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Error estimates of a high order numerical method for solving linear fractional differential equationsLi, Zhiqiang; Yan, Yubin; Ford, Neville J.; Luliang University; University of Chester (Elsevier, IMACS, 20160429)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 firstdegree compound quadrature formula was used to approximate the Hadamard finitepart 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 seconddegree compound quadrature formula was used to approximate the Hadamard finitepart 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.

A finite element method for time fractional partial differential equationsFord, Neville J.; Xiao, Jingyu; Yan, Yubin; University of Chester ; Harbin Institute of Technology ; University of Chester (20110901)This article considers the finite element method for time fractional differential equations.

Fractional boundary value problems: Analysis and numerical methodsFord, Neville J.; Morgado, Maria L.; University of Chester ; University of TrasosMontes e Alto Douro (Springer, 20110728)This journal article discusses nonlinear boundary value problems.

Fractional pennes' bioheat equation: Theoretical and numerical studiesFerras, Luis L.; Ford, Neville J.; Morgado, Maria L.; Rebelo, Magda S.; Nobrega, Joao M.; University of Minho & University of Chester, University of Chester, UTAD, UNL Lisboa, University of Minho (de Gruyter, 20150804)In this work we provide a new mathematical model for the Pennes’ bioheat equation, assuming a fractional time derivative of single order. Alternative versions of the bioheat equation are studied and discussed, to take into account the temperaturedependent variability in the tissue perfusion, and both finite and infinite speed of heat propagation. The proposed bio heat model is solved numerically using an implicit finite difference scheme that we prove to be convergent and stable. The numerical method proposed can be applied to general reaction diffusion equations, with a variable diffusion coefficient. The results obtained with the single order fractional model, are compared with the original models that use classical derivatives.

A geneticalgorithm approach to simulating human immunodeficiency virus evolution reveals the strong impact of multiply infected cells and recombinationBocharov, Gennady; Ford, Neville J.; Edwards, John T.; Breinig, Tanja; WainHobson, Simon; Meyerhans, Andreas; Institute of Numerical Mathematics, Russian Academy of Sciences ; University of Chester ; University of Chester ; University of the Saarland ; Unité de Rétrovirologie Moléculaire, Institut Pasteur ; University of the Saarland (Society for General Microbiology / High Wire Press, 20051101)It has been previously shown that the majority of human immunodeficiency virus type 1 (HIV1)infected splenocytes can harbour multiple, divergent proviruses with a copy number ranging from one to eight. This implies that, besides point mutations, recombination should be considered as an important mechanism in the evolution of HIV within an infected host. To explore in detail the possible contributions of multiinfection and recombination to HIV evolution, the effects of major microscopic parameters of HIV replication (i.e. the pointmutation rate, the crossover number, the recombination rate and the provirus copy number) on macroscopic characteristics (such as the Hamming distance and the abundance of npoint mutants) have been simulated in silico. Simulations predict that multiple provirus copies per infected cell and recombination act in synergy to speed up the development of sequence diversity. Point mutations can be fixed for some time without fitness selection. The time needed for the selection of multiple mutations with increased fitness is highly variable, supporting the view that stochastic processes may contribute substantially to the kinetics of HIV variation in vivo.

Higher order numerical methods for solving fractional differential equationsYan, Yubin; Pal, Kamal; Ford, Neville J.; University of Chester (Springer, 20131005)In this paper we introduce higher order numerical methods for solving fractional differential equations. We use two approaches to this problem. The first approach is based on a direct discretisation of the fractional differential operator: we obtain a numerical method for solving a linear fractional differential equation with order 0 < α < 1. The order of convergence of the numerical method is O(h^(3−α)). Our second approach is based on discretisation of the integral form of the fractional differential equation and we obtain a fractional Adamstype method for a nonlinear fractional differential equation of any order α >0. The order of convergence of the numerical method is O(h^3) for α ≥ 1 and O(h^(1+2α)) for 0 < α ≤ 1 for sufficiently smooth solutions. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

How do numerical methods perform for delay differential equations undergoing a Hopf bifurcation?Ford, Neville J.; Wulf, Volker (Manchester Centre for Computational Mathematics, 19990930)This paper discusses the numerical solution of delay differential equations undergoing a Hopf birufication. Three distinct and complementary approaches to the analysis are presented.

