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Gcodes over Formal Power Series Rings and Finite Chain RingsIn this work, we define $G$codes over the infinite ring $R_\infty$ as ideals in the group ring $R_\infty G$. We show that the dual of a $G$code is again a $G$code in this setting. We study the projections and lifts of $G$codes over the finite chain rings and over the formal power series rings respectively. We extend known results of constructing $\gamma$adic codes over $R_\infty$ to $\gamma$adic $G$codes over the same ring. We also study $G$codes over principal ideal rings.

Galerkin methods for a Schroedingertype equation with a dynamical boundary condition in two dimensionsIn this paper, we consider a twodimensional Schodingertype equation with a dynamical boundary condition. This model describes the longrange sound propagation in naval environments of variable rigid bottom topography. Our choice for a regular enough finite element approximation is motivated by the dynamical condition and therefore, consists of a cubic splines implicit Galerkin method in space. Furthermore, we apply a CrankNicolson time stepping for the evolutionary variable. We prove existence and stability of the semidiscrete and fully discrete solution.

A geneticalgorithm approach to simulating human immunodeficiency virus evolution reveals the strong impact of multiply infected cells and recombinationIt 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.

Group Rings, GCodes and Constructions of SelfDual and Formally SelfDual CodesWe describe Gcodes, which are codes that are ideals in a group ring, where the ring is a finite commutative Frobenius ring and G is an arbitrary finite group. We prove that the dual of a Gcode is also a Gcode. We give constructions of selfdual and formally selfdual codes in this setting and we improve the existing construction given in [13] by showing that one of the conditions given in the theorem is unnecessary and, moreover, it restricts the number of selfdual codes obtained by the construction. We show that several of the standard constructions of selfdual codes are found within our general framework. We prove that our constructed codes must have an automorphism group that contains G as a subgroup. We also prove that a common construction technique for producing selfdual codes cannot produce the putative [72, 36, 16] Type II code. Additionally, we show precisely which groups can be used to construct the extremal Type II codes over length 24 and 48. We define quasiG codes and give a construction of these codes.

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.

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 Fractional Order Differential EquationsThis thesis explores higher order numerical methods for solving fractional differential equations.

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.

Insights from the parallel implementation of efficient algorithms for the fractional calculusThis thesis concerns the development of parallel algorithms to solve fractional differential equations using a numerical approach. The methodology adopted is to adapt existing numerical schemes and to develop prototype parallel programs using the MatLab Parallel Computing Toolbox (MPCT). The approach is to build on existing insights from parallel implementation of ordinary differential equations methods and to test a range of potential candidates for parallel implementation in the fractional case. As a consequence of the work, new insights on the use of MPCT for prototyping are presented, alongside conclusions and algorithms for the effective implementation of parallel methods for the fractional calculus. The principal parallel approaches considered in the work include:  A RungeKutta Method for Ordinary Differential Equations including the application of an adapted Richardson Extrapolation Scheme  An implementation of the DiethelmChern Algorithm for Fractional Differential Equations  A parallel version of the wellestablished Fractional Adams Method for Fractional Differential Equations  The adaptation for parallel implementation of Lubich's Fractional Multistep Method for Fractional Differential Equations An important aspect of the work is an improved understanding of the comparative diffi culty of using MPCT for obtaining fair comparisons of parallel implementation. We present details of experimental results which are not satisfactory, and we explain how the problems may be overcome to give meaningful experimental results. Therefore, an important aspect of the conclusions of this work is the advice for other users of MPCT who may be planning to use the package as a prototyping tool for parallel algorithm development: by understanding how implicit multithreading operates, controls can be put in place to allow likeforlike performance comparisons between sequential and parallel programs.

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.