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Halanaytype theory in the context of evolutionary equations with timelagWe consider extensions and modifications of a theory due to Halanay, and the context in which such results may be applied. Our emphasis is on a mathematical framework for Halanaytype analysis of problems with time lag and simulations using discrete versions or numerical formulae. We present selected (linear and nonlinear, discrete and continuous) results of Halanay type that can be used in the study of systems of evolutionary equations with various types of delayed argument, and the relevance and application of our results is illustrated, by reference to delaydifferential equations, difference equations, and methods.

High order algorithms for numerical solution of fractional differential equationsIn this paper, two novel high order numerical algorithms are proposed for solving fractional differential equations where the fractional derivative is considered in the Caputo sense. The total domain is discretized into a set of small subdomains and then the unknown functions are approximated using the piecewise Lagrange interpolation polynomial of degree three and degree four. The detailed error analysis is presented, and it is analytically proven that the proposed algorithms are of orders 4 and 5. The stability of the algorithms is rigorously established and the stability region is also achieved. Numerical examples are provided to check the theoretical results and illustrate the efficiency and applicability of the novel algorithms.

A high order numerical method for solving nonlinear fractional differential equation with nonuniform meshesWe introduce a highorder numerical method for solving nonlinear fractional differential equation with nonuniform meshes. We first transform the fractional nonlinear differential equation into the equivalent Volterra integral equation. Then we approximate the integral by using the quadratic interpolation polynomials. On the first subinterval $[t_{0}, t_{1}]$, we approximate the integral with the quadratic interpolation polynomials defined on the nodes $t_{0}, t_{1}, t_{2}$ and in the other subinterval $[t_{j}, t_{j+1}], j=1, 2, \dots N1$, we approximate the integral with the quadratic interpolation polynomials defined on the nodes $t_{j1}, t_{j}, t_{j+1}$. A highorder numerical method is obtained. Then we apply this numerical method with the nonuniform meshes with the step size $\tau_{j}= t_{j+1} t_{j}= (j+1) \mu$ where $\mu= \frac{2T}{N (N+1)}$. Numerical results show that this method with the nonuniform meshes has the higher convergence order than the standard numerical methods obtained by using the rectangle and the trapzoid rules with the same nonuniform meshes.

HighOrder Numerical Methods for Solving Time Fractional Partial Differential EquationsIn this paper we introduce a new numerical method for solving time fractional partial differential equation. The time discretization is based on Diethelm’s method where the Hadamard finitepart integral is approximated by using the piecewise quadratic interpolation polynomials. The space discretization is based on the standard finite element method. The error estimates with the convergence order O(τ^(3−α) +h^2 ),0

A highorder scheme to approximate the Caputo fractional derivative and its application to solve the fractional diffusion wave equationA new highorder finite difference scheme to approximate the Caputo fractional derivative $\frac{1}{2} \big ( \, _{0}^{C}D^{\alpha}_{t}f(t_{k})+ \, _{0}^{C}D^{\alpha}_{t}f(t_{k1}) \big ), k=1, 2, \dots, N, $ with the convergence order $O(\Delta t^{4\alpha}), \, \alpha\in(1,2)$ is obtained when $f^{\prime \prime \prime} (t_{0})=0$, where $\Delta t$ denotes the time step size. Based on this scheme we introduce a finite difference method for solving fractional diffusion wave equation with the convergence order $O(\Delta t^{4\alpha} + h^2)$, where $h$ denotes the space step size. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

A higher order numerical method for time fractional partial differential equations with nonsmooth dataGao et al. (2014) introduced a numerical scheme to approximate the Caputo fractional derivative with the convergence rate $O(k^{3\alpha}), 0< \alpha <1$ by directly approximating the integerorder derivative with some finite difference quotients in the definition of the Caputo fractional derivative, see also Lv and Xu (2016), where $k$ is the time step size. Under the assumption that the solution of the time fractional partial differential equation is sufficiently smooth, Lv and Xu (2016) proved by using energy method that the corresponding numerical method for solving time fractional partial differential equation has the convergence rate $O(k^{3\alpha}), 0< \alpha <1$ uniformly with respect to the time variable $t$. However, in general the solution of the time fractional partial differential equation has low regularity and in this case the numerical method fails to have the convergence rate $O(k^{3 \alpha}), 0 < \alpha <1$ uniformly with respect to the time variable $t$. In this paper, we first obtain a similar approximation scheme to the RiemannLiouville fractional derivative with the convergence rate $O(k^{3 \alpha}), 0 < \alpha <1$ as in Gao \et \cite{gaosunzha} (2014) by approximating the Hadamard finitepart integral with the piecewise quadratic interpolation polynomials. Based on this scheme, we introduce a time discretization scheme to approximate the time fractional partial differential equation and show by using Laplace transform methods that the time discretization scheme has the convergence rate $O(k^{3 \alpha}), 0 < \alpha <1$ for any fixed $t_{n}>0$ for smooth and nonsmooth data in both homogeneous and inhomogeneous cases. Numerical examples are given to show that the theoretical results are consistent with the numerical results.

Higher order numerical methods for solving fractional differential equationsIn 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.

