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Subjectsmodelling (2)Characteristic functions (1)Chebyshev method (1)computation (1)computational mathematics (1)delay (1)Delay differential equations (1)differential equations (1)Differential equations with deviating arguments (1)Discontinuous Galerkin method (1)View MoreJournal

Applied Numerical Mathematics (8)

AuthorsFord, Neville J. (4)Baker, Christopher T. H. (2)Roberts, Jason A. (2)Yan, Yubin (2)Bocharov, Gennady (1)Ferras, Luis L. (1)Kavallaris, Nikos I. (1)Khan, Monzorul (1)Li, Zhiqiang (1)Lima, Pedro M. (1)View MoreTypesArticle (8)

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Computational methods for a mathematical model of propagation of nerve impulses in myelinated axons

Lima, Pedro M.; Ford, Neville J.; Lumb, Patricia M. (Elsevier, 2014-07-07)

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.

Introducing delay dynamics to Bertalanffy's spherical tumour growth model

Roberts, Jason A.; Themairi, Asmaa A. (Elsevier, 2016-10-21)

We introduce delay dynamics to an ordinary differential equation model of tumour growth based upon von Bertalanffy's growth model, a model which has received little attention in comparison to other models, such as Gompterz, Greenspan and logistic models. Using existing, previously published data sets we show that our delay model can perform better than delay models based on a Gompertz, Greenspan or logistic formulation. We look for replication of the oscillatory behaviour in the data, as well as a low error value (via a Least-Squares approach) when comparing. We provide the necessary analysis to show that a unique, continuous, solution exists for our model equation and consider the qualitative behaviour of a solution near a point of equilibrium.

Numerical solution for diffusion equations with distributed order in time using a Chebyshev collocation method

Morgado, Maria L.; Rebelo, Magda S.; Ferras, Luis L.; Ford, Neville J. (Elsevier, 2016-11-09)

In this work we present a new numerical method for the solution of the distributed order time fractional diffusion equation. The method is based on the approximation of the solution by a double Chebyshev truncated series, and the subsequent collocation of the resulting discretised system of equations at suitable collocation points. An error analysis is provided and a comparison with other methods used in the solution of this type of equation is also performed.

Error estimates of a high order numerical method for solving linear fractional differential equations

Li, Zhiqiang; Yan, Yubin; Ford, Neville J. (Elsevier, IMACS, 2016-04-29)

In this paper, we first introduce an alternative proof of the error estimates of the numerical methods for solving linear fractional differential equations proposed in Diethelm [6] where a first-degree compound quadrature formula was used to approximate the Hadamard finite-part integral and the convergence order of the proposed numerical method is O(∆t 2−α ), 0 < α < 1, where α is the order of the fractional derivative and ∆t is the step size. We then use the similar idea to prove the error estimates of a high order numerical method for solving linear fractional differential equations proposed in Yan et al. [37], where a second-degree compound quadrature formula was used to approximate the Hadamard finite-part integral and we show that the convergence order of the numerical method is O(∆t 3−α ), 0 < α < 1. The numerical examples are given to show that the numerical results are consistent with the theoretical results.

Discontinuous Galerkin time stepping method for solving linear space fractional partial differential equations

Liu, Yanmei; Yan, Yubin; Khan, Monzorul (Elsevier, 2017-01-23)

In this paper, we consider the discontinuous Galerkin time stepping method for solving the linear space fractional partial differential equations. The space fractional derivatives are defined by using Riesz fractional derivative. The space variable is discretized by means of a Galerkin finite element method and the time variable is discretized by the discontinuous Galerkin method. The approximate solution will be sought as a piecewise polynomial function in $t$ of degree at most $q-1, q \geq 1$, which is not necessarily continuous at the nodes of the defining partition. The error estimates in the fully discrete case are obtained and the numerical examples are given.

Characteristic functions of differential equations with deviating arguments

Baker, Christopher T. H.; Ford, Neville J. (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.

A discrete mutualism model: analysis and exploration of a financial application

Roberts, Jason A.; Kavallaris, Nikos I.; Rowntree, Andrew P. (Elsevier, 2019-09-16)

We perform a stability analysis on a discrete analogue of a known, continuous model of
mutualism. We illustrate how the introduction of delays affects the asymptotic stability
of the system’s positive nontrivial equilibrium point. In the second part of the paper we
explore the insights that the model can provide when it is used in relation to interacting
financial markets. We also note the limitations of such an approach.

Computational modelling with functional differential equations: Identification, selection, and sensitivity

Baker, Christopher T. H.; Bocharov, Gennady; Paul, C. A. H.; Rihan, F. A. R. (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).

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