• Addendum to the article: On the Dirichlet to Neumann Problem for the 1-dimensional Cubic NLS Equation on the Half-Line

      Antonopoulou, Dimitra; Kamvissis, Spyridon (IOPSCIENCE Published jointly with the London Mathematical Society, 2016-08-31)
      We present a short note on the extension of the results of [1] to the case of non-zero initial data. More specifically, the defocusing cubic NLS equation is considered on the half-line with decaying (in time) Dirichlet data and sufficiently smooth and decaying (in space) initial data. We prove that for this case also, and for a large class of decaying Dirichlet data, the Neumann data are sufficiently decaying so that the Fokas unified method for the solution of defocusing NLS is applicable.
    • An algorithm for the numerical solution of two-sided space-fractional partial differential equations.

      Ford, Neville J.; Pal, Kamal; Yan, Yubin; University of Chester (de Gruyter, 2015-08-20)
      We introduce an algorithm for solving two-sided space-fractional partial differential equations. The space-fractional derivatives we consider here are left-handed and right-handed Riemann–Liouville fractional derivatives which are expressed by using Hadamard finite-part integrals. We approximate the Hadamard finite-part integrals by using piecewise quadratic interpolation polynomials and obtain a numerical approximation of the space-fractional derivative with convergence order
    • An algorithm to detect small solutions in linear delay differential equations

      Ford, Neville J.; Lumb, Patricia M. (Elsevier, 2006-08-15)
      This preprint discusses an algorithm that provides a simple reliable mechanism for the detection of small solutions in linear delay differential equations.
    • Algorithms for the fractional calculus: A selection of numerical methods

      Diethelm, Kai; Ford, Neville J.; Freed, Alan D.; Luchko, Yury (Elsevier Science, 2005-02)
      This article discusses how numerical algorithms can help engineers work with fractional models in an efficient way.
    • Analysis of fractional differential equations

      Diethelm, Kai; Ford, Neville J. (Elsevier Science, 2002)
    • An analysis of the modified L1 scheme for time-fractional partial differential equations with nonsmooth data

      Yan, Yubin; Khan, Monzorul; Ford, Neville J.; University of Chester (Society for Industrial and Applied Mathematics, 2018-01-11)
      We introduce a modified L1 scheme for solving time fractional partial differential equations and obtain error estimates for smooth and nonsmooth initial data in both homogeneous and inhomogeneous cases. Jin \et (2016, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA J. of Numer. Anal., 36, 197-221) established an $O(k)$ convergence rate for the L1 scheme for smooth and nonsmooth initial data for the homogeneous problem, where $k$ denotes the time step size. We show that the modified L1 scheme has convergence rate $O(k^{2-\alpha}), 0< \alpha <1$ for smooth and nonsmooth initial data in both homogeneous and inhomogeneous cases. Numerical examples are given to show that the numerical results are consistent with the theoretical results.
    • Analysis via integral equations of an identification problem for delay differential equations

      Baker, Christopher T. H.; Parmuzin, Evgeny I.; University College Chester ; Institute of Numerical Mathematics, Russian Academy of Sciences (Rocky Mountain Mathematics Consortium, 2004)
    • An analytic approach to the normalized Ricci flow-like equation: Revisited

      Kavallaris, Nikos I.; Suzuki, Takashi; University of Chester ; Osaka University (Elsevier, 2015-01-07)
      In this paper we revisit Hamilton’s normalized Ricci flow, which was thoroughly studied via a PDE approach in Kavallaris and Suzuki (2010). Here we provide an improved convergence result compared to the one presented Kavallaris and Suzuki (2010) for the critical case λ=8πλ=8π. We actually prove that the convergence towards the stationary normalized Ricci flow is realized through any time sequence.
    • Analytical and numerical investigation of mixed-type functional differential equations

      Lima, Pedro M.; Teodoro, M. Filomena; Ford, Neville J.; Lumb, Patricia M.; Instituto Superior Tecnico UTL, Lisbon : Instituto Politecnico de Setubal, Lisbon : University of Chester : University of Chester (Elsevier, 2009-11-09)
      This journal article is concerned with the approximate solution of a linear non-autonomous functional differential equation, with both advanced and delayed arguments.
    • Analytical and numerical treatment of oscillatory mixed differential equations with differentiable delays and advances

      Ferreira, José M.; Ford, Neville J.; Malique, Md A.; Pinelas, Sandra; Yan, Yubin; Instituto Superior Técnico, Lisbon : University of Chester : University of Chester : Universidade dos Açores : University of Chester (Elsevier, 2011-04-12)
      This article discusses the oscillatory behaviour of the differential equation of mixed type.
    • An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data

      Ford, Neville J.; Yan, Yubin; University of Chester (De Gruyter, 2017-09)
      In this paper, we shall review an approach by which we can seek higher order time discretisation schemes for solving time fractional partial differential equations with nonsmooth data. The low regularity of the solutions of time fractional partial differential equations implies standard time discretisation schemes only yield first order accuracy. To obtain higher order time discretisation schemes when the solutions of time fractional partial differential equations have low regularities, one may correct the starting steps of the standard time discretisation schemes to capture the singularities of the solutions. We will consider these corrections of some higher order time discretisation schemes obtained by using Lubich's fractional multistep methods, L1 scheme and its modification, discontinuous Galerkin methods, etc. Numerical examples are given to show that the theoretical results are consistent with the numerical results.
    • 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, 2003-12)
    • 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.
    • Boundedness and stability of solutions to difference equations

      Edwards, John T.; Ford, Neville J. (Elsevier Science, 2002)
      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)
      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.
    • 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)