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An algorithm for the numerical solution of twosided spacefractional partial differential equations.Ford, Neville J.; Pal, Kamal; Yan, Yubin; University of Chester (de Gruyter, 20150820)We introduce an algorithm for solving twosided spacefractional partial differential equations. The spacefractional derivatives we consider here are lefthanded and righthanded Riemann–Liouville fractional derivatives which are expressed by using Hadamard finitepart integrals. We approximate the Hadamard finitepart integrals by using piecewise quadratic interpolation polynomials and obtain a numerical approximation of the spacefractional derivative with convergence order

An algorithm to detect small solutions in linear delay differential equationsFord, Neville J.; Lumb, Patricia M. (Elsevier, 20060815)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 methodsDiethelm, Kai; Ford, Neville J.; Freed, Alan D.; Luchko, Yury (Elsevier Science, 20050225)This article discusses how numerical algorithms can help engineers work with fractional models in an efficient way.

Analysis of fractional differential equationsDiethelm, Kai; Ford, Neville J. (Elsevier Science, 20020115)

An analysis of the modified L1 scheme for timefractional partial differential equations with nonsmooth dataYan, Yubin; Khan, Monzorul; Ford, Neville J.; University of Chester (Society for Industrial and Applied Mathematics, 20180111)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, 197221) 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.

Analytical and numerical investigation of mixedtype functional differential equationsLima, 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, 20091109)This journal article is concerned with the approximate solution of a linear nonautonomous functional differential equation, with both advanced and delayed arguments.

Analytical and numerical treatment of oscillatory mixed differential equations with differentiable delays and advancesFerreira, 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, 20110412)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 dataFord, Neville J.; Yan, Yubin; University of Chester (De Gruyter, 20171031)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 equationsBaker, 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 integrodifferential equationsEdwards, 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.

Boundedness and stability of solutions to difference equationsEdwards, John T.; Ford, Neville J. (Elsevier Science, 20020301)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 equationsEdwards, John T.; Ford, Neville J. (Manchester Centre for Computational Mathematics, 20030523)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 approximationsFord, Neville J.; Lunel, Sjoerd M. V. (Manchester Centre for Computational Mathematics, 20030523)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 approximationsFord, Neville J.; Lunel, Sjoerd M. V. (Elsevier Science, 20020925)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 argumentsBaker, Christopher T. H.; Ford, Neville J.; University of Manchester; University of Chester (Elsevier, 20190424)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 realvalued 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 RiemannStieltjes 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 nonoscillatory 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.

Comparison of numerical methods for fractional differential equationsFord, Neville J.; Connolly, Joseph A. (American Institute of Mathematical Sciences/Shanghai Jiao Tong University, 200606)This article discusses and evaluates the merits of five numerical methods for the solution of single term fractional differential equations.

Computational methods for a mathematical model of propagation of nerve impulses in myelinated axonsLima, Pedro M.; Ford, Neville J.; Lumb, Patricia M.; CEMAT, IST, Lisbon ; University of Chester ; University of Chester (Elsevier, 20140626)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.

Detailed error analysis for a fractional Adams methodDiethelm, Kai; Ford, Neville J.; Freed, Alan D. (Springer, 200405)This preprint discusses a method for a numerical solution of a nonlinear fractional differential equation, which can be seen as a generalisation of the Adams–Bashforth–Moulton scheme.

Determining control parameters for dendritic cellcytotoxic T lymphocyte interactionLudewig, Burkhard; Krebs, Philippe; Junt, Tobias; Metters, Helen; Ford, Neville J.; Anderson, Roy M.; Bocharov, Gennady; University of Zürich ; University of Zürich ; University of Zürich ; University of Zürich ; University College Chester ; Imperial College, University of London ; Institute of Numerical Mathematics, Russian Academy of Sciences (WILEYVCH Verlag GmbH & Co. KGaA, 20040805)Dendritic cells (DC) are potent immunostimulatory cells facilitating antigen transport to lymphoid tissues and providing efficient stimulation of T cells. A series of experimental studies in mice demonstrated that cytotoxic T lymphocytes (CTL) can be efficiently induced by adoptive transfer of antigenpresenting DC. However, the success of DCbased immunotherapeutic treatment of human cancer, for example, is still limited because the details of the regulation and kinetics of the DCCTL interaction are not yet completely understood. Using a combination of experimental mouse studies, mathematical modeling, and nonlinear parameter estimation, we analyzed the population dynamics of DCinduced CTL responses. The model integrates a predatorpreytype interaction of DC and CTL with the nonlinear compartmental dynamics of T cells. We found that T cell receptor avidity, the halflife of DC, and the rate of CTLmediated DCelimination are the major control parameters for optimal DCinduced CTL responses. For induction of high avidity CTL, the number of adoptively transferred DC was of minor importance once a minimal threshold of approximately 200 cells per spleen had been reached. Taken together, our study indicates that the availability of high avidity T cells in the recipient in combination with the optimal application regimen is of prime importance for successful DCbased immunotherapy.

Distributed order equations as boundary value problemsFord, Neville J.; Morgado, Maria L.; University of Chester ; University of TrasosMontes e Alto Douro (Elsevier, 20120120)This preprint discusses the existence and uniqueness of solutions and proposes a numerical method for their approximation in the case where the initial conditions are not known and, instead, some Caputotype conditions are given away from the origin.