Browsing Mathematics by Title
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Optimal convergence rates for semidiscrete finite element approximations of linear spacefractional partial differential equations under minimal regularity assumptionsWe consider the optimal convergence rates of the semidiscrete finite element approximations for solving linear spacefractional partial differential equations by using the regularity results for the fractional elliptic problems obtained recently by Jin et al. \cite{jinlazpasrun} and Ervin et al. \cite{ervheuroo}. The error estimates are proved by using two approaches. One approach is to apply the duality argument in Johnson \cite{joh} for the heat equation to consider the error estimates for the linear spacefractional partial differential equations. This argument allows us to obtain the optimal convergence rates under the minimal regularity assumptions for the solution. Another approach is to use the approximate solution operators of the corresponding fractional elliptic problems. This argument can be extended to consider more general linear spacefractional partial differential equations. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

Orthogonality for a class of generalised Jacobi polynomial $P^{\alpha,\beta}_{\nu}(x)$This work considers gJacobi polynomials, a fractional generalisation of the classical Jacobi polynomials. We discuss the polynomials and compare some of their properties to the classical case. The main result of the paper is to show that one can derive an orthogonality property for a subclass of gJacobi polynomials $P^{\alpha,\beta}_{\nu}(x)$ The paper concludes with an application in modelling of ophthalmic surfaces.

Periodic solutions of discrete Volterra equationsThis article investigates periodic solutions of linear and nonlinear discrete Volterra equations of convolution or nonconvolution type with unbounded memory. For linear discrete Volterra equations of convolution type, we establish Fredholm’s alternative theorem and for equations of nonconvolution type, and we prove that a unique periodic solution exists for a particular bounded initial function under appropriate conditions. Further, this unique periodic solution attracts all other solutions with bounded initial function. All solutions of linear discrete Volterra equations with bounded initial functions are asymptotically periodic under certain conditions. A condition for periodic solutions in the nonlinear case is established.

Perturbation of Volterra difference equationsA fixed point theorem is used to investigate nonlinear Volterra difference equations that are perturbed versions of linear equations. Sufficient conditions are established to ensure that the stability properties of linear Volterra difference equations are preserved under perturbation. The existence of asymptotically periodic solutions of perturbed Volterra difference equations is also proved.

Pitfalls in fast numerical solvers for fractional differential equationsThis preprint discusses the properties of high order methods for the solution of fractional differential equations. A number of fractional multistep methods are are discussed.

A Posteriori Analysis for SpaceTime, discontinuous in time Galerkin approximations for parabolic equations in a variable domainThis paper presents an a posteriori error analysis for the discontinuous in time spacetime scheme proposed by Jamet for the heat equation in multidimensional, noncylindrical domains [25]. Using a Cl ementtype interpolant, we prove abstract a posteriori error bounds for the numerical error. Furthermore, in the case of twodimensional spatial domains we transform the problem into an equivalent one, of parabolic type, with spacetime dependent coe cients but posed on a cylindrical domain. We formulate a discontinuous in time space{time scheme and prove a posteriori error bounds of optimal order. The a priori estimates of [19] for general parabolic initial and boundary value problems are used in the derivation of the upper bound. Our lower bound coincides with that of Picasso [36], proposed for adaptive, RungeKutta finite element methods for linear parabolic problems. Our theoretical results are verified by numerical experiments.

A posteriori error estimates for fully discrete fractionalstep ϑapproximations for parabolic equationsWe derive optimal order a posteriori error estimates for fully discrete approximations of initial and boundary value problems for linear parabolic equations. For the discretisation in time we apply the fractionalstep #scheme and for the discretisation in space the finite element method with finite element spaces that are allowed to change with time.

A posteriori error estimates for fully discrete schemes for the time dependent Stokes problemThis work is devoted to a posteriori error analysis of fully discrete finite element approximations to the time dependent Stokes system. The space discretization is based on popular stable spaces, including Crouzeix–Raviart and Taylor–Hood finite element methods. Implicit Euler is applied for the time discretization. The finite element spaces are allowed to change with time steps and the projection steps include alternatives that is hoped to cope with possible numerical artifices and the loss of the discrete incompressibility of the schemes. The final estimates are of optimal order in L∞(L2) for the velocity error.

Predicting changes in dynamical behaviour in solutions to stochastic delay differential equationsThis article considers numerical approximations to parameterdependent linear and logistic stochastic delay differential equations with multiplicative noise. The aim of the investigation is to explore the parameter values at which there are changes in qualitative behaviour of the solutions. One may use a phenomenological approach but a more analytical approach would be attractive. A possible tool in this analysis is the calculation of the approximate local Lyapunov exponents. In this paper we show that the phenomenological approach can be used effectively to estimate bifurcation parameters for deterministic linear equations but one needs to use the dynamical approach for stochastic equations.

