony School of Technology, University of Chester, Uzhgorod National University
      Many generator matrices for constructing extremal binary self-dual codes of different lengths have the form G = (In | A); where In is the n x n identity matrix and A is the n x n matrix fully determined by the first row. In this work, we define a generator matrix in which A is a block matrix, where the blocks come from group rings and also, A is not fully determined by the elements appearing in the first row. By applying our construction over F2 +uF2 and by employing the extension method for codes, we were able to construct new extremal binary self-dual codes of length 68. Additionally, by employing a generalised neighbour method to the codes obtained, we were able to con- struct many new binary self-dual [68,34,12]-codes with the rare parameters $\gamma = 7$; $8$ and $9$ in $W_{68,2}$: In particular, we find 92 new binary self-dual [68,34,12]-codes.
    • Noise induced changes to dynamic behaviour of stochastic delay differential equations

      Ford, Neville J.; Norton, Stewart J. (University of Liverpool (University of Chester)University of Chester, 2008-02)
      This thesis is concerned with changes in the behaviour of solutions to parameter-dependent stochastic delay differential equations.
    • Noise-induced changes to the behaviour of semi-implicit Euler methods for stochastic delay differential equations undergoing bifurcation

      Ford, Neville J.; Norton, Stewart J.; University of Chester (Elsevier, 2009-07-15)
      This article discusses estimating parameter values at which bifurcations occur in stochastic delay differential equations. After a brief review of bifurcation, we employ a numerical approach and consider how bifurcation values are influenced by the choice of numerical scheme and the step length and by the level of white noise present in the equation. In this paper we provide a formulaic relationship between the estimated bifurcation value, the level of noise, the choice of numerical scheme and the step length. We are able to show that in the presence of noise there may be some loss of order in the accuracy of the approximation to the true bifurcation value compared to the use of the same approach in the absence of noise.
    • Noise-induced changes to the bifurcation behaviour of semi-implicit Euler methods for stochastic delay differential equations

      Ford, Neville J.; Norton, Stewart J.; University of Chester (University of Chester, 2007)
      We are concerned with estimating parameter values at which bifurcations occur in stochastic delay differential equations. After a brief review of bifurcation, we employ a numerical approach and consider how bifurcation values are influenced by the choice of numerical scheme and the step length and by the level of white noise present in the equation. In this paper we provide a formulaic relationship between the estimated bifurcation value, the level of noise, the choice of numerical scheme and the step length. We are able to show that in the presence of noise there maybe some loss of order in the accuracy of the approximation to the true bifurcation value compared to the use of the same approach in the absence of noise.
    • Non-Local Partial Differential Equations for Engineering and Biology: Mathematical Modeling and Analysis

      Kavallaris, Nikos I.; Suzuki, Takashi; University of Chester; Osaka University (Springer, 2017-12-14)
      This book presents new developments in non-local mathematical modeling and mathematical analysis on the behavior of solutions with novel technical tools. Theoretical backgrounds in mechanics, thermo-dynamics, game theory, and theoretical biology are examined in details. It starts off with a review and summary of the basic ideas of mathematical modeling frequently used in the sciences and engineering. The authors then employ a number of models in bio-science and material science to demonstrate applications, and provide recent advanced studies, both on deterministic non-local partial differential equations and on some of their stochastic counterparts used in engineering. Mathematical models applied in engineering, chemistry, and biology are subject to conservation laws. For instance, decrease or increase in thermodynamic quantities and non-local partial differential equations, associated with the conserved physical quantities as parameters. These present novel mathematical objects are engaged with rich mathematical structures, in accordance with the interactions between species or individuals, self-organization, pattern formation, hysteresis. These models are based on various laws of physics, such as mechanics of continuum, electro-magnetic theory, and thermodynamics. This is why many areas of mathematics, calculus of variation, dynamical systems, integrable systems, blow-up analysis, and energy methods are indispensable in understanding and analyzing these phenomena. This book aims for researchers and upper grades students in mathematics, engineering, physics, economics, and biology.
    • Nonpolynomial approximation of solutions to delay fractional differential equations

      Ford, Neville J.; Morgado, Maria L.; Rebelo, Magda S.; University of Chester ; University of Tras-os-Montes e Alto Douro ; Univeridade Nova de Lisboa (University of Oviedo, 2013)
    • A nonpolynomial collocation method for fractional terminal value problems

      Ford, Neville J.; Morgado, Maria L.; Rebelo, Magda S.; University of Chester ; UTAD, Portugal; Universidade de Nova Lisboa, Portugal (Elsevier, 2014-06-14)
      In this paper we propose a non-polynomial collocation method for solving a class of terminal (or boundary) value problems for differential equations of fractional order α, 0 < α < 1. The approach used is based on the equivalence between a problem of this type and a Fredholm integral equation of a particular form. Taking into account the asymptotic behaviour of the solution of this problem, we propose a non-polynomial collocation method on a uniform mesh. We study the order of convergence of the proposed algorithm and a result on optimal order of convergence is obtained. In order to illustrate the theoretical results and the performance of the method we present several numerical examples.
    • A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes

