Browsing Mathematics by Title
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On a degenerate nonlocal parabolic problem describing infinite dimensional replicator dynamicsWe establish the existence of locally positive weak solutions to the homogeneous Dirichlet problem for \[ u_t = u \Delta u + u \int_\Omega \nabla u^2 \] in bounded domains $\Om\sub\R^n$ which arises in game theory. We prove that solutions converge to $0$ if the initial mass is small, whereas they undergo blowup in finite time if the initial mass is large. In particular, it is shown that in this case the blowup set coincides with $\overline{\Omega}$, i.e. the finitetime blowup is global.

On hereditary reducibility of 2monomial matrices over commutative ringsA 2monomial matrix over a commutative ring $R$ is by definition any matrix of the form $M(t,k,n)=\Phi\left(\begin{smallmatrix}I_k&0\\0&tI_{nk}\end{smallmatrix}\right)$, $0<k<n$, where $t$ is a noninvertible element of $R$, $\Phi$ the compa\nion matrix to $\lambda^n1$ and $I_k$ the identity $k\times k$matrix. In this paper we introduce the notion of hereditary reducibility (for these matrices) and indicate one general condition of the introduced reducibility.

On some aspects of casual and neutral equations used in mathematical modellingThe problem that motivates the considerations here is the construction of mathematical models of natural phenomena that depend upon past states. The paper divides naturally into two parts: in the first, we expound the interconnection between ordinary differential equations, delay differential equations, neutral delaydifferential equations and integral equations (with emphasis on certain linear cases). As we show, this leads to a natural hierarchy of model complexity when such equations are used in mathematical and computational modelling, and to the possibility of reformulating problems either to facilitate their numerical solution or to provide mathematical insight, or both. Volterra integral equations include as special cases the others we consider. In the second part, we develop some practical and theoretical consequences of results given in the first part. In particular, we consider various approaches to the definition of an adjoint, we establish (notably, in the context of sensitivity analysis for neutral delaydifferential equations) roles for welldefined adjoints and ‘quasiadjoints’, and we explore relationships between sensitivity analysis, the variation of parameters formulae, the fundamental solution and adjoints.

On the behavior of the solutions for linear autonomous mixed type difference equationA class of linear autonomous mixed type difference equations is considered, and some new results on the asymptotic behavior and the stability are given, via a positive root of the corresponding characteristic equation.

On the decay of the elements of inverse triangular Toeplitz matricesWe consider half–infinite triangular Toeplitz matrices with slow decay of the elements and prove under a monotonicity condition that the elements of the inverse matrix, as well as the elements of the fundamental matrix, decay to zero. We provide a quantitative description of the decay of the fundamental matrix in terms of p–norms. The results add to the classical results of Jaffard and Vecchio, and are illustrated by numerical examples.

On the Dirichlet to Neumann Problem for the 1dimensional Cubic NLS Equation on the halflineInitialboundary value problems for 1dimensional `completely integrable' equations can be solved via an extension of the inverse scattering method, which is due to Fokas and his collaborators. A crucial feature of this method is that it requires the values of more boundary data than given for a wellposed problem. In the case of cubic NLS, knowledge of the Dirichet data su ces to make the problem wellposed but the Fokas method also requires knowledge of the values of Neumann data. The study of the Dirichlet to Neumann map is thus necessary before the application of the `Fokas transform'. In this paper, we provide a rigorous study of this map for a large class of decaying Dirichlet data. We show that the Neumann data are also su ciently decaying and that, hence, the Fokas method can be applied.

On the dynamics of a nonlocal parabolic equation arising from the GiererMeinhardt systemThe purpose of the current paper is to contribute to the comprehension of the dynamics of the shadow system of an activatorinhibitor system known as a GiererMeinhardt model. Shadow systems are intended to work as an intermediate step between single equations and reactiondiffusion systems. In the case where the inhibitor's response to the activator's growth is rather weak, then the shadow system of the GiererMeinhardt model is reduced to a single though nonlocal equation whose dynamics will be investigated. We mainly focus on the derivation of blowup results for this nonlocal equation which can be seen as instability patterns of the shadow system. In particular, a {\it diffusion driven instability (DDI)}, or {\it Turing instability}, in the neighbourhood of a constant stationary solution, which it is destabilised via diffusiondriven blowup, is obtained. The latter actually indicates the formation of some unstable patterns, whilst some stability results of globalintime solutions towards nonconstant steady states guarantee the occurrence of some stable patterns.

On the quenching of a nonlocal parabolic problem arising in electrostatic MEMS controlWe consider a nonlocal parabolic model for a microelectromechanical system. Specifically, for a radially symmetric problem with monotonic initial data, it is shown that the solution quenches, so that touchdown occurs in the device, in a situation where there is no steady state. It is also shown that quenching occurs at a single point and a bound on the approach to touchdown is obtained. Numerical simulations illustrating the results are given.

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.