A predictor corrector approach for the numerical solution of fractional differential equationsThis article discusses an Adamstype predictorcorrector method for the numerical solution of fractional differential equations.

Quadruple Bordered Constructions of SelfDual Codes from Group RingsIn this paper, we introduce a new bordered construction for selfdual codes using group rings. We consider constructions over the binary field, the family of rings Rk and the ring F4 + uF4. We use groups of order 4, 12 and 20. We construct some extremal selfdual codes and nonextremal selfdual codes of length 16, 32, 48, 64 and 68. In particular, we construct 33 new extremal selfdual codes of length 68.

Qualitative behaviour of numerical approximations to Volterra integrodifferential equationsThis article investigates the qualitative behaviour of numerical approximations to a nonlinear Volterra integrodifferential equation with unbounded delay.

Quenching solutions of a stochastic parabolic problem arising in electrostatic MEMS controlIn the current paper, we consider a stochastic parabolic equation which actually serves as a mathematical model describing the operation of an electrostatic actuated microelectromechanical system (MEMS). We first present the derivation of the mathematical model. Then after establishing the local wellposedeness of the problem we investigate under which circumstances a {\it finitetime quenching} for this SPDE, corresponding to the mechanical phenomenon of {\it touching down}, occurs. For that purpose the Kaplan's eigenfunction method adapted in the context of SPDES is employed.

Recombination: Multiply infected spleen cells in HIV patientsThe genome of the human immunodeficiency virus is highly prone to recombination, although it is not obvious whether recombinants arise infrequently or whether they are constantly being spawned but escape identification because of the massive and rapid turnover of virus particles. Here we use fluorescence in situ hybridization to estimate the number of proviruses harboured by individual splenocytes from two HIV patients, and determine the extent of recombination by sequencing amplified DNA from these cells. We find an average of three or four proviruses per cell and evidence for huge numbers of recombinants and extensive genetic variation. Although this creates problems for phylogenetic analyses, which ignore recombination effects, the intracellular variation may help to broaden immune recognition.

SelfDual Codes using Bisymmetric Matrices and Group RingsIn this work, we describe a construction in which we combine together the idea of a bisymmetric matrix and group rings. Applying this construction over the ring F4 + uF4 together with the well known extension and neighbour methods, we construct new selfdual codes of length 68: In particular, we find 41 new codes of length 68 that were not known in the literature before.

The sharp interface limit for the stochastic CahnHilliard EquationWe study the two and three dimensional stochastic CahnHilliard equation in the sharp interface limit, where the positive parameter \eps tends to zero, which measures the width of transition layers generated during phase separation. We also couple the noise strength to this parameter. Using formal asymptotic expansions, we identify the limit. In the right scaling we indicate that the solutions of stochastic CahnHilliard converge to a solution of a HeleShaw problem with stochastic forcing. In the case when the noise is sufficiently small, we rigorously prove that the limit is a deterministic HeleShaw problem. Finally, we discuss which estimates are necessary in order to extend the rigorous result to larger noise strength.

Solution of a singular integral equation by a splitinterval methodThis article discusses a new numerical method for the solution of a singular integral equation of Volterra type that has an infinite class of solutions. The splitinterval method is discussed and examples demonstrate its effectiveness.

Some time stepping methods for fractional diffusion problems with nonsmooth dataWe consider error estimates for some time stepping methods for solving fractional diffusion problems with nonsmooth data in both homogeneous and inhomogeneous cases. McLean and Mustapha \cite{mclmus} (Timestepping error bounds for fractional diffusion problems with nonsmooth initial data, Journal of Computational Physics, 293(2015), 201217) established an $O(k)$ convergence rate for the piecewise constant discontinuous Galerkin method with nonsmooth initial data for the homogeneous problem when the linear operator $A$ is assumed to be selfadjoint, positive semidefinite and densely defined in a suitable Hilbert space, where $k$ denotes the time step size. In this paper, we approximate the RiemannLiouville fractional derivative by Diethelm's method (or $L1$ scheme) and obtain the same time discretisation scheme as in McLean and Mustapha \cite{mclmus}. We first prove that this scheme has also convergence rate $O(k)$ with nonsmooth initial data for the homogeneous problem when $A$ is a closed, densely defined linear operator satisfying some certain resolvent estimates. We then introduce a new time discretization scheme for the homogeneous problem based on the convolution quadrature and prove that the convergence rate of this new scheme is $O(k^{1+ \alpha}), 0<\alpha <1 $ with the nonsmooth initial data. Using this new time discretization scheme for the homogeneous problem, we define a time stepping method for the inhomogeneous problem and prove that the convergence rate of this method is $O(k^{1+ \alpha}), 0<\alpha <1 $ with the nonsmooth data. Numerical examples are given to show that the numerical results are consistent with the theoretical results.