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Addendum to the article: On the Dirichlet to Neumann Problem for the 1dimensional Cubic NLS Equation on the HalfLine(IOPSCIENCE Published jointly with the London Mathematical Society, 20160831)We present a short note on the extension of the results of [1] to the case of nonzero initial data. More specifically, the defocusing cubic NLS equation is considered on the halfline with decaying (in time) Dirichlet data and sufficiently smooth and decaying (in space) initial data. We prove that for this case also, and for a large class of decaying Dirichlet data, the Neumann data are sufficiently decaying so that the Fokas unified method for the solution of defocusing NLS is applicable.

An algorithm for the numerical solution of twosided spacefractional partial differential equations.(de Gruyter, 20150820)We introduce an algorithm for solving twosided spacefractional partial differential equations. The spacefractional derivatives we consider here are lefthanded and righthanded Riemann–Liouville fractional derivatives which are expressed by using Hadamard finitepart integrals. We approximate the Hadamard finitepart integrals by using piecewise quadratic interpolation polynomials and obtain a numerical approximation of the spacefractional derivative with convergence order

An algorithm to detect small solutions in linear delay differential equations(Elsevier, 20060815)This preprint discusses an algorithm that provides a simple reliable mechanism for the detection of small solutions in linear delay differential equations.

Algorithms for the fractional calculus: A selection of numerical methods(Elsevier Science, 200502)This article discusses how numerical algorithms can help engineers work with fractional models in an efficient way.

Analysis of fractional differential equations(Elsevier Science, 2002)

An analysis of the modified L1 scheme for timefractional partial differential equations with nonsmooth data(Society for Industrial and Applied Mathematics, 20180111)We introduce a modified L1 scheme for solving time fractional partial differential equations and obtain error estimates for smooth and nonsmooth initial data in both homogeneous and inhomogeneous cases. Jin \et (2016, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA J. of Numer. Anal., 36, 197221) established an $O(k)$ convergence rate for the L1 scheme for smooth and nonsmooth initial data for the homogeneous problem, where $k$ denotes the time step size. We show that the modified L1 scheme has convergence rate $O(k^{2\alpha}), 0< \alpha <1$ for smooth and nonsmooth initial data in both homogeneous and inhomogeneous cases. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

Analysis via integral equations of an identification problem for delay differential equations(Rocky Mountain Mathematics Consortium, 2004)

An analytic approach to the normalized Ricci flowlike equation: Revisited(Elsevier, 20150107)In this paper we revisit Hamilton’s normalized Ricci flow, which was thoroughly studied via a PDE approach in Kavallaris and Suzuki (2010). Here we provide an improved convergence result compared to the one presented Kavallaris and Suzuki (2010) for the critical case λ=8πλ=8π. We actually prove that the convergence towards the stationary normalized Ricci flow is realized through any time sequence.

Analytical and numerical investigation of mixedtype functional differential equations(Elsevier, 20091109)This journal article is concerned with the approximate solution of a linear nonautonomous functional differential equation, with both advanced and delayed arguments.

Analytical and numerical treatment of oscillatory mixed differential equations with differentiable delays and advances(Elsevier, 20110412)This article discusses the oscillatory behaviour of the differential equation of mixed type.

An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data(De Gruyter, 201709)In this paper, we shall review an approach by which we can seek higher order time discretisation schemes for solving time fractional partial differential equations with nonsmooth data. The low regularity of the solutions of time fractional partial differential equations implies standard time discretisation schemes only yield first order accuracy. To obtain higher order time discretisation schemes when the solutions of time fractional partial differential equations have low regularities, one may correct the starting steps of the standard time discretisation schemes to capture the singularities of the solutions. We will consider these corrections of some higher order time discretisation schemes obtained by using Lubich's fractional multistep methods, L1 scheme and its modification, discontinuous Galerkin methods, etc. Numerical examples are given to show that the theoretical results are consistent with the numerical results.

Bifurcations in approximate solutions of stochastic delay differential equations(World Scientific Publishing Company, 2004)

Bifurcations in numerical methods for volterra integrodifferential equations(World Scientific Publishing Company, 2003)This article discusses changes in bifurcations in the solutions. It extends the work of Brunner and Lambert and Matthys to consider other bifurcations.

A black box at the end of the rainbow: Searching for the perfect preconditioner(Royal Society, 200312)

Blending loworder stabilised finite element methods: a positivity preserving local projection method for the convectiondiffusion equation(Elsevier, 20170120)In 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.

Boundedness and stability of solutions to difference equations(Elsevier Science, 2002)This 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 equations(Manchester Centre for Computational Mathematics, 20030523)This 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 approximations(Manchester Centre for Computational Mathematics, 20030523)This 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 approximations(Elsevier Science, 2002)This 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.

Combining Kronecker product approximation with discrete wavelet transforms to solve dense, functionrelated linear systems(Society for Industrial and Applied Mathematics, 200311)