Browsing Mathematics by Title
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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.

SpaceTime Discontinuous Galerkin Methods for the '\eps'dependent Stochastic AllenCahn Equation with mild noiseWe consider the $\eps$dependent stochastic AllenCahn equation with mild space time noise posed on a bounded domain of R^2. The positive parameter $\eps$ is a measure for the inner layers width that are generated during evolution. This equation, when the noise depends only on time, has been proposed by Funaki in [15]. The noise although smooth becomes white on the sharp interface limit as $\eps$ tends to zero. We construct a nonlinear dG scheme with spacetime finite elements of general type which are discontinuous in time. Existence of a unique discrete solution is proven by application of Brouwer's Theorem. We first derive abstract error estimates and then for the case of piecewise polynomial finite elements we prove an error in expectation of optimal order. All the appearing constants are estimated in terms of the parameter $\eps$. Finally, we present a linear approximation of the nonlinear scheme for which we prove existence of solution and optimal error in expectation in piecewise linear finite element spaces. The novelty of this work is based on the use of a finite element formulation in space and in time in 2+1dimensional subdomains for a nonlinear parabolic problem. In addition, this problem involves noise. These type of schemes avoid any RungeKutta type discretization for the evolutionary variable and seem to be very effective when applied to equations of such a difficulty.

Stability analysis of a continuous model of mutualism with delay dynamicsIn this paper we introduce delay dynamics to a coupled system of ordinary differential equations which represent two interacting species exhibiting facultative mutualistic behaviour. The delays are represen tative of the beneficial effects of the indirect, interspecies interactions not being realised immediately. We show that the system with delay possesses a continuous solution, which is unique. Furthermore we show that, for suitablybehaved, positive initial functions that this unique solution is bounded and remains positive, i.e. both of the components representing the two species remain greater than zero. We show that the system has a positive equilibrium point and prove that this point is asymptotically stable for positive solutions and that this stability property is not conditional upon the delays.

Stability of a numerical method for a fractional telegraph equationIn this paper, we introduce a numerical method for solving the timespace fractional telegraph equations. The numerical method is based on a quadrature formula approach and a stability condition for the numerical method is obtained. Two numerical examples are given and the stability regions are plotted.

Stabilizing a mathematical model of plant species interactionIn this paper, we will consider how to stabilize a mathematical model of plant species interaction which is modelled by using LotkaVolterra system. We first identify the unstable steady states of the system, then we use the feedback control based on the solutions of the Riccati equation to stabilize the linearized system. We further stabilize the nonlinear system by using the feedback controller obtained in the stabilization of the linearized system. We introduce the backward Euler method to approximate the feedback control nonlinear system and obtain the error estimates. Four numerical examples are given which come from the application areas.

Superfast solution of linear convolutional Volterra equations using QTT approximationThis article address a linear fractional differential equation and develop effective solution methods using algorithms for the inversion of triangular Toeplitz matrices and the recently proposed QTT format. The inverses of such matrices can be computed by the divide and conquer and modified Bini’s algorithms, for which we present the versions with the QTT approximation. We also present an efficient formula for the shift of vectors given in QTT format, which is used in the divide and conquer algorithm. As a result, we reduce the complexity of inversion from the fast Fourier level O(nlogn) to the speed of superfast Fourier transform, i.e., O(log^2n). The results of the paper are illustrated by numerical examples.

Systemsbased decomposition schemes for the approximate solution of multiterm fractional differential equationsWe give a comparison of the efficiency of three alternative decomposition schemes for the approximate solution of multiterm fractional differential equations using the Caputo form of the fractional derivative. The schemes we compare are based on conversion of the original problem into a system of equations. We review alternative approaches and consider how the most appropriate numerical scheme may be chosen to solve a particular equation.

Theoretical and numerical analysis of unsteady fractional viscoelastic flows in simple geometries.In this work we discuss the connection between classical and fractional viscoelastic Maxwell models, presenting the basic theory supporting these constitutive equations, and establishing some background on the admissibility of the fractional Maxwell model. We then develop a numerical method for the solution of two coupled fractional differential equations (one for the velocity and the other for the stress), that appear in the pure tangential annular ow of fractional viscoelastic fluids. The numerical method is based on finite differences, with the approximation of fractional derivatives of the velocity and stress being inspired by the method proposed by Sun and Wu for the fractional diffusionwave equation [ Z.Z. Sun, X. Wu, A fully discrete difference scheme for a diffusionwave system, Applied Numerical Mathematics 56 (2006) 193209]. We prove solvability, study numerical convergence of the method, and also discuss the applicability of this method for simulating the rheological response of complex fluids in a real concentric cylinder rheometer. By imposing a torsional stepstrain, we observe the different rates of stress relaxation obtained with different values of \alpha and \beta (the fractional order exponents that regulate the viscoelastic response of the complex fluids).

Theory and numerics for multiterm periodic delay differential equations, small solutions and their detectionWe summarise a theoretical treatment that analyses whether the equation has small solutions. We consider discrete equations that arise when a numerical method with fixed step size is applied to approximate the solution to (†) and we develop a corresponding theory. Our results show that small solutions can be detected reliably by the numerical scheme. We conclude with some numerical examples.

A time discretization scheme for a nonlocal degenerate problem modelling resistance spot weldingIn the current work we construct a nonlocal mathematical model describing the phase transition occurs during the resistance spot welding process in the industry of metallurgy. We then consider a time discretization scheme for solving the resulting nonlocal moving boundary problem. The scheme consists of solving at each time step a linear elliptic partial differential equation and then making a correction to account for the nonlinearity. The stability and error estimates of the developed scheme are investigated. Finally some numerical results are presented confirming the efficiency of the developed numerical algorithm.

Torsion Units for a Ree group, Tits group and a Steinberg triality groupWe investigate the Zassenhaus conjecture for the Steinberg triality group ${}^3D_4(2^3)$, Tits group ${}^2F_4(2)'$ and the Ree group ${}^2F_4(2)$. Consequently, we prove that the Prime Graph question is true for all three groups.

Torsion Units for Some Almost Simple GroupsWe prove that the Zassenhaus conjecture is true for $Aut(PSL(2,11))$. Additionally we prove that the Prime graph question is true for $Aut(PSL(2,13))$.

Torsion Units for some Projected Special Linear GroupsIn this paper, we investigate the Zassenhaus conjecture for $PSL(4,3)$ and $PSL(5,2)$. Consequently, we prove that the Prime graph question is true for both groups.