Higher Order Time Stepping Methods for Subdiffusion Problems Based on Weighted and Shifted Grünwald–Letnikov Formulae with Nonsmooth DataTwo higher order time stepping methods for solving subdiffusion problems are studied in this paper. The Caputo time fractional derivatives are approximated by using the weighted and shifted Gr\"unwaldLetnikov formulae introduced in Tian et al. [Math. Comp. 84 (2015), pp. 27032727]. After correcting a few starting steps, the proposed time stepping methods have the optimal convergence orders $O(k^2)$ and $ O(k^3)$, respectively for any fixed time $t$ for both smooth and nonsmooth data. The error estimates are proved by directly bounding the approximation errors of the kernel functions. Moreover, we also present briefly the applicabilities of our time stepping schemes to various other fractional evolution equations. Finally, some numerical examples are given to show that the numerical results are consistent with the proven theoretical results.

High‐order ADI orthogonal spline collocation method for a new 2D fractional integro‐differential problemWe use the generalized L1 approximation for the Caputo fractional derivative, the secondorder fractional quadrature rule approximation for the integral term, and a classical CrankNicolson alternating direction implicit (ADI)scheme for the time discretization of a new twodimensional (2D) fractionalintegrodifferential equation, in combination with a space discretization by anarbitraryorder orthogonal spline collocation (OSC) method. The stability of aCrankNicolson ADI OSC scheme is rigourously established, and error estimateis also derived. Finally, some numerical tests are given

How do numerical methods perform for delay differential equations undergoing a Hopf bifurcation?This paper discusses the numerical solution of delay differential equations undergoing a Hopf birufication. Three distinct and complementary approaches to the analysis are presented.

Identification of the initial function for discretized delay differential equationsIn the present work, we analyze a discrete analogue for the problem of the identification of the initial function for a delay differential equation (DDE) discussed by Baker and Parmuzin in 2004. The basic problem consists of finding an initial function that gives rise to a solution of a discretized DDE, which is a close fit to observed data.

Identification of the initial function for nonlinear delay differential equationsWe consider a 'data assimilation problem' for nonlinear delay differential equations. Our problem is to find an initial function that gives rise to a solution of a given nonlinear delay differential equation, which is a close fit to observed data. A role for adjoint equations and fundamental solutions in the nonlinear case is established. A 'pseudoNewton' method is presented. Our results extend those given by the authors in [(C. T. H. Baker and E. I. Parmuzin, Identification of the initial function for delay differential equation: Part I: The continuous problem & an integral equation analysis. NA Report No. 431, MCCM, Manchester, England, 2004.), (C. T. H. Baker and E. I. Parmuzin, Analysis via integral equations of an identification problem for delay differential equations. J. Int. Equations Appl. (2004) 16, 111–135.)] for the case of linear delay differential equations.

An implicit finite difference approximation for the solution of the diffusion equation with distributed order in timeIn 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.

Introducing delay dynamics to Bertalanffy's spherical tumour growth modelWe 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 LeastSquares 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.

Malliavin Calculus for the stochastic Cahn Hilliard/AllenCahn equation with unbounded noise diffusionThe stochastic partial di erential equation analyzed in this work, is motivated by a simplified mesoscopic physical model for phase separation. It describes pattern formation due to adsorption and desorption mechanisms involved in surface processes, in the presence of a stochastic driving force. This equation is a combination of CahnHilliard and AllenCahn type operators with a multiplicative, white, spacetime noise of unbounded di usion. We apply Malliavin calculus, in order to investigate the existence of a density for the stochastic solution u. In dimension one, according to the regularity result in [5], u admits continuous paths a.s. Using this property, and inspired by a method proposed in [8], we construct a modi ed approximating sequence for u, which properly treats the new second order AllenCahn operator. Under a localization argument, we prove that the Malliavin derivative of u exists locally, and that the law of u is absolutely continuous, establishing thus that a density exists.

Mathematical modelling and numerical simulations in nerve conductionIn the present work we analyse a functionaldifferential equation, sometimes known as the discrete FitzHughNagumo equation, arising in nerve conduction theory.

Mathematical models of DNA methylation dynamics: Implications for health and ageingDNA methylation status is a key epigenetic process which has been intimately associated with gene regulation. In recent years growing evidence has associated DNA methylation status with a variety of diseases including cancer, Alzheimers disease and cardiovascular disease. Moreover, changes to DNA methylation have also recently been implicated in the ageing process. The factors which underpin DNA methylation are complex, and remain to be fully elucidated. Over the years mathematical modelling has helped to shed light on the dynamics of this important molecular system. Although the existing models have contributed significantly to our overall understanding of DNA methylation, they fallshort of fully capturing the dynamics of this process. In this paper we develop a linear and nonlinear model which captures more fully the dynamics of the key intracellular events which characterise DNA methylation. In particular the outcomes of our linear model result in gene promoter specific methylation levels which are more biologically plausible than those revealed by previous mathematical models. In addition, our nonlinear model predicts DNA methylation promoter bistability which is commonly observed experimentally. The findings from our models have implications for our current understanding of how changes to the dynamics which underpin DNA methylation affect ageing and health.