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Noise induced changes to dynamic behaviour of stochastic delay differential equationsThis thesis is concerned with changes in the behaviour of solutions to parameterdependent stochastic delay differential equations.

Noiseinduced changes to the behaviour of semiimplicit Euler methods for stochastic delay differential equations undergoing bifurcationThis article discusses estimating parameter values at which bifurcations occur in stochastic delay differential equations. After a brief review of bifurcation, we employ a numerical approach and consider how bifurcation values are influenced by the choice of numerical scheme and the step length and by the level of white noise present in the equation. In this paper we provide a formulaic relationship between the estimated bifurcation value, the level of noise, the choice of numerical scheme and the step length. We are able to show that in the presence of noise there may be some loss of order in the accuracy of the approximation to the true bifurcation value compared to the use of the same approach in the absence of noise.

Noiseinduced changes to the bifurcation behaviour of semiimplicit Euler methods for stochastic delay differential equationsWe are concerned with estimating parameter values at which bifurcations occur in stochastic delay differential equations. After a brief review of bifurcation, we employ a numerical approach and consider how bifurcation values are influenced by the choice of numerical scheme and the step length and by the level of white noise present in the equation. In this paper we provide a formulaic relationship between the estimated bifurcation value, the level of noise, the choice of numerical scheme and the step length. We are able to show that in the presence of noise there maybe some loss of order in the accuracy of the approximation to the true bifurcation value compared to the use of the same approach in the absence of noise.

NonExhaust Vehicle Emissions of Particulate Matter and VOC from Road Traffic: A ReviewAs exhaust emissions of particles and volatile organic compounds (VOC) from road vehicles have progressively come under greater control, nonexhaust emissions have become an increasing proportion of the total emissions, and in many countries now exceed exhaust emissions. Nonexhaust particle emissions arise from abrasion of the brakes and tyres and wear of the road surface, as well as from resuspension of road dusts. The national emissions, particle size distributions and chemical composition of each of these sources is reviewed. Most estimates of airborne concentrations derive from the use of chemical tracers of specific emissions; the tracers and airborne concentrations estimated from their use are considered. Particle size distributions have been measured both in the laboratory and in field studies, and generally show particles to be in both the coarse (PM2.510) and fine (PM2.5) fractions, with a larger proportion in the former. The introduction of battery electric vehicles is concluded to have only a small effect on overall road traffic particle emissions. Approaches to numerical modelling of nonexhaust particles in the atmosphere are reviewed. Abatement measures include engineering controls, especially for brake wear, improved materials (e.g. for tyre wear) and road surface cleaning and dust suppressants for resuspension. Emissions from solvents in screen wash and deicers now dominate VOC emissions from traffic in the UK, and exhibit a very different composition to exhaust VOC emissions. Likely future trends in nonexhaust particle emissions are described.

NonLocal Partial Differential Equations for Engineering and Biology: Mathematical Modeling and AnalysisThis book presents new developments in nonlocal mathematical modeling and mathematical analysis on the behavior of solutions with novel technical tools. Theoretical backgrounds in mechanics, thermodynamics, game theory, and theoretical biology are examined in details. It starts off with a review and summary of the basic ideas of mathematical modeling frequently used in the sciences and engineering. The authors then employ a number of models in bioscience and material science to demonstrate applications, and provide recent advanced studies, both on deterministic nonlocal partial differential equations and on some of their stochastic counterparts used in engineering. Mathematical models applied in engineering, chemistry, and biology are subject to conservation laws. For instance, decrease or increase in thermodynamic quantities and nonlocal partial differential equations, associated with the conserved physical quantities as parameters. These present novel mathematical objects are engaged with rich mathematical structures, in accordance with the interactions between species or individuals, selforganization, pattern formation, hysteresis. These models are based on various laws of physics, such as mechanics of continuum, electromagnetic theory, and thermodynamics. This is why many areas of mathematics, calculus of variation, dynamical systems, integrable systems, blowup analysis, and energy methods are indispensable in understanding and analyzing these phenomena. This book aims for researchers and upper grades students in mathematics, engineering, physics, economics, and biology.