      Yanzhi, Liu; Roberts, Jason A.; Yan, Yubin; Lvliang University; University of Chester (Taylor & Francis, 2017-10-09)
      We consider finite difference methods for solving nonlinear fractional differential equations in the Caputo fractional derivative sense with non-uniform meshes. Under the assumption that the Caputo derivative of the solution of the fractional differential equation is suitably smooth, Li et al. \lq \lq Finite difference methods with non-uniform meshes for nonlinear fractional differential equations\rq\rq, Journal of Computational Physics, 316(2016), 614-631, obtained the error estimates of finite difference methods with non-uniform meshes. However the Caputo derivative of the solution of the fractional differential equation in general has a weak singularity near the initial time. In this paper, we obtain the error estimates of finite difference methods with non-uniform meshes when the Caputo fractional derivative of the solution of the fractional differential equation has lower smoothness. The convergence result shows clearly how the regularity of the Caputo fractional derivative of the solution affect the order of convergence of the finite difference methods. Numerical results are presented that confirm the sharpness of the error analysis.
    • A Note on the Well-Posedness of Terminal Value Problems for Fractional Differential Equations.

      Diethelm, Kai; Ford, Neville J.; GNS & TU-BS, Braunschweig, Germany; Univerity of Chester (Journal of Integral Equations and Applications, Rocky Mountains Mathematics Consortium, 2018-11-08)
      This note is intended to clarify some im- portant points about the well-posedness of terminal value problems for fractional di erential equations. It follows the recent publication of a paper by Cong and Tuan in this jour- nal in which a counter-example calls into question the earlier results in a paper by this note's authors. Here, we show in the light of these new insights that a wide class of terminal value problems of fractional differential equations is well- posed and we identify those cases where the well-posedness question must be regarded as open.
    • A novel high-order algorithm for the numerical estimation of fractional differential equations

      Asl, Mohammad S.; Javidi, Mohammad; Yan, Yubin; University of Tabriz; University of Chester (Elsevier, 2018-01-09)
      This paper uses polynomial interpolation to design a novel high-order algorithm for the numerical estimation of fractional differential equations. The Riemann-Liouville fractional derivative is expressed by using the Hadamard finite-part integral and the piecewise cubic interpolation polynomial is utilized to approximate the integral. The detailed error analysis is presented and it is established that the convergence order of the algorithm is O(h4−a). Asymptotic expansion of the error for the presented algorithm is also investigated. Some numerical examples are provided and compared with the exact solution to show that the numerical results are in well agreement with the theoretical ones and also to illustrate the accuracy and efficiency of the proposed algorithm.
    • Numerical analysis for distributed order differential equations

      Diethelm, Kai; Ford, Neville J.; University of Chester (University of Chester, 2001-04)
      In this paper we present and analyse a numerical method for the solution of a distributed order differential equation.
    • Numerical analysis of a singular integral equation

      Diogo, Teresa; Edwards, John T.; Ford, Neville J.; Thomas, Sophy M.
      This preprint discusses the numerical analysis of an integral equation to which convential analytical and numerical theory does not apply.
    • Numerical analysis of a two-parameter fractional telegraph equation

      Ford, Neville J.; Rodrigues, M. M.; Xiao, Jingyu; Yan, Yubin; University of Chester, Harbin Institute of Technology, University of Aveiro, Campus Universitario de Santiago (Elsevier, 2013-09-26)
      In this paper we consider the two-parameter fractional telegraph equation of the form $$-\, ^CD_{t_0^+}^{\alpha+1} u(t,x) + \, ^CD_{x_0^+}^{\beta+1} u (t,x)- \, ^CD_{t_0^+}^{\alpha}u (t,x)-u(t,x)=0.$$ Here $\, ^CD_{t_0^+}^{\alpha}$, $\, ^CD_{t_0^+}^{\alpha+1}$, $\, ^CD_{x_0^+}^{\beta+1}$ are operators of the Caputo-type fractional derivative, where $0\leq \alpha < 1$ and $0 \leq \beta < 1$. The existence and uniqueness of the equations are proved by using the Banach fixed point theorem. A numerical method is introduced to solve this fractional telegraph equation and stability conditions for the numerical method are obtained. Numerical examples are given in the final section of the paper.
    • Numerical analysis of some integral equations with singularities

      Ford, Neville J.; Thomas, Sophy M. (University of Liverpool (Chester College of Higher Education), 2006-04)
      In this thesis we consider new approaches to the numerical solution of a class of Volterra integral equations, which contain a kernel with singularity of non-standard type. The kernel is singular in both arguments at the origin, resulting in multiple solutions, one of which is differentiable at the origin. We consider numerical methods to approximate any of the (infinitely many) solutions of the equation. We go on to show that the use of product integration over a short primary interval, combined with the careful use of extrapolation to improve the order, may be linked to any suitable standard method away from the origin. The resulting split-interval algorithm is shown to be reliable and flexible, capable of achieving good accuracy, with convergence to the one particular smooth solution.
    • Numerical approaches to bifurcations in solutions to integro-differential equations

      Edwards, John T.; Ford, Neville J.; Roberts, Jason A. (Lea Press, 2002)
      This conference paper discusses the qualitative behaviour of numerical approximations of a carefully chosen class of integro-differential equations of the Volterra type. The results are illustrated with some numerical experiments.