An implicit finite difference approximation for the solution of the diffusion equation with distributed order in timeFord, Neville J.; Morgado, Maria L.; Rebelo, Magda S.; University of Chester, UTAD, Portugal, Universidade Nova de Lisboa, Portugal (Kent State University/Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences, 20150610)In this paper we are concerned with the numerical solution of a diffusion equation in which the time order derivative is distributed over the interval [0,1]. An implicit numerical method is presented and its unconditional stability and convergence are proved. A numerical example is provided to illustrate the obtained theoretical results.

Mathematical modelling and numerical simulations in nerve conductionFord, Neville J.; Lima, Pedro M.; Lumb, Patricia M.; University of Chester ; University of Lisbon / University of Linz ; University of Chester (Scitepress, 20150112)In the present work we analyse a functionaldifferential equation, sometimes known as the discrete FitzHughNagumo equation, arising in nerve conduction theory.

Mixedtype functional differential equations: A numerical approachFord, Neville J.; Lumb, Patricia M.; University of Chester (Elsevier, 20071029)This preprint discusses mixedtype functional equations.

Mixedtype functional differential equations: A numerical approach (extended version)Ford, Neville J.; Lumb, Patricia M.; University of Chester (University of Chester, 2007)

Multiorder fractional differential equations and their numerical solutionDiethelm, Kai; Ford, Neville J.; Technische Universität Braunschweig ; University College Chester (Elsevier, 20040715)This article considers the numerical solution of (possibly nonlinear) fractional differential equations of the form y(α)(t)=f(t,y(t),y(β1)(t),y(β2)(t),…,y(βn)(t)) with α>βn>βn−1>>β1 and α−βn1, βj−βj−11, 0

Noiseinduced changes to the behaviour of semiimplicit Euler methods for stochastic delay differential equations undergoing bifurcationFord, Neville J.; Norton, Stewart J.; University of Chester (Elsevier, 20090715)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.

Noiseinduced changes to the bifurcation behaviour of semiimplicit Euler methods for stochastic delay differential equationsFord, 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.

Nonpolynomial approximation of solutions to delay fractional differential equationsFord, Neville J.; Morgado, Maria L.; Rebelo, Magda S.; University of Chester ; University of TrasosMontes e Alto Douro ; Univeridade Nova de Lisboa (University of Oviedo, 2013)

A nonpolynomial collocation method for fractional terminal value problemsFord, Neville J.; Morgado, Maria L.; Rebelo, Magda S.; University of Chester ; UTAD, Portugal; Universidade de Nova Lisboa, Portugal (Elsevier, 20140614)In this paper we propose a nonpolynomial 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 nonpolynomial 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 the WellPosedness of Terminal Value Problems for Fractional Differential Equations.Diethelm, Kai; Ford, Neville J.; GNS & TUBS, Braunschweig, Germany; Univerity of Chester (Journal of Integral Equations and Applications, Rocky Mountains Mathematics Consortium, 20181108)This note is intended to clarify some im portant points about the wellposedness 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 counterexample 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 wellposedness question must be regarded as open.

Numerical analysis for distributed order differential equationsDiethelm, Kai; Ford, Neville J.; University of Chester (University of Chester, 200104)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 equationDiogo, 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 twoparameter fractional telegraph equationFord, Neville J.; Rodrigues, M. M.; Xiao, Jingyu; Yan, Yubin; University of Chester, Harbin Institute of Technology, University of Aveiro, Campus Universitario de Santiago (Elsevier, 20130926)In this paper we consider the twoparameter 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 Caputotype 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.