A nonpolynomial collocation method for fractional terminal value problemsIn this paper we propose a nonpolynomial collocation method for solving a class of terminal (or boundary) value problems for differential equations of fractional order α, 0 < α < 1. The approach used is based on the equivalence between a problem of this type and a Fredholm integral equation of a particular form. Taking into account the asymptotic behaviour of the solution of this problem, we propose a nonpolynomial collocation method on a uniform mesh. We study the order of convergence of the proposed algorithm and a result on optimal order of convergence is obtained. In order to illustrate the theoretical results and the performance of the method we present several numerical examples.

A note on finite difference methods for nonlinear fractional differential equations with nonuniform meshesWe consider finite difference methods for solving nonlinear fractional differential equations in the Caputo fractional derivative sense with nonuniform meshes. Under the assumption that the Caputo derivative of the solution of the fractional differential equation is suitably smooth, Li et al. \lq \lq Finite difference methods with nonuniform meshes for nonlinear fractional differential equations\rq\rq, Journal of Computational Physics, 316(2016), 614631, obtained the error estimates of finite difference methods with nonuniform meshes. However the Caputo derivative of the solution of the fractional differential equation in general has a weak singularity near the initial time. In this paper, we obtain the error estimates of finite difference methods with nonuniform meshes when the Caputo fractional derivative of the solution of the fractional differential equation has lower smoothness. The convergence result shows clearly how the regularity of the Caputo fractional derivative of the solution affect the order of convergence of the finite difference methods. Numerical results are presented that confirm the sharpness of the error analysis.

A Note on the WellPosedness of Terminal Value Problems for Fractional Differential Equations.This note is intended to clarify some im portant points about the wellposedness of terminal value problems for fractional di erential equations. It follows the recent publication of a paper by Cong and Tuan in this jour nal in which a counterexample calls into question the earlier results in a paper by this note's authors. Here, we show in the light of these new insights that a wide class of terminal value problems of fractional differential equations is well posed and we identify those cases where the wellposedness question must be regarded as open.

A novel highorder algorithm for the numerical estimation of fractional differential equationsThis paper uses polynomial interpolation to design a novel highorder algorithm for the numerical estimation of fractional differential equations. The RiemannLiouville fractional derivative is expressed by using the Hadamard finitepart integral and the piecewise cubic interpolation polynomial is utilized to approximate the integral. The detailed error analysis is presented and it is established that the convergence order of the algorithm is O(h4−a). Asymptotic expansion of the error for the presented algorithm is also investigated. Some numerical examples are provided and compared with the exact solution to show that the numerical results are in well agreement with the theoretical ones and also to illustrate the accuracy and efficiency of the proposed algorithm.

A novel ‘bottomup’ synthesis of few and multilayer graphene platelets with partial oxidation via cavitationThe transient cavitation of diaromatic components such as 1methylnaphthalene has been used to produce graphene platelets in a ‘bottomup’ synthesis via the high temperature (>5000 K) conditions that are generated inside collapsing bubbles. Acoustic cavitation produced yields of 5.7×10−11 kgJ−1 at a production rate of 2.2×10−9 kgs−1. This can be improved by generating cavitation hydrodynamically, thus making commercial scale production viable. Hydrodynamic cavitation produced platelets with larger lateral dimensions (≥2 μm) than those formed by acoustic cavitation (10–200 nm). The partially oxidised nature of the platelets enables their covalent chemical functionalisation, which was achieved by combining suitable molecules in the reaction medium to affect a onepot formation and functionalisation of graphene

Numerical analysis for distributed order differential equationsIn this paper we present and analyse a numerical method for the solution of a distributed order differential equation.

Numerical analysis of a singular integral equationThis preprint discusses the numerical analysis of an integral equation to which convential analytical and numerical theory does not apply.

Numerical analysis of a twoparameter fractional telegraph equationIn this paper we consider the twoparameter fractional telegraph equation of the form $$\, ^CD_{t_0^+}^{\alpha+1} u(t,x) + \, ^CD_{x_0^+}^{\beta+1} u (t,x) \, ^CD_{t_0^+}^{\alpha}u (t,x)u(t,x)=0.$$ Here $\, ^CD_{t_0^+}^{\alpha}$, $\, ^CD_{t_0^+}^{\alpha+1}$, $\, ^CD_{x_0^+}^{\beta+1}$ are operators of the Caputotype fractional derivative, where $0\leq \alpha < 1$ and $0 \leq \beta < 1$. The existence and uniqueness of the equations are proved by using the Banach fixed point theorem. A numerical method is introduced to solve this fractional telegraph equation and stability conditions for the numerical method are obtained. Numerical examples are given in the final section of the paper.

Numerical analysis of some integral equations with singularitiesIn this thesis we consider new approaches to the numerical solution of a class of Volterra integral equations, which contain a kernel with singularity of nonstandard type. The kernel is singular in both arguments at the origin, resulting in multiple solutions, one of which is differentiable at the origin. We consider numerical methods to approximate any of the (infinitely many) solutions of the equation. We go on to show that the use of product integration over a short primary interval, combined with the careful use of extrapolation to improve the order, may be linked to any suitable standard method away from the origin. The resulting splitinterval algorithm is shown to be reliable and flexible, capable of achieving good accuracy, with convergence to the one particular smooth solution.

Numerical approaches to bifurcations in solutions to integrodifferential equationsThis conference paper discusses the qualitative behaviour of numerical approximations of a carefully chosen class of integrodifferential equations of the Volterra type. The results are illustrated with some numerical experiments.

Numerical approaches to delay equations with small solutionsThis book chapter discusses the use of numerical schemes to find whether dalay differential equations have small solutions. Two questions  can the onset of small solutions be predicted for a wider range of delay differential equations in a similar way and how should one chose the appropriate numerical method for the investigation  are discussed.

Numerical Approximation of Stochastic TimeFractional DiffusionWe develop and analyze a numerical method for stochastic timefractional diffusion driven by additive fractionally integrated Gaussian noise. The model involves two nonlocal terms in time, i.e., a Caputo fractional derivative of order $\alpha\in(0,1)$, and fractionally integrated Gaussian noise (with a RiemannLiouville fractional integral of order $\gamma \in[0,1]$ in the front). The numerical scheme approximates the model in space by the standard Galerkin method with continuous piecewise linear finite elements and in time by the classical Gr\"unwaldLetnikov method, and the noise by the $L^2$projection. Sharp strong and weak convergence rates are established, using suitable nonsmooth data error estimates for the deterministic counterpart. One and twodimensional numerical results are presented to support the theoretical findings.

Numerical approximation of the Stochastic CahnHilliard Equation near the Sharp Interface LimitAbstract. We consider the stochastic CahnHilliard equation with additive noise term that scales with the interfacial width parameter ε. We verify strong error estimates for a gradient flow structureinheriting timeimplicit discretization, where ε only enters polynomially; the proof is based on highermoment estimates for iterates, and a (discrete) spectral estimate for its deterministic counterpart. For γ sufficiently large, convergence in probability of iterates towards the deterministic HeleShaw/MullinsSekerka problem in the sharpinterface limit ε → 0 is shown. These convergence results are partly generalized to a fully discrete finite element based discretization. We complement the theoretical results by computational studies to provide practical evidence concerning the effect of noise (depending on its ’strength’ γ) on the geometric evolution in the sharpinterface limit. For this purpose we compare the simulations with those from a fully discrete finite element numerical scheme for the (stochastic) MullinsSekerka problem. The computational results indicate that the limit for γ ≥ 1 is the deterministic problem, and for γ = 0 we obtain agreement with a (new) stochastic version of the MullinsSekerka problem.