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Blending loworder stabilised finite element methods: a positivity preserving local projection method for the convectiondiffusion equationIn this work we propose a nonlinear blending of two loworder stabilisation mechanisms for the convection–diffusion equation. The motivation for this approach is to preserve monotonicity without sacrificing accuracy for smooth solutions. The approach is to blend a firstorder artificial diffusion method, which will be active only in the vicinity of layers and extrema, with an optimal order local projection stabilisation method that will be active on the smooth regions of the solution. We prove existence of discrete solutions, as well as convergence, under appropriate assumptions on the nonlinear terms, and on the exact solution. Numerical examples show that the discrete solution produced by this method remains within the bounds given by the continuous maximum principle, while the layers are not smeared significantly.

Bordered Constructions of SelfDual Codes from Group Rings and New Extremal Binary SelfDual CodesWe introduce a bordered construction over group rings for selfdual codes. We apply the constructions over the binary field and the ring $\F_2+u\F_2$, using groups of orders 9, 15, 21, 25, 27, 33 and 35 to find extremal binary selfdual codes of lengths 20, 32, 40, 44, 52, 56, 64, 68, 88 and best known binary selfdual codes of length 72. In particular we obtain 41 new binary extremal selfdual codes of length 68 from groups of orders 15 and 33 using neighboring and extensions. All the numerical results are tabulated throughout the paper.

Boundedness and stability of solutions to difference equationsThis 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 equationsThis 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 approximationsThis 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 approximationsThis 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 argumentsThe 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 equationsThis article discusses and evaluates the merits of five numerical methods for the solution of single term fractional differential equations.

Composite Constructions of SelfDual Codes from Group Rings and New Extremal SelfDual Binary Codes of Length 68We describe eight composite constructions from group rings where the orders of the groups are 4 and 8, which are then applied to find selfdual codes of length 16 over F4. These codes have binary images with parameters [32, 16, 8] or [32, 16, 6]. These are lifted to codes over F4 + uF4, to obtain codes with Gray images extremal selfdual binary codes of length 64. Finally, we use a buildingup method over F2 + uF2 to obtain new extremal binary selfdual codes of length 68. We construct 11 new codes via the buildingup method and 2 new codes by considering possible neighbors.

Composite Matrices from Group Rings, Composite GCodes and Constructions of SelfDual CodesIn this work, we define composite matrices which are derived from group rings. We extend the idea of Gcodes to composite Gcodes. We show that these codes 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 composite Gcode is also a composite Gcode. We also define quasicomposite Gcodes. Additionally, we study generator matrices, which consist of the identity matrices and the composite matrices. Together with the generator matrices, the well known extension method, the neighbour method and its generalization, we find extremal binary selfdual codes of length 68 with new weight enumerators for the rare parameters $\gamma$ = 7; 8 and 9: In particular, we find 49 new such codes. Moreover, we show that the codes we find are inaccessible from other constructions.

Computational approaches to parameter estimation and model selection in immunologyThis article seeks to illustrate the computational implementation of an informationtheoretic approach (associated with a maximum likelihood treatment) to modelling in immunology.

Computational aspects of timelag models of Marchuk type that arise in immunologyIn his book published in English translation in 1983, Marchuk proposed a set of evolutionary equations incorporating delaydifferential equations, and the corresponding initial conditions as a model ('Marchuk's model') for infectious diseases. The parameters in this model (and its subsequent extensions) represent scientifically meaningful characteristics. For a given infection, the parameters can be estimated using observational data on the course of the infection. Sensitivity analysis is an important tool for understanding a particular model; this can be viewed as an issue of stability with respect to structural perturbations in the model. Examining the sensitivity of the models based on delay differential equations leads to systems of neutral delay differential equations. Below we formulate a general set of equations for the sensitivity coefficients for models comprising neutral delay differential equations. We discuss computational approaches to the sensitivity of solutions — (i) sensitivity to the choice of model, in particular, to the lag parameter τ > 0 and (ii) sensitivity to the initial function — of dynamical systems with time lag and illustrate them by considering the sensitivity of solutions of timelag models of Marchuk type.

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

Computational modelling with functional differential equations: Identification, selection, and sensitivityMathematical 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 timelag which is entirely natural from the scientific perspective. The timelag 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 timelag) 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 bestfit models, we are able to employ certain indicators based on informationtheoretic 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).

Concerning periodic solutions to nonlinear discrete Volterra equations with finite memoryIn this paper we discuss the existence of periodic solutions of discrete (and discretized) nonlinear Volterra equations with finite memory. The literature contains a number of results on periodic solutions of non linear Volterra integral equations with finite memory, of a type that arises in biomathematics. The “summation” equations studied here can arise as discrete models in their own right but are (as we demonstrate) of a type that arise from the discretization of such integral equations. Our main results are in two parts: (i) results for discrete equations and (ii) consequences for quadrature methods applied to integral equations. The first set of results are obtained using a variety of fixed point theorems. The second set of results address the preservation of properties of integral equations on discretizing them. An expository style is adopted and examples are given to illustrate the discussion.

Constructing SelfDual Codes from Group Rings and Reverse Circulant MatricesIn this work, we describe a construction for selfdual codes in which we employ group rings and reverse circulant matrices. By applying the construction directly over different alphabets, and by employing the well known extension and neighbor methods we were able to obtain extremal binary selfdual codes of different lengths of which some have parameters that were not known in the literature before. In particular, we constructed three new codes of length 64, twentytwo new codes of length 68, twelve new codes of length 80 and four new codes of length 92.

Constructions for SelfDual Codes Induced from Group RingsIn this work, we establish a strong connection between group rings and selfdual codes. We prove that a group ring element corresponds to a selfdual code if and only if it is a unitary unit. We also show that the doublecirculant and fourcirculant constructions come from cyclic and dihedral groups, respectively. Using groups of order 8 and 16 we find many new construction methods, in addition to the wellknown methods, for selfdual codes. We establish the relevance of these new constructions by finding many extremal binary selfdual codes using them, which we list in several tables. In particular, we construct 10 new extremal binary selfdual codes of length 68.

CrankNicolson finite element discretizations for a twodimenional linear Schroedingertype equation posed in noncylindrical domainMotivated by the paraxial narrow–angle approximation of the Helmholtz equation in domains of variable topography, we consider an initialand boundaryvalue problem for a general Schr¨odingertype equation posed on a two spacedimensional noncylindrical domain with mixed boundary conditions. The problem is transformed into an equivalent one posed on a rectangular domain, and we approximate its solution by a Crank–Nicolson finite element method. For the proposed numerical method, we derive an optimal order error estimate in the L2 norm, and to support the error analysis we prove a global elliptic regularity theorem for complex elliptic boundary value problems with mixed boundary conditions. Results from numerical experiments are presented which verify the optimal order of convergence